NIT #7 CORE ALGE COMMON IALS

Size: px
Start display at page:

Download "NIT #7 CORE ALGE COMMON IALS"

Transcription

1 UN NIT #7 ANSWER KEY POLYNOMIALS Lesson #1 Introduction too Polynomials Lesson # Multiplying Polynomials Lesson # Factoring Polynomials Lesson # Factoring Based on Conjugate Pairs Lesson #5 Factoring Trinomials Lesson #6 More Work Factoring Trinomials COMMON CORE ALGE emathinstruction, EBRA I, UNIT #77 POLYNOMI, RED HOOK, NY 1571, 01 IALS

2 COMMON CORE ALGEBRA I, UNIT #7 POLYNOMIALS emathinstruction, RED HOOK, NY 1571, 01

3 Answer Key Name: Date: INTRODUCTION TO POLYNOMIALS COMMON CORE ALGEBRA I The way we write numbers in our systems is interesting because with only 10 digits, i.e. 0, 1,,,, 5, 6, 7, 8, and 9, we are able to write whole numbers as large as we would like. This is because what we really are doing is counting how many powers of 10 that we have. Eercise #1: Write each of the following numbers as a sum of multiples of powers of 10. The first is done as an eample (b) 7 (c), (d) 5, (e) 1,78 0, 000 1, , We can now use algebra to replace the base of 10 with a generic base of (or whatever variable you like). Eercise #: Consider the number 6,75. As in #1, write this number as the sum of (b) If 10, write this number in terms of an multiples of powers of 10. equivalent epression involving. 60, 000, If 10 then , The base of a polynomial certainly doesn t have to be 10. But, all polynomials have a form similar to your answer in letter (b). Let s define them a little more definitively. POLYNOMIAL EXPRESSIONS n n1 n Any epression of the form: a b c constant, where the eponents, n, n 1, n, etcetera are all positive integers. Note that not all powers need to be presents because the coefficients, i.e. a, b, c, etcetera can be zero. Eercise #: Of the epressions shown below, circle all of them that represent polynomials. Discuss why the ones that aren t polynomials fail the definition above Not a polynomial because is 1 equivalent to. In a polynomial all powers are non-negative integers. 1 This is not a polynomial because the bases are constants and the powers are variable. This would be a type of eponential epression. COMMON CORE ALGEBRA I, UNIT #7 POLYNOMIALS LESSON #1 emathinstruction, RED HOOK, NY 1571,

4 It is often important to place polynomials in their standard form. The standard form of a polynomial is simply achieved by writing it as an equivalent epression where the powers on the variables always descend. Eercise #: Write each of the following polynomials in standard form (b) 9 1 (c) Polynomials are simply abstract representations of numbers that we see every day and they behave like these numbers as well. Let s look at adding polynomials together. Eercise #5: Consider the numbers 5 and 71. Write each as the sum of multiples of powers of 10 as previously done (b) Add these numbers by adding each individual power of (c) Use this idea to add: (d) Find the sum of the polynomials and Finding sums of polynomials is fairly easy. Subtracting them, though, can lead to a lot of errors. Eercise #6: Find each of the following differences. Be careful and rewrite as an equivalent addition problem if necessary (b) Eercise #7: For each of the following, write an equivalent polynomial in simplest standard form. 6 1 (b) COMMON CORE ALGEBRA I, UNIT #7 POLYNOMIALS LESSON #1 emathinstruction, RED HOOK, NY 1571, 01

5 Answer Key Name: Date: INTRODUCTION TO POLYNOMIALS COMMON CORE ALGEBRA I HOMEWORK FLUENCY 1. Write each of the following integers as multiples of powers of 10. The first is done as a reminder of this process. 56 (b) 78 (c) (d) 5,78 (f) 19, , 000 9, ,000 91, Consider the number 5,6. Write this number as the sum of multiples of powers of 10 as in # (b) If 10, write an epression in terms of for the number 5,6. If 10 then the epression is equivalently epressed as 5 6. Which of the following would be the value of the epression 5 8 when 10? (1) 6, () 5,8 (),85 () 58. Which of the following would be the value of the epression 8 when 10? (1) 8 () 8,0 () 8,0 () 8,0 5. Which of the following is not a polynomial epression? , () () (1) () 1 () () 6 1 Choice () is not a polynomial because terms in a polynomial must have constant eponents. Here the eponent is a variable. () COMMON CORE ALGEBRA I, UNIT #7 POLYNOMIALS LESSON #1 emathinstruction, RED HOOK, NY 1571, 01

6 6. Write each of the following polynomial epressions in standard form. 7 5 (b) 5 (c) (d) 1 5 (e) (f) y y y y y y y y 7. Find each of the following sums and differences. Write your answer in simplest standard form (b) (c) (d) (e) (f) APPLICATIONS 7 8. A bo has a width that is inches greater than its height and a length that is 6 inches greater than its height. It s volume is given by the polynomial epression 8 1, where is the bo s height. What is the bo s volume, in cubic inches, if its height is 10 inches? (1) 1,81 () 18 If 10 then the epression 8 1 is equal to: () 1,90 (), REASONING , 90 () 9. Polynomial epressions act a lot like integers because the structure of polynomials is based on the structure of integers. Based on the statement below about integers, make a statement about polynomials. Statement About Integers: An integer added to an integer gives an integer. A polynomial added to a polynomial gives a polynomial. Statement About Polynomials: BONUS: This means the polynomials are closed under addition. COMMON CORE ALGEBRA I, UNIT #7 POLYNOMIALS LESSON #1 emathinstruction, RED HOOK, NY 1571, 01

7 Answer Key Name: Date: MULTIPLYING POLYNOMIALS COMMON CORE ALGEBRA I Polynomials, as we saw in the last lesson, behave a lot like integers (whole numbers including the negatives). We saw that just like integers, adding one polynomial to another polynomial results in a third polynomial. The same will occur with multiplying them. First, a review problem. Eercise #1: Monomials are the simplest of polynomials. They consists of one term (terms are separated by addition and subtraction). Find the following products of monomials (b) 8 (c) y y y y y 8 We have also used the Distributive Property in previous lessons to multiply polynomials that are more complicated. Eercise #: Find each of the following products in simplest form by using the distributive property once or twice. 1 (b) 1 6 (c) y y 5 y y y 5 y 10 y (d) 6 (e) 7 (f) Never forget that as we do these manipulations we are using properties of equality to produce equivalent epressions. Eercise #: Consider the product of the two binomial polynomials 1 Find this product and epress it as a trinomial polynomial written in standard form. Fill in the result in the first row (third column) of table (b). 1. COMMON CORE ALGEBRA I, UNIT #7 POLYNOMIALS LESSON # emathinstruction, RED HOOK, NY 1571, 01 (b) Fill out the table below using TABLES on your calculator to show they are equivalent

8 We can evaluate more complicated products, just as we have done in the past with normal numbers. The key will always be the careful use of the distributive property. Eercise #: Find each of the following more challenging products. (b) (c) 5 (d) Eercise #5: Consider the product 1. Write this product as an equivalent trinomial epression in standard form (b) How can you use your answer from to evaluate the product 1? Find the product and check using your calculator If 10 then So COMMON CORE ALGEBRA I, UNIT #7 POLYNOMIALS LESSON # emathinstruction, RED HOOK, NY 1571, 01

9 Answer Key Name: Date: FLUENCY MULTIPLYING POLYNOMIALS COMMON CORE ALGEBRA I HOMEWORK 1. Write the following products as polynomials in either or t. The first is done as an eample for you. 5 (b) tt 7 (c) (d) t 5t tt t7 t 1 t 51 0 (e) (f) 5tt t 7 t t 5 t 0t 8 5t t 5t t 5t 7 10t 15t 5t. Perhaps the most important type of polynomial multiplication is that of two binomials. Make sure you are fluent with this skill. Write each of the following products as an equivalent polynomial written in standard form. The first problem is done as an eample using repeated distribution. 5 (b) 10 (c) (d) 5 8 (e) 1 (f) Never forget that squaring a binomial also a process of repeated distribution. Write each of the following perfect squares as trinomials in standard form. (b) 10 (c) t COMMON CORE ALGEBRA I, UNIT #7 POLYNOMIALS LESSON # emathinstruction, RED HOOK, NY 1571, 01 t t tt t t 1t9 t 6t 6t 9

10 . An interesting thing happens when you multiply two conjugate binomials. Conjugates have the property of having the same terms but differ by the operation between the two terms (in one case addition and in one case subtraction). Multiply each of the following conjugate pairs and state your answers in standard form. The first is done as an eample (b) 5 5 (c) (d) tt (e) 5t15t 1 (f) 8t8 t ttt 5t5t1 15t1 t 6t6t9 5t 5t5t1 t 9 5t 1 5. Write each of the following products in standard polynomial form. 8 (b) REASONING COMMON CORE ALGEBRA I, UNIT #7 POLYNOMIALS LESSON # emathinstruction, RED HOOK, NY 1571, 01 (Hint: try to use #) 6. Notice again how similar polynomials are to integers, i.e. the set...,, 1, 0,1,,.... Write a statement below for polynomials based on the statement about integers. Statement About Integers: An integer times an integer produces an integer. Statement About Polynomials: 7. Consider the product Write this product in standard trinomial form t t t tt9t 6 9t A polynomial times a polynomial produces a polynomial. BONUS: The polynomials are closed under multiplication. (b) Use your answer in part to determine the value of 1 without your calculator. If 10 then So

11 Answer Key Name: Date: FACTORING POLYNOMIALS COMMON CORE ALGEBRA I Factoring epressions is one of the gateway skills that is necessary for much of what we do in algebra for the rest of the course. The word factor has two meanings and both are important. THE TWO MEANINGS OF FACTOR 1. Factor (verb): To rewrite an algebraic epression as an equivalent product.. Factor (noun): An algebraic epression that is one part of a larger factored epression. Eercise #1: Consider the epression Write the individual terms 6 and 15 as completely factored epressions. Determine their greatest common factor. (b) Using the Distributive Property, rewrite 6 15 as a product involving the gcf from gcf 5 (c) Evaluate both 6 15 and the factored epression you wrote in (b) for. What do you find? What does this support about the two epressions? Evaluating 6 15 at : Evaluating 5 at : We find the two epressions have the same value at =. This supports the fact that the two epressions are equivalent, i.e. have the same values for any given replacement value. It is important that you are fluent reversing the distributive property in order to factor out a common factor (most often the greatest common factor). Let s get some practice in the net eercise just identifying the greatest common factors. Eercise #: For each of the following sets of monomials, identify the greatest common factor of each. Write each term as an etended product (if necessary). (d) 1 and 18 (b) 1 18 gcf 6,16, and 8 (e) 16 8 gcf 8 5 and (c) 1 y and 1y y 7 y y y y y y 7 y y y y y y y gcf gcf 7 y 7y 0, 1, and 8 (f) 18 y, 5 y, and 90y y y y 1 5 y 5 y y 5 y y gcf gcf y 9y COMMON CORE ALGEBRA I, UNIT #7 POLYNOMIALS LESSON # emathinstruction, RED HOOK, NY 1571, 01

12 Once you can identify the greatest common factor of a set of monomials, you can then easily use it and the distributive property to write equivalent factored epressions. Eercise #: Write each polynomial below as a factored epression involving the greatest common factor of the polynomial (b) (c) gcf gcf gcf (d) 8 (e) 6 8 (f) 10 5 gcf gcf 1 6 gcf (g) (h) 8 (i) 8 gcf gcf gcf 8 Being able to fluently factor out a gcf is an essential skill. Sometimes greatest common factors are more complicated than simple monomials. We have done this type of factoring back in Unit #1. Eercise #: Rewrite each of the following epressions as the product of two binomials by factoring out a common binomial factor. 515 (b) 17 1 gcf gcf COMMON CORE ALGEBRA I, UNIT #7 POLYNOMIALS LESSON # emathinstruction, RED HOOK, NY 1571, 01

13 Name: Answer Key FACTORING POLYNOMIALS COMMON CORE ALGEBRA I HOMEWORK Date: FLUENCY 1. Identify the greatest common factor for each of the following sets of monomials. (d) 6 and (b) 6, 6, and 1 (e) 15 and 10 (c) 5 16 t, 8 t, and 80 (f) and t,1 t, and 16t 16 t. Which of the following is the greatest common factor of the terms 7 6 y and y? (1) () 1y () 7 y 6 y () y 6 y y y y y y y y y y y y 7 y 1y (). Write each of the following as equivalent products of the polynomial s greatest common factor with another polynomial (of the same number of terms). The first is done as an eample. 8 8 (b) 50 0 (c) gcf 10 gcf (d) 18 1 (e) gcf 6 6 (g) (h) gcf (f) gcf gcf 1 (i) 6 8 gcf 7 gcf 7 9 (j) 0 75 (k) gcf t t 6 16t 96t (l) 1 t t t gcf 16t gcf t tt 8t. Which of the following is not a correct factorization of the binomial 10 0? (1) 10 () 5 () 10 () 5 8 Choice () is not correct because: () COMMON CORE ALGEBRA I, UNIT #7 POLYNOMIALS LESSON # emathinstruction, RED HOOK, NY 1571, 01

14 5. Rewrite each of the following epressions as the product of two binomials by factoring out a common binomial factor. Watch out for the subtraction problems (b) and (d) (b) 15 1 (c) (d) APPLICATIONS 6. The area of a rectangle is represented by the polynomial The width of the rectangle is given by the binomial 7. Give a monomial epression in terms of for the length of the rectangle. Show how you arrived at your answer. Area Length Width REASONING gcf gcf 7 7. These crazy polynomials keep acting like integers. We can factor integers to determine their factors. We can also do the same for polynomials. List all of the positive factors of the integer 1 by writing all possible positive integer products (such as 1 ) Which of the following is not a factor of (1) () () () Length 8 1? gcf (b) If the length of the rectangle is 80, what is the width of the rectangle? Eplain your thinking. (b) List all of the factors of 6 by also writing all possible products, such as. gcf width 7 7 Factors: 1,,,, 6, Factors: 1,,,, 6,, and 6 () COMMON CORE ALGEBRA I, UNIT #7 POLYNOMIALS LESSON # emathinstruction, RED HOOK, NY 1571, 01

15 Answer Key Name: Date: FACTORING BASED ON CONJUGATES COMMON CORE ALGEBRA I There are a number of different types of factoring techniques. But, each one of them boils down to reversing a product. We begin the lesson today by looking at products of conjugate binomials, or binomials of the form a b and a b. Eercise #1: Find each of the following products of conjugate pairs. See if you can work out a pattern. 5 5 (b) (c) (d) y y (e) (f) 5 y5 y y y y 5 5y y 5y y y y y y10y y 5 y What we should see is that if we multiply conjugates, opposites always cancel and instead of getting our epected trinomial, we still get a binomial. Specifically. MULTIPLYING CONJUGATE PAIRS abab a b Eercise #: Use the pattern from Eercise #1 to quickly rewrite the following products. 6 6 (b) 5 5 (c) 7y 7y y 9y (d) (e) 65y6 5y (f) 10 y10 y y 6 5 y 10 y y COMMON CORE ALGEBRA I, UNIT #7 POLYNOMIALS LESSON # emathinstruction, RED HOOK, NY 1571, 01

16 We now should be able to reverse this multiplication in order to rewrite epressions that are the difference of perfect squares into products. Eercise #: Write each of the following first in the form a pairs. 81 (b) b and then as equivalent products of conjugate 9 (c) 5 y y 5 y5 y (d) 81y (e) 11 1 (f) 1 9y 9y 9y Never forget that when we factor, we are always rewriting an epression in a form that might look different, but it is ultimately still equivalent to the original. Eercise #: Let s take a look at the binomial 9. can be factored as Amelia believes that while her friend Isabel believes that it is factored as. Fill out the table below to develop evidence as to who is correct. Use technology on your calculator to help (b) By multiplying out their respective factors, show which of the two friends has the correct factorization. Use the Distributive Property Twice. Amelia: 1 9 Isabel: COMMON CORE ALGEBRA I, UNIT #7 POLYNOMIALS LESSON # emathinstruction, RED HOOK, NY 1571, 01

17 Answer Key Name: Date: FACTORING BASED ON CONJUGATE PAIRS COMMON CORE ALGEBRA I HOMEWORK FLUENCY 1. Use the fact that the product of conjugates follows the following pattern, a bab a b, to quickly find the following products in standard form. 5 5 (b) 7 7 (c) 5 9 (d) (e) 1 1 (f) (h) (i) (g) Write each of the following binomials as an equivalent product of conjugates. 16 (b) 100 (c) (d) 5 (e) (f) 9 (g) (h) 16 9 (i) (j) 9y (k) 81 t (l) 6 y y 9t9t 6 6 COMMON CORE ALGEBRA I, UNIT #7 POLYNOMIALS LESSON # emathinstruction, RED HOOK, NY 1571, 01

18 APPLICATIONS. A square is changed into a new rectangle by increasing its width by inches and decreasing its length by inches. Make sure to draw pictures to help you solve these problems! If the original square had a side length of 8 inches, find its area and the area of the new rectangle. How many square inches larger is the square s area? 8 in 10 in (b) If the original square had a side length of 0 inches, find its area and the area of the new rectangle. How many square inches larger is the square s area? 0 in in 8 in 6 in 0 in 18 in A 6 in in larger A 60 in A 00 in in larger A 96 in (c) If the square had a side length of inches, show that its area will always be four square inches more than the area of the new rectangle. + in in - in in in larger area always REASONING A in in A A in. Consider the numerical epression Use your calculator to find the numerical value of this epression Consider the following epression Using your calculator, determine the value of this epression for various values of (b) Can you used facts about conjugate pairs to show why this difference should work out to be the answer from? (b) Algebraically show that this product has a constant value (seen in ) regardless of the value of COMMON CORE ALGEBRA I, UNIT #7 POLYNOMIALS LESSON # emathinstruction, RED HOOK, NY 1571, 01

19 Answer Key Name: Date: FACTORING TRINOMIALS COMMON CORE ALGEBRA I So far we have two factoring techniques: (1) Factoring out a g.c.f. and () Factoring based on conjugate pairs (factoring the difference of perfect squares). Today we will tackle the most difficult of the factoring techniques, and that is of factoring trinomials. First, let s make sure we can multiply binomials. Eercise #1: Write each of the following products in equivalent trinomial form. 5 (b) 7 (c) Try a few that are a bit more difficult. It is critical that you are fluent with this type of multiplication before we try to reverse the process. Eercise #: Write each of the following products in equivalent trinomial form (b) 6 7 (c) Now, we need to reverse this process to take a trinomial and write it as the product of the binomials. There is truly only one fail proof method for this type of factoring and that is GUESS AND CHECK. This method often frustrates students, but look at it as a puzzle and make smart guesses and quick checks. Eercise #: Consider the trinomial 1 0. We want to write it as the product of two binomials, in other words, reverse what we did in Eercises #1 and #. Fill in the missing blanks with the only pair that makes sense (that is a smart guess).. (c) List pairs of factors that can produce a 0. 1, 0, 10, COMMON CORE ALGEBRA I, UNIT #7 POLYNOMIALS LESSON #5 emathinstruction, RED HOOK, NY 1571, (b) Why does it make sense that both of the binomials will be addition (as shown in )? This makes sense because the product of the second terms is a positive 0 and when we combine the linear terms we have a positive 1. Look at the binomial in Eercise #(b) to see how this works. (d) Try various pairs from (c), checking only that the linear terms will sum to 1 until you find the correct factorization. Guess #1: 1 0 Quick Check: Guess #: 5 Quick Check: 5 1 CORRECT GUESS!

20 There is absolutely no substitution for rote practice with factoring trinomials. To be able to do this critical skill, you must be smart with your guesses and patient with your checks. You will find the correct answer, but you must be willing to guess incorrectly multiple times. Eercise #: Write each of the following trinomials in equivalent factored form. Show your checks and don t worry if your first guess isn t correct. You will get better at these! Note for yourself which ones seemed easier and which were harder. 11 (b) Guess #1: 1 Quick Check: Guess #: 1 Quick Check: 1 11 CORRECT GUESS! 7 10 Guess #1: 5 Quick Check: Guess #: 5 Quick Check: 5 7 CORRECT GUESS! (c) 10 1 (d) 10 9 Guess #1: 1 1 Quick Check: Guess #1: 5 1 Quick Check: Guess #: 7 Quick Check: 7 10 CORRECT GUESS! (e) (f) Guess #1: 8 Quick Check: Guess #: Quick Check: 18 CORRECT GUESS! 5 Guess #: 5 1 Quick Check: CORRECT GUESS! Guess #1: 11 Quick Check: Guess #: 11 Quick Check: 11 5 CORRECT GUESS! (g) 8 1 (h) Guess #1: Quick Check: Guess #1: 7 Quick Check: Guess #: 6 Quick Check: 6 8 CORRECT GUESS! Guess #: 7 Quick Check: 7 6 CORRECT GUESS! COMMON CORE ALGEBRA I, UNIT #7 POLYNOMIALS LESSON #5 emathinstruction, RED HOOK, NY 1571, 01

21 Answer Key Name: Date: FACTORING TRINOMIALS COMMON CORE ALGEBRA I HOMEWORK FLUENCY 1. Which of the following products is equivalent to the trinomial 5? (1) 1 () () 1 () Written in factored form, the trinomial (1) 7 () 1 () 7 () can be epressed equivalently as. (Easier) Write each of the following trinomials in equivalent factored form. Keep at it!!! Use etra scrap paper for etra guesses (b) (c) () (1) (d) 5 6 (e) 11 8 (f) (g) 18 (h) 0 (i) (j) 8 15 (k) 0 00 (l) COMMON CORE ALGEBRA I, UNIT #7 POLYNOMIALS LESSON #5 emathinstruction, RED HOOK, NY 1571, 01

22 . (Medium) Write each of the following trinomials in equivalent factored form. Keep at it!!! Use etra scrap paper for etra guesses. 1 1 (b) 5 1 (c) (d) (e) 10 (f) (Hardest) Write each of the following trinomials in equivalent factored form. Keep at it!!! These might require quite a few tries. You will get better at them as you practice. Use etra scrap paper for etra guesses (b) 6 5 (c) APPLICATIONS 6. A rectangle has dimensions as shown below in terms of an unknown variable,. Find a binomial epression for the width of the rectangle in terms of. Justify your answer based on the epressions for the rectangle s length and area. Area Length Width 9 Area 1 Width 1 (b) If the width of the rectangle is 1 inches, what is the length and the area? Use appropriate units and eplain how you found your answer. Length Area 9 Width =? If width 1 and width 1 then 10. Length 1 in Area 9 9 in COMMON CORE ALGEBRA I, UNIT #7 POLYNOMIALS LESSON #5 emathinstruction, RED HOOK, NY 1571, 01

23 Answer Key Name: Date: MORE WORK FACTORING TRINOMIALS COMMON CORE ALGEBRA I Factoring trinomials, which we first practiced in the last lesson, is a trying eperience. All algebra students must learn how to do this procedure because of its immense number of practical applications. We will eventually see these applications, but for now, we need to get more practice factoring these trinomials. We begin by looking at a process known as complete factoring. Eercise #1: Consider the trinomial 0. Write this trinomial as an equivalent epression involving the product of its term s gcf and another trinomial. gcf 56 (b) Factor this additional trinomial to epress the original in completely factored form. 56 Whenever we factor, we should always look to see if a greatest common factor eists that can be factored out to begin the problem. This will always make any subsequent factoring easier. Eercise #: Rewrite each of the following trinomials in completely factored form (b) 1 6 gcf gcf 7 1 (c) (d) 6 gcf gcf 1 COMMON CORE ALGEBRA I, UNIT #7 POLYNOMIALS LESSON #6 emathinstruction, RED HOOK, NY 1571, 01

24 Complete factoring can also involve factoring the difference of perfect squares. Try the net eercise to see how this works. Eercise #: Write each of the following binomials in completely factored form. 18 (b) gcf gcf (c) 1 (d) 5 gcf gcf If you understand factoring as breaking an epression into an equivalent product, then essentially you can always check to see if you have factored correctly. Complete factoring actually leads to a nice way to eliminate some guesses from trinomial guess and check methods. Eercise #: Consider the trinomial Do the three terms of this trinomial have a gcf other than 1? (c) Fill in the statement: If a trinomial does not have a gcf, then neither binomial of its factors will have a gcf. NO (b) Why would the guesses 6, and 1 1, not make sense given your answer to? Each of these guess makes no sense because in each, there is a binomial with a gcf, but the overall trinomial does not have a gcf, thus none of its binomial factors can have a gcf. (d) Factor this trinomial by limiting your guesses. Eercise #5: Use the Smart Guessing Tip from the last problem to factor Unreasonable Guess: Unreasonable Guess: 6 9 Final Answer: 6 gcf gcf COMMON CORE ALGEBRA I, UNIT #7 POLYNOMIALS LESSON #6 emathinstruction, RED HOOK, NY 1571, 01

25 Answer Key Name: Date: MORE WORK FACTORING TRINOMIALS COMMON CORE ALGEBRA I HOMEWORK FLUENCY 1. Rewrite each of the following trinomials in completely factored form. 0 (b) gcf gcf (c) (e) gcf 5 (d) (f) gcf gcf gcf 5 1 (g) 5 5 (h) gcf 5 gcf (i) (j) gcf 9. Which of the following is not a factor of (1) 9 () () () gcf ? Show work that justifies your choice. gcf 18 (1) 6 COMMON CORE ALGEBRA I, UNIT #7 POLYNOMIALS LESSON #6 emathinstruction, RED HOOK, NY 1571, 01

26 . Which of the following is the missing factor in the product 1? if it is equivalent to the trinomial 10 1? (1) 1 () () 6 () 5 gcf 5 6 () 1 6. Use the Smart Guessing Tip from Eercise # to help factor the following challenging trinomials. Note that they do not have a greatest common factor (b) REASONING 5. Consider the cubic trinomial 8 7. Write this trinomial as an equivalent product in completely factored form. gcf (b) How can the original trinomial and your answer to (b) help you determine the value of without a calculator? What is the value? Let 10 then Use the complete factorization of your reasoning. 8 8 to determine the value of the product First we will factor to eplore: If 10 then: , Eplain COMMON CORE ALGEBRA I, UNIT #7 POLYNOMIALS LESSON #6 emathinstruction, RED HOOK, NY 1571, 01

LESSON #1: VARIABLES, TERMS, AND EXPRESSIONS COMMON CORE ALGEBRA II

LESSON #1: VARIABLES, TERMS, AND EXPRESSIONS COMMON CORE ALGEBRA II 1 LESSON #1: VARIABLES, TERMS, AND EXPRESSIONS COMMON CORE ALGEBRA II Mathematics has developed a language all to itself in order to clarify concepts and remove ambiguity from the analysis of problems.

More information

VARIABLES, TERMS, AND EXPRESSIONS COMMON CORE ALGEBRA II

VARIABLES, TERMS, AND EXPRESSIONS COMMON CORE ALGEBRA II Name: Date: VARIABLES, TERMS, AND EXPRESSIONS COMMON CORE ALGEBRA II Mathematics has developed a language all to itself in order to clarify concepts and remove ambiguity from the analysis of problems.

More information

In this unit we will study exponents, mathematical operations on polynomials, and factoring.

In this unit we will study exponents, mathematical operations on polynomials, and factoring. GRADE 0 MATH CLASS NOTES UNIT E ALGEBRA In this unit we will study eponents, mathematical operations on polynomials, and factoring. Much of this will be an etension of your studies from Math 0F. This unit

More information

UNIT 9 (Chapter 7 BI) Polynomials and Factoring Name:

UNIT 9 (Chapter 7 BI) Polynomials and Factoring Name: UNIT 9 (Chapter 7 BI) Polynomials and Factoring Name: The calendar and all assignments are subject to change. Students will be notified of any changes during class, so it is their responsibility to pay

More information

Lesson #33 Solving Incomplete Quadratics

Lesson #33 Solving Incomplete Quadratics Lesson # Solving Incomplete Quadratics A.A.4 Know and apply the technique of completing the square ~ 1 ~ We can also set up any quadratic to solve it in this way by completing the square, the technique

More information

Algebra Review C H A P T E R. To solve an algebraic equation with one variable, find the value of the unknown variable.

Algebra Review C H A P T E R. To solve an algebraic equation with one variable, find the value of the unknown variable. C H A P T E R 6 Algebra Review This chapter reviews key skills and concepts of algebra that you need to know for the SAT. Throughout the chapter are sample questions in the style of SAT questions. Each

More information

ACCUPLACER MATH 0311 OR MATH 0120

ACCUPLACER MATH 0311 OR MATH 0120 The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 0 OR MATH 00 http://www.academics.utep.edu/tlc MATH 0 OR MATH 00 Page Factoring Factoring Eercises 8 Factoring Answer to Eercises

More information

Algebra I Notes Unit Eleven: Polynomials

Algebra I Notes Unit Eleven: Polynomials Syllabus Objective: 9.1 The student will add, subtract, multiply, and factor polynomials connecting the arithmetic and algebraic processes. Teacher Note: A nice way to illustrate operations with polynomials

More information

Secondary Math 2H Unit 3 Notes: Factoring and Solving Quadratics

Secondary Math 2H Unit 3 Notes: Factoring and Solving Quadratics Secondary Math H Unit 3 Notes: Factoring and Solving Quadratics 3.1 Factoring out the Greatest Common Factor (GCF) Factoring: The reverse of multiplying. It means figuring out what you would multiply together

More information

Practical Algebra. A Step-by-step Approach. Brought to you by Softmath, producers of Algebrator Software

Practical Algebra. A Step-by-step Approach. Brought to you by Softmath, producers of Algebrator Software Practical Algebra A Step-by-step Approach Brought to you by Softmath, producers of Algebrator Software 2 Algebra e-book Table of Contents Chapter 1 Algebraic expressions 5 1 Collecting... like terms 5

More information

Review: Properties of Exponents (Allow students to come up with these on their own.) m n m n. a a a. n n n m. a a a. a b a

Review: Properties of Exponents (Allow students to come up with these on their own.) m n m n. a a a. n n n m. a a a. a b a Algebra II Notes Unit Si: Polynomials Syllabus Objectives: 6. The student will simplify polynomial epressions. Review: Properties of Eponents (Allow students to come up with these on their own.) Let a

More information

COUNCIL ROCK HIGH SCHOOL MATHEMATICS. A Note Guideline of Algebraic Concepts. Designed to assist students in A Summer Review of Algebra

COUNCIL ROCK HIGH SCHOOL MATHEMATICS. A Note Guideline of Algebraic Concepts. Designed to assist students in A Summer Review of Algebra COUNCIL ROCK HIGH SCHOOL MATHEMATICS A Note Guideline of Algebraic Concepts Designed to assist students in A Summer Review of Algebra [A teacher prepared compilation of the 7 Algebraic concepts deemed

More information

Polynomials and Factoring

Polynomials and Factoring 7.6 Polynomials and Factoring Basic Terminology A term, or monomial, is defined to be a number, a variable, or a product of numbers and variables. A polynomial is a term or a finite sum or difference of

More information

ACCUPLACER MATH 0310

ACCUPLACER MATH 0310 The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 00 http://www.academics.utep.edu/tlc MATH 00 Page Linear Equations Linear Equations Eercises 5 Linear Equations Answer to

More information

Chapter 9: Roots and Irrational Numbers

Chapter 9: Roots and Irrational Numbers Chapter 9: Roots and Irrational Numbers Index: A: Square Roots B: Irrational Numbers C: Square Root Functions & Shifting D: Finding Zeros by Completing the Square E: The Quadratic Formula F: Quadratic

More information

Introduction. So, why did I even bother to write this?

Introduction. So, why did I even bother to write this? Introduction This review was originally written for my Calculus I class, but it should be accessible to anyone needing a review in some basic algebra and trig topics. The review contains the occasional

More information

LESSON #34 - FINDING RESTRICTED VALUES AND SIMPLIFYING RATIONAL EXPRESSIONS COMMON CORE ALGEBRA II

LESSON #34 - FINDING RESTRICTED VALUES AND SIMPLIFYING RATIONAL EXPRESSIONS COMMON CORE ALGEBRA II LESSON #4 - FINDING RESTRICTED VALUES AND SIMPLIFYING RATIONAL EXPRESSIONS COMMON CORE ALGEBRA II A rational epression is a fraction that contains variables. A variable is very useful in mathematics. In

More information

STARTING WITH CONFIDENCE

STARTING WITH CONFIDENCE STARTING WITH CONFIDENCE A- Level Maths at Budmouth Name: This booklet has been designed to help you to bridge the gap between GCSE Maths and AS Maths. Good mathematics is not about how many answers you

More information

6.2 Multiplying Polynomials

6.2 Multiplying Polynomials Locker LESSON 6. Multiplying Polynomials PAGE 7 BEGINS HERE Name Class Date 6. Multiplying Polynomials Essential Question: How do you multiply polynomials, and what type of epression is the result? Common

More information

Algebra II Notes Polynomial Functions Unit Introduction to Polynomials. Math Background

Algebra II Notes Polynomial Functions Unit Introduction to Polynomials. Math Background Introduction to Polynomials Math Background Previously, you Identified the components in an algebraic epression Factored quadratic epressions using special patterns, grouping method and the ac method Worked

More information

Divisibility Rules Algebra 9.0

Divisibility Rules Algebra 9.0 Name Period Divisibility Rules Algebra 9.0 A Prime Number is a whole number whose only factors are 1 and itself. To find all of the prime numbers between 1 and 100, complete the following eercise: 1. Cross

More information

7.2 Multiplying Polynomials

7.2 Multiplying Polynomials Locker LESSON 7. Multiplying Polynomials Teas Math Standards The student is epected to: A.7.B Add, subtract, and multiply polynomials. Mathematical Processes A.1.E Create and use representations to organize,

More information

Honours Advanced Algebra Unit 2: Polynomial Functions What s Your Identity? Learning Task (Task 8) Date: Period:

Honours Advanced Algebra Unit 2: Polynomial Functions What s Your Identity? Learning Task (Task 8) Date: Period: Honours Advanced Algebra Name: Unit : Polynomial Functions What s Your Identity? Learning Task (Task 8) Date: Period: Introduction Equivalent algebraic epressions, also called algebraic identities, give

More information

5.2 Polynomial Operations

5.2 Polynomial Operations 5.2 Polynomial Operations At times we ll need to perform operations with polynomials. At this level we ll just be adding, subtracting, or multiplying polynomials. Dividing polynomials will happen in future

More information

LESSON #48 - INTEGER EXPONENTS COMMON CORE ALGEBRA II

LESSON #48 - INTEGER EXPONENTS COMMON CORE ALGEBRA II LESSON #8 - INTEGER EXPONENTS COMMON CORE ALGEBRA II We just finished our review of linear functions. Linear functions are those that grow b equal differences for equal intervals. In this unit we will

More information

Chapter 6: Polynomials

Chapter 6: Polynomials Chapter : Polynomials Chapter : Polynomials POLYNOMIALS Definition: A polynomial is an algebraic epression that is a sum of terms, where each term contains only variables with whole number eponents and

More information

LESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II

LESSON #28 - POWER FUNCTIONS COMMON CORE ALGEBRA II 1 LESSON #8 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The

More information

INTRODUCTION TO RATIONAL FUNCTIONS COMMON CORE ALGEBRA II

INTRODUCTION TO RATIONAL FUNCTIONS COMMON CORE ALGEBRA II Name: Date: INTRODUCTION TO RATIONAL FUNCTIONS COMMON CORE ALGEBRA II Rational functions are simply the ratio of polynomial functions. They take on more interesting properties and have more interesting

More information

Mini-Lecture 5.1 Exponents and Scientific Notation

Mini-Lecture 5.1 Exponents and Scientific Notation Mini-Lecture.1 Eponents and Scientific Notation Learning Objectives: 1. Use the product rule for eponents.. Evaluate epressions raised to the zero power.. Use the quotient rule for eponents.. Evaluate

More information

Edexcel AS and A Level Mathematics Year 1/AS - Pure Mathematics

Edexcel AS and A Level Mathematics Year 1/AS - Pure Mathematics Year Maths A Level Year - Tet Book Purchase In order to study A Level Maths students are epected to purchase from the school, at a reduced cost, the following tetbooks that will be used throughout their

More information

Unit 7: Factoring Quadratic Polynomials

Unit 7: Factoring Quadratic Polynomials Unit 7: Factoring Quadratic Polynomials A polynomial is represented by: where the coefficients are real numbers and the exponents are nonnegative integers. Side Note: Examples of real numbers: Examples

More information

Polynomial one or more monomials added or subtracted. (i.e. : 5x or 6xy-3 or 6xy - 5x + 3 or

Polynomial one or more monomials added or subtracted. (i.e. : 5x or 6xy-3 or 6xy - 5x + 3 or Polynomials Necessary Terms for Success Welcome back! We will now tackle the world of polynomials. Before we get started with performing operations or using polynomials for applications, we will need some

More information

Core Connections Algebra 2 Checkpoint Materials

Core Connections Algebra 2 Checkpoint Materials Core Connections Algebra 2 Note to Students (and their Teachers) Students master different skills at different speeds. No two students learn eactly the same way at the same time. At some point you will

More information

SOLVING QUADRATIC EQUATIONS USING GRAPHING TOOLS

SOLVING QUADRATIC EQUATIONS USING GRAPHING TOOLS GRADE PRE-CALCULUS UNIT A: QUADRATIC EQUATIONS (ALGEBRA) CLASS NOTES. A definition of Algebra: A branch of mathematics which describes basic arithmetic relations using variables.. Algebra is just a language.

More information

LESSON 7.2 FACTORING POLYNOMIALS II

LESSON 7.2 FACTORING POLYNOMIALS II LESSON 7.2 FACTORING POLYNOMIALS II LESSON 7.2 FACTORING POLYNOMIALS II 305 OVERVIEW Here s what you ll learn in this lesson: Trinomials I a. Factoring trinomials of the form x 2 + bx + c; x 2 + bxy +

More information

Algebra. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

Algebra. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed. This document was written and copyrighted by Paul Dawkins. Use of this document and its online version is governed by the Terms and Conditions of Use located at. The online version of this document is

More information

Bishop Kelley High School Summer Math Program Course: Algebra 2 A

Bishop Kelley High School Summer Math Program Course: Algebra 2 A 06 07 Bishop Kelley High School Summer Math Program Course: Algebra A NAME: DIRECTIONS: Show all work in packet!!! The first 6 pages of this packet provide eamples as to how to work some of the problems

More information

Polynomials. This booklet belongs to: Period

Polynomials. This booklet belongs to: Period HW Mark: 10 9 8 7 6 RE-Submit Polynomials This booklet belongs to: Period LESSON # DATE QUESTIONS FROM NOTES Questions that I find difficult Pg. Pg. Pg. Pg. Pg. Pg. Pg. Pg. Pg. Pg. REVIEW TEST Your teacher

More information

Bishop Kelley High School Summer Math Program Course: Algebra 2 A

Bishop Kelley High School Summer Math Program Course: Algebra 2 A 015 016 Bishop Kelley High School Summer Math Program Course: Algebra A NAME: DIRECTIONS: Show all work in packet!!! The first 16 pages of this packet provide eamples as to how to work some of the problems

More information

Eby, MATH 0310 Spring 2017 Page 53. Parentheses are IMPORTANT!! Exponents only change what they! So if a is not inside parentheses, then it

Eby, MATH 0310 Spring 2017 Page 53. Parentheses are IMPORTANT!! Exponents only change what they! So if a is not inside parentheses, then it Eby, MATH 010 Spring 017 Page 5 5.1 Eponents Parentheses are IMPORTANT!! Eponents only change what they! So if a is not inside parentheses, then it get raised to the power! Eample 1 4 b) 4 c) 4 ( ) d)

More information

We will work with two important rules for radicals. We will write them for square roots but they work for any root (cube root, fourth root, etc.).

We will work with two important rules for radicals. We will write them for square roots but they work for any root (cube root, fourth root, etc.). College algebra We will review simplifying radicals, exponents and their rules, multiplying polynomials, factoring polynomials, greatest common denominators, and solving rational equations. Pre-requisite

More information

A summary of factoring methods

A summary of factoring methods Roberto s Notes on Prerequisites for Calculus Chapter 1: Algebra Section 1 A summary of factoring methods What you need to know already: Basic algebra notation and facts. What you can learn here: What

More information

Unit 5 AB Quadratic Expressions and Equations 1/9/2017 2/8/2017

Unit 5 AB Quadratic Expressions and Equations 1/9/2017 2/8/2017 Unit 5 AB Quadratic Expressions and Equations 1/9/2017 2/8/2017 Name: By the end of this unit, you will be able to Add, subtract, and multiply polynomials Solve equations involving the products of monomials

More information

Bridging the gap between GCSE and A level mathematics

Bridging the gap between GCSE and A level mathematics Bridging the gap between GCSE and A level mathematics This booklet is designed to help you revise important algebra topics from GCSE and make the transition from GCSE to A level a smooth one. You are advised

More information

Name Class Date. Multiplying Two Binomials Using Algebra Tiles

Name Class Date. Multiplying Two Binomials Using Algebra Tiles Name Class Date Multiplying Polynomials Going Deeper Essential question: How do you multiply polynomials? 6-5 A monomial is a number, a variable, or the product of a number and one or more variables raised

More information

Multiplying a Polynomial by a Monomial

Multiplying a Polynomial by a Monomial Lesson -3 Multiplying a Polynomial by a Monomial Lesson -3 BIG IDEA To multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial and add the products. In earlier chapters,

More information

Multiplying Monomials

Multiplying Monomials 320 Chapter 5 Polynomials Eample 1 Multiplying Monomials Multiply the monomials. a. 13 2 y 7 215 3 y2 b. 1 3 4 y 3 21 2 6 yz 8 2 a. 13 2 y 7 215 3 y2 13 521 2 3 21y 7 y2 15 5 y 8 Group coefficients and

More information

Lesson 21 Not So Dramatic Quadratics

Lesson 21 Not So Dramatic Quadratics STUDENT MANUAL ALGEBRA II / LESSON 21 Lesson 21 Not So Dramatic Quadratics Quadratic equations are probably one of the most popular types of equations that you ll see in algebra. A quadratic equation has

More information

LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II

LESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also,

More information

Algebra II Polynomials: Operations and Functions

Algebra II Polynomials: Operations and Functions Slide 1 / 276 Slide 2 / 276 Algebra II Polynomials: Operations and Functions 2014-10-22 www.njctl.org Slide 3 / 276 Table of Contents click on the topic to go to that section Properties of Exponents Review

More information

Honors Advanced Algebra Unit 2 Polynomial Operations September 14, 2016 Task 7: What s Your Identity?

Honors Advanced Algebra Unit 2 Polynomial Operations September 14, 2016 Task 7: What s Your Identity? Honors Advanced Algebra Name Unit Polynomial Operations September 14, 016 Task 7: What s Your Identity? MGSE9 1.A.APR.4 Prove polynomial identities and use them to describe numerical relationships. MGSE9

More information

Conceptual Explanations: Radicals

Conceptual Explanations: Radicals Conceptual Eplanations: Radicals The concept of a radical (or root) is a familiar one, and was reviewed in the conceptual eplanation of logarithms in the previous chapter. In this chapter, we are going

More information

Lesson 9-5 Completing the Square Name

Lesson 9-5 Completing the Square Name Lesson 9-5 Completing the Square Name In the next two lessons, you will be solving quadratic equations by a couple of new methods that often produce solutions involving square roots. In Lesson 9-5A, you

More information

Algebra III and Trigonometry Summer Assignment

Algebra III and Trigonometry Summer Assignment Algebra III and Trigonometry Summer Assignment Welcome to Algebra III and Trigonometry! This summer assignment is a review of the skills you learned in Algebra II. Please bring this assignment with you

More information

A2T. Rational Expressions/Equations. Name: Teacher: Pd:

A2T. Rational Expressions/Equations. Name: Teacher: Pd: AT Packet #1: Rational Epressions/Equations Name: Teacher: Pd: Table of Contents o Day 1: SWBAT: Review Operations with Polynomials Pgs: 1-3 HW: Pages -3 in Packet o Day : SWBAT: Factor using the Greatest

More information

Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1

Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1 Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1 Chapter 5 Lesson 1 Use Properties of Exponents Vocabulary Learn these! Love these! Know these! 1 Example 1: Evaluate Numerical Expressions

More information

Core Connections Algebra 2 Checkpoint Materials

Core Connections Algebra 2 Checkpoint Materials Core Connections Algebra 2 Note to Students (and their Teachers) Students master different skills at different speeds. No two students learn eactly the same way at the same time. At some point you will

More information

Pre-Algebra Notes Unit 12: Polynomials and Sequences

Pre-Algebra Notes Unit 12: Polynomials and Sequences Pre-Algebra Notes Unit 1: Polynomials and Sequences Polynomials Syllabus Objective: (6.1) The student will write polynomials in standard form. Let s review a definition: monomial. A monomial is a number,

More information

Adding and Subtracting Polynomials

Adding and Subtracting Polynomials Adding and Subtracting Polynomials Polynomial A monomial or sum of monomials. Binomials and Trinomial are also polynomials. Binomials are sum of two monomials Trinomials are sum of three monomials Degree

More information

Algebra I Polynomials

Algebra I Polynomials Slide 1 / 217 Slide 2 / 217 Algebra I Polynomials 2014-04-24 www.njctl.org Slide 3 / 217 Table of Contents Definitions of Monomials, Polynomials and Degrees Adding and Subtracting Polynomials Multiplying

More information

1 Rational Exponents and Radicals

1 Rational Exponents and Radicals Introductory Algebra Page 1 of 11 1 Rational Eponents and Radicals 1.1 Rules of Eponents The rules for eponents are the same as what you saw earlier. Memorize these rules if you haven t already done so.

More information

Math 10-C Polynomials Concept Sheets

Math 10-C Polynomials Concept Sheets Math 10-C Polynomials Concept Sheets Concept 1: Polynomial Intro & Review A polynomial is a mathematical expression with one or more terms in which the exponents are whole numbers and the coefficients

More information

Name Period Date. polynomials of the form x ± bx ± c. Use guess and check and logic to factor polynomials of the form 2

Name Period Date. polynomials of the form x ± bx ± c. Use guess and check and logic to factor polynomials of the form 2 Name Period Date POLYNOMIALS Student Packet 3: Factoring Polynomials POLY3 STUDENT PAGES POLY3.1 An Introduction to Factoring Polynomials Understand what it means to factor a polynomial Factor polynomials

More information

INTEGER EXPONENTS HOMEWORK. 1. For each of the following, determine the integer value of n that satisfies the equation. The first is done for you.

INTEGER EXPONENTS HOMEWORK. 1. For each of the following, determine the integer value of n that satisfies the equation. The first is done for you. Name: Date: INTEGER EXPONENTS HOMEWORK Algebra II INTEGER EXPONENTS FLUENCY. For each of the following, determine the integer value of n that satisfies the equation. The first is done for ou. n = 8 n =

More information

Adding and Subtracting Rational Expressions

Adding and Subtracting Rational Expressions COMMON CORE Locker LESSON 9.1 Adding and Subtracting Rational Epressions Name Class Date 9.1 Adding and Subtracting Rational Epressions Essential Question: How can you add and subtract rational epressions?

More information

3.1 Solving Quadratic Equations by Taking Square Roots

3.1 Solving Quadratic Equations by Taking Square Roots COMMON CORE -8-16 1 1 10 8 6 0 y Locker LESSON.1 Solving Quadratic Equations by Taking Square Roots Name Class Date.1 Solving Quadratic Equations by Taking Square Roots Essential Question: What is an imaginary

More information

Multiplying Polynomials. The rectangle shown at the right has a width of (x + 2) and a height of (2x + 1).

Multiplying Polynomials. The rectangle shown at the right has a width of (x + 2) and a height of (2x + 1). Page 1 of 6 10.2 Multiplying Polynomials What you should learn GOAL 1 Multiply two polynomials. GOAL 2 Use polynomial multiplication in real-life situations, such as calculating the area of a window in

More information

Section 6.2 Long Division of Polynomials

Section 6.2 Long Division of Polynomials Section 6. Long Division of Polynomials INTRODUCTION In Section 6.1 we learned to simplify a rational epression by factoring. For eample, + 3 10 = ( + 5)( ) ( ) = ( + 5) 1 = + 5. However, if we try to

More information

Finite Mathematics : A Business Approach

Finite Mathematics : A Business Approach Finite Mathematics : A Business Approach Dr. Brian Travers and Prof. James Lampes Second Edition Cover Art by Stephanie Oxenford Additional Editing by John Gambino Contents What You Should Already Know

More information

Unit 2-1: Factoring and Solving Quadratics. 0. I can add, subtract and multiply polynomial expressions

Unit 2-1: Factoring and Solving Quadratics. 0. I can add, subtract and multiply polynomial expressions CP Algebra Unit -1: Factoring and Solving Quadratics NOTE PACKET Name: Period Learning Targets: 0. I can add, subtract and multiply polynomial expressions 1. I can factor using GCF.. I can factor by grouping.

More information

A-2. Polynomials and Factoring. Section A-2 1

A-2. Polynomials and Factoring. Section A-2 1 A- Polynomials and Factoring Section A- 1 What you ll learn about Adding, Subtracting, and Multiplying Polynomials Special Products Factoring Polynomials Using Special Products Factoring Trinomials Factoring

More information

2 P a g e. Essential Questions:

2 P a g e. Essential Questions: NC Math 1 Unit 5 Quadratic Functions Main Concepts Study Guide & Vocabulary Classifying, Adding, & Subtracting Polynomials Multiplying Polynomials Factoring Polynomials Review of Multiplying and Factoring

More information

Essential Question How can you cube a binomial? Work with a partner. Find each product. Show your steps. = (x + 1) Multiply second power.

Essential Question How can you cube a binomial? Work with a partner. Find each product. Show your steps. = (x + 1) Multiply second power. 4.2 Adding, Subtracting, and Multiplying Polynomials COMMON CORE Learning Standards HSA-APR.A.1 HSA-APR.C.4 HSA-APR.C.5 Essential Question How can you cube a binomial? Cubing Binomials Work with a partner.

More information

Chapter 5 Simplifying Formulas and Solving Equations

Chapter 5 Simplifying Formulas and Solving Equations Chapter 5 Simplifying Formulas and Solving Equations Look at the geometry formula for Perimeter of a rectangle P = L W L W. Can this formula be written in a simpler way? If it is true, that we can simplify

More information

Algebra 1 Skills Needed for Success in Math

Algebra 1 Skills Needed for Success in Math Algebra 1 Skills Needed for Success in Math A. Simplifing Polnomial Epressions Objectives: The student will be able to: Appl the appropriate arithmetic operations and algebraic properties needed to simplif

More information

Unit 13: Polynomials and Exponents

Unit 13: Polynomials and Exponents Section 13.1: Polynomials Section 13.2: Operations on Polynomials Section 13.3: Properties of Exponents Section 13.4: Multiplication of Polynomials Section 13.5: Applications from Geometry Section 13.6:

More information

Grade 9 Mathematics Unit #2 Patterns & Relations Sub-Unit #1 Polynomials

Grade 9 Mathematics Unit #2 Patterns & Relations Sub-Unit #1 Polynomials Grade 9 Mathematics Unit #2 Patterns & Relations Sub-Unit #1 Polynomials Lesson Topic I Can 1 Definitions Define Polynomials Identify Polynomials Identify different parts of a polynomial Identify monomials,

More information

Quadratic Expressions and Equations

Quadratic Expressions and Equations Unit 5 Quadratic Expressions and Equations 1/9/2017 2/8/2017 Name: By the end of this unit, you will be able to Add, subtract, and multiply polynomials Solve equations involving the products of monomials

More information

Algebra I. Book 2. Powered by...

Algebra I. Book 2. Powered by... Algebra I Book 2 Powered by... ALGEBRA I Units 4-7 by The Algebra I Development Team ALGEBRA I UNIT 4 POWERS AND POLYNOMIALS......... 1 4.0 Review................ 2 4.1 Properties of Exponents..........

More information

EXPONENT REVIEW!!! Concept Byte (Review): Properties of Exponents. Property of Exponents: Product of Powers. x m x n = x m + n

EXPONENT REVIEW!!! Concept Byte (Review): Properties of Exponents. Property of Exponents: Product of Powers. x m x n = x m + n Algebra B: Chapter 6 Notes 1 EXPONENT REVIEW!!! Concept Byte (Review): Properties of Eponents Recall from Algebra 1, the Properties (Rules) of Eponents. Property of Eponents: Product of Powers m n = m

More information

Algebra Final Exam Review Packet

Algebra Final Exam Review Packet Algebra 1 00 Final Eam Review Packet UNIT 1 EXPONENTS / RADICALS Eponents Degree of a monomial: Add the degrees of all the in the monomial together. o Eample - Find the degree of 5 7 yz Degree of a polynomial:

More information

Common Core State Standards for Activity 14. Lesson Postal Service Lesson 14-1 Polynomials PLAN TEACH

Common Core State Standards for Activity 14. Lesson Postal Service Lesson 14-1 Polynomials PLAN TEACH Postal Service Lesson 1-1 Polynomials Learning Targets: Write a third-degree equation that represents a real-world situation. Graph a portion of this equation and evaluate the meaning of a relative maimum.

More information

Module 3 Study Guide. GCF Method: Notice that a polynomial like 2x 2 8 xy+9 y 2 can't be factored by this method.

Module 3 Study Guide. GCF Method: Notice that a polynomial like 2x 2 8 xy+9 y 2 can't be factored by this method. Module 3 Study Guide The second module covers the following sections of the textbook: 5.4-5.8 and 6.1-6.5. Most people would consider this the hardest module of the semester. Really, it boils down to your

More information

LESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II

LESSON #24 - POWER FUNCTIONS COMMON CORE ALGEBRA II 1 LESSON #4 - POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The

More information

Chapter Three: Translations & Word Problems

Chapter Three: Translations & Word Problems Chapter Three: Translations & Word Problems Index: A: Literal Equations B: Algebraic Translations C: Consecutive Word Problems D: Linear Word Problems Name: Date: Period: Algebra I Literal Equations 3A

More information

Algebra Introduction to Polynomials

Algebra Introduction to Polynomials Introduction to Polynomials What is a Polynomial? A polynomial is an expression that can be written as a term or a sum of terms, each of which is the product of a scalar (the coefficient) and a series

More information

Bridging the gap between GCSE and AS/A Level Mathematics A student guide

Bridging the gap between GCSE and AS/A Level Mathematics A student guide Bridging the gap between GCSE and AS/A Level Mathematics A student guide Welcome to A Level mathematics! The object of these pages is to help you get started with the A Level course, and to smooth your

More information

12x y (4) 2x y (4) 5x y is the same as

12x y (4) 2x y (4) 5x y is the same as Name: Unit #6 Review Quadratic Algebra Date: 1. When 6 is multiplied b the result is 0 1 () 9 1 () 9 1 () 1 0. When is multiplied b the result is 10 6 1 () 7 1 () 7 () 10 6. Written without negative eponents

More information

Geometry 21 Summer Work Packet Review and Study Guide

Geometry 21 Summer Work Packet Review and Study Guide Geometry Summer Work Packet Review and Study Guide This study guide is designed to accompany the Geometry Summer Work Packet. Its purpose is to offer a review of the ten specific concepts covered in the

More information

Quadratic Equations Part I

Quadratic Equations Part I Quadratic Equations Part I Before proceeding with this section we should note that the topic of solving quadratic equations will be covered in two sections. This is done for the benefit of those viewing

More information

Algebra/Trigonometry Review Notes

Algebra/Trigonometry Review Notes Algebra/Trigonometry Review Notes MAC 41 Calculus for Life Sciences Instructor: Brooke Quinlan Hillsborough Community College ALGEBRA REVIEW FOR CALCULUS 1 TOPIC 1: POLYNOMIAL BASICS, POLYNOMIAL END BEHAVIOR,

More information

Table of Contents. Unit 3: Rational and Radical Relationships. Answer Key...AK-1. Introduction... v

Table of Contents. Unit 3: Rational and Radical Relationships. Answer Key...AK-1. Introduction... v These materials may not be reproduced for any purpose. The reproduction of any part for an entire school or school system is strictly prohibited. No part of this publication may be transmitted, stored,

More information

Algebra. Robert Taggart

Algebra. Robert Taggart Algebra Robert Taggart Table of Contents To the Student.............................................. v Unit 1: Algebra Basics Lesson 1: Negative and Positive Numbers....................... Lesson 2: Operations

More information

Lesson 3: Advanced Factoring Strategies for Quadratic Expressions

Lesson 3: Advanced Factoring Strategies for Quadratic Expressions Advanced Factoring Strategies for Quadratic Expressions Student Outcomes Students develop strategies for factoring quadratic expressions that are not easily factorable, making use of the structure of the

More information

Multiplication of Polynomials

Multiplication of Polynomials Summary 391 Chapter 5 SUMMARY Section 5.1 A polynomial in x is defined by a finite sum of terms of the form ax n, where a is a real number and n is a whole number. a is the coefficient of the term. n is

More information

Intermediate Algebra. Gregg Waterman Oregon Institute of Technology

Intermediate Algebra. Gregg Waterman Oregon Institute of Technology Intermediate Algebra Gregg Waterman Oregon Institute of Technology c August 2013 Gregg Waterman This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

More information

1. A definition of Algebra: a branch of mathematics which describes basic arithmetic relations using variables.

1. A definition of Algebra: a branch of mathematics which describes basic arithmetic relations using variables. MA30S PRECALC UNIT C: ALGEBRA CLASS NOTES. A definition of Algebra: a branch of mathematics which describes basic arithmetic relations using variables.. It is just a language. Instead of saying Jason is

More information

Algebra I. Polynomials.

Algebra I. Polynomials. 1 Algebra I Polynomials 2015 11 02 www.njctl.org 2 Table of Contents Definitions of Monomials, Polynomials and Degrees Adding and Subtracting Polynomials Multiplying a Polynomial by a Monomial Multiplying

More information

Lecture 5: Finding limits analytically Simple indeterminate forms

Lecture 5: Finding limits analytically Simple indeterminate forms Lecture 5: Finding its analytically Simple indeterminate forms Objectives: (5.) Use algebraic techniques to resolve 0/0 indeterminate forms. (5.) Use the squeeze theorem to evaluate its. (5.3) Use trigonometric

More information