Divisibility Rules Algebra 9.0

Transcription

1 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 out 1 by Shading in the bo completely. 1 is neither prime nor composite. It has only 1 factor - itself.. Use a forward Slash \ to cross out all multiples of, starting with 4. is the first prime number.. Use a backward Slash / to cross out all multiples of starting with Multiples of 4 have been crossed out already when we did #. 5. Draw a Square on all multiples of 5 starting with is prime. 6. Multiples of 6 should be X d already from # and #. 7. Circle all multiples of 7 starting with is prime. 8. Multiples of 8 were crossed out already when we did #. 9. Multiples of 9 were crossed out already when we did #. 10. Multiples of 10 were crossed out when we did # and #5. All of the remaining numbers are prime. How many prime numbers are left between 1 and 100? Answer: use your chart for help. Is 51 prime? If not, what are its factors? Is 59 prime? If not, what are its factors? Is 87 prime? If not, what are its factors? Is 91 prime? If not, what are its factors?

2 Name Period Divisibility Rules Algebra 9.0 There are some easy tricks you can use to determine if a number is divisible by,, 4, 5, 6, 8, 9 and 10. A number is divisible by: - if it is even. - if the sum of its digits is divisible by. 4 - if the number formed by the last digits is divisible by if the ones digit is 5 or if it is divisible by AND. (All even multiples of.) 7 - there is no good trick for if the number formed by the last digits is divisible by if the sum of the digits is divisible by if the last digit is a 0. 11: We will learn this trick separately. Practice: Write yes or no in each blank. Determine whether 1,408 is divisible by: Determine whether 1,45,866 is divisible by: Determine whether,,5 is divisible by: Write the complete prime factorization for each number below. Use a factor tree if necessary: E: 1,

3 GCF and LCM Algebra 9.0 The GCF is the Greatest Common Factor between two or more numbers. Sometimes the GCF is obvious: Find the GCF for each pair of numbers and and and 60 When the GCF is not obvious: E. Find the GCF between 405 and Common factors are 5 45, the GCF is 45. notes: The GCF between a pair or set of numbers is the product of their common prime factors. Practice: Find the GCF and and and 88 The LCM is the Least Common Multiple. This means the smallest number that both numbers divide with no remainder. The LCM is rarely obvious: Find the LCM for each pair of numbers and and and 4 When the LCM is not obvious: E. Find the LCM between 144 and Shared factors are other factors are 7..., so GCF = 7 1, 008

4 GCF and LCM: Venn Diagrams Algebra 9.0 Review Practice: Find the GCF and LCM for each: 1. 6 and , 4, and 105 Venn Diagrams are a great way to solve GCF and LCM problems. Eample: Use a Venn diagram to find the GCF and LCM between 84 and Eample: Use a Venn diagram to find the GCF and LCM for 75 and 105: Practice: Use a Venn diagram to find the GCF and LCM for each and and 11. 8, 4, and 105

5 Name Period GCF and LCM Algebra 9.0 Find the GCF and LCM for each pair or set of numbers: You may use a calculator, and Venn diagrams are encouraged but not required and and 16 GCF LCM GCF LCM. 90 and and 7 GCF LCM GCF LCM and and 180 GCF LCM GCF LCM 7. 0 and and 88 GCF LCM GCF LCM and and 1 (neither one is prime) GCF LCM GCF LCM

6 Factoring the GCF Algebra 9.1 You can find the GCF of epressions which include variables and eponents: Eamples: 1. Find the GCF of 7 y 5 and 10 y 7 :. Find the GCF of 9 b 11 a and 14 b 5 7 a : Practice: Find the GCF for each pair or set: and. 78 n n m 4 and. a b m 5 8 and 1ab and 0a b We have learned to Factor. Factoring is like Reverse Distribution. To factor an epression: a. Look for the GCF of all terms, including the variables. b. Place the GCF outside of the parenthesis. c. Divide each original term by the GCF to get the terms inside the parenthesis. Eamples: Factor each y y 1 y Practice: Factor each y y Practice: Factor each y 91 y 1y y 144 y

7 Name Period Factoring the GCF Algebra For each polynomial, factor the GCF from the epression. These should be easy enough to factor the GCF in your head a 6a b 9ab.. 16a ab 0b m 8m 1m a 8a b 14a b y 18 1y y 0y 15y 7.

8 Factoring the GCF For each polynomial, factor the GCF from the epression. You will likely need to find the GCF separately with these problems. Name Period Algebra a 68a b 60ab a 56ab 4b m 48m 108m a b 70a b y 57 76y 1.

9 Polynomials Algebra 9.1 A polynomial is a sum of one or more terms called monomials. Eamples: y y 4y 1 A monomial is the product of variables and constants (numbers). Eamples: abc A binomial is the sum of two monomials. Eamples: a a 4 5 A trinomial is the sum of three monomials. Eamples: 7 5 a b c y The degree of a monomial is the sum of the eponents of its variables. Eamples: a nd degree 5 ab 6 c = = 8 th degree note: The degree of a constant is zero. The degree of a polynomial is the largest degree of its monomial terms. Eamples: a 7a is a nd degree binomial 5 5a b c is a 5 th degree trinomial

10 Polynomials Algebra 9.1 Ordering Polynomials: The general rule for ordering a polynomial is to write the terms in descending order by powers of a given variable: 5 Eample: Arrange by descending powers of : 5 Eample: Arrange by descending powers of : y y y Practice: Order the following polynomials by descending powers of. 1. 5y y. 4 y 5 y. a a a a y Answer: What degree is each of the polynomials above? Other tiebreakers: Alphabetical order. E: Arrange by descending powers of a: a c a b5a b a c Practice: Order the following polynomials by descending powers of a. 1. ab a c b. a a b5a b a c. a a a y a y 4. 7az a ay a

11 Multiplying Polynomials Algebra 9. Find the area of each rectangle below: 4 5 y Multiplying Binomials: Setup a grid like the one above to solve the following: 1. ( )( 5). ( 5)( ) The FOIL Method: First ac Outer ad Inner bc Last bd ( a b)( c d) ac ad bcbd Eamples: Epand each using the FOIL Method. 1. ( 1)( a ) a. ( y)( y) Practice: Epand each using the FOIL Method. 1. ( )( 5 ). ( a c)(a c) 4. ( y )( 4). ( 1 a )( a )

12 Multiplying Polynomials Algebra 9. Find the area of each rectangle below: y a b 4 ( y)( y ) ( a4b )( ab 4) Multiplying longer polynomials is easier using the grid method: Setup a grid like the one above to solve the following: Remember to combine like terms and place answers in descending order. 1. ( y )( y 5) ( a 5b )( ab. ) Practice: Epand each. 1. ( y )( y 5). ( a 5b )( ab) Practice: Epress the area of the shaded region below as a polynomial in simplest form: 9 5

13 Beyond the Grid Algebra Once you have learned the grid and FOIL methods, you should begin to see multiplying polynomials is just distribution. Eamples: 1. ( y)( y 5). ( 5a 4b)(a b 1) Practice: Multiply each. 1. ( y)( y 5). ( a b)(a b 4) Now, multiply each using the Distributive Property. You should notice something about the answers.. a( a b) 6b(a b) 1. ( y)( 5y). ( 5y) y( 5y) 4. ( a 6b)(a b) Work Backwards: write each as a product of binomials. 1. ( ) y( ). a( a b) b(a b). ( y) 5( y) 4. a( a 7) 5b( a 7) Use the grid method when problems get more comple: 1. ( y 5)( y ). ( a a 5a)( a a )

14 Special Cases Algebra 9.4 Multiply each pair of binomials using the FOIL Method. Simplify answers. 1. ( )( 5). ( a 5b)( a b). ( )( ) 4. ( a 5b )( a 5b ) 5. ( )( ) 6. ( a 5b ) #1 and # are typical trinomials. # and #4 are called DIFFERENCE OF SQUARES. Why? #5 and #6 are called PERFECT SQUARE trinomials. More practice: Difference of Squares. Solve each using FOIL, try to recognize a shortcut. 1. ( )( ). ( a b)(a b) 4. ( a )( a ). ( 5)( 5) How do you recognize a difference of squares? More practice: Perfect Squares. Solve each using FOIL, try to recognize a shortcut. 1. ( ). ( a b). ( 5 ) 4. ( a ) Challenge: Epand ( a b)( ab)( ab)( ab) in 1 minute.

15 Quiz Review Algebra 9.4 Factor out the GCF for each trinomial y 0 y 10y a b 15a b 6a b a 7a 48a Multiply each: Order your answers by descending powers of or a y( y) ( y) 00. (a b)( b a) 00. ( 5 )( 4) 400. ( a b c)(a b c) Multiply each. Order your answers by descending powers of or a ( 0 )(0) ( )( ) 400. ( a 5b) ( a ) ( a )

16 Name Period Factoring and FOIL Practice Quiz Algebra 9.4 Factor each epression (Reverse distribution): y 1 y 0y a b 80a b y 95 y 15. Simplify each (Distribute, combine like terms, and then reorder the terms by descending powers of or a): 4. 4( y y) ( y y) 5. ab( a ) a(4 b 5ab) 6. (y ) ( 7) Multiply (FOIL or Grid method) 7. ( 4)( ) 8. ( a )( a ) 9. ( )( y )

17 Name Period Factoring and FOIL Practice Quiz Algebra 9.4 Multiply each (Look for perfect squares and difference of squares): 10. ( y )( y ) ( a ) ( a)( a) ( y )( y ) ( a b)(a b)(a b) ( ) 15. ( 1)( 1)

18 Polynomial Applications Algebra 9.4 A common use for multiplying polynomials involves finding area. Eample: Epress the area of the shaded regions in terms of I will call these frame problems because the diagrams usually look like frames. Practice: Epress the area of the shaded regions in terms of

19 Polynomial Applications Algebra 9.4 Word problems can involve similar area problems, but the diagrams must be given. Eample: You are matting a photograph that is twice as tall as it is wide. You want to have five inches of matting around the entire photograph. Epress the area of matting you will need based on the width (w) of the photograph. Eample: Barry bought a new rectangular rug for his rectangular dining room. The rug is three feet longer than it is wide. The room is si feet wider than his rug, and seven feet longer than the rug. Epress the area of bare floor that will be showing in terms of the rug s width (w). Answer: If there are 190 square feet of bare floor showing, what ar the dimensions of the rug? Practice: Jeremy has a backyard pool surrounded by a tiled walkway that is two yards wide. The pool is 5 yards longer than it is wide. Epress the area of the walkway in terms of the width (w) of the pool. Answer: If the walkway is 196 square yards, how long is the pool? Practice: A painting has a frame that is 7 inches wider and 8 inches taller than the artwork it surrounds. The artwork is 5 inches taller than it is wide. Epress the area of the frame in terms of the painting s width (w). Answer: If the area of the frame is 196 square inches, what is the height of the painting?

20 + 10 Name Period Polynomial Applications Algebra 9.4 Epress the area of each shaded region in terms of Epress the area of each shaded region in terms of

21 Name Period Polynomial Applications Algebra 9.4 Solve each. Include a sketch for each. 7. Connor is planting a garden surrounded by 1-foot square concrete blocks. The garden will be 10 feet longer than it is wide. Epress the number of square blocks he will need based on the width (w) of the garden. If he uses 56 blocks, how many square feet is the area enclosed by the blocks? 8. Kerry takes a sheet of paper that is inches shorter than it is wide. He cuts a hole out of the paper that leaves inches of paper on all sides of the hole. Epress the area of the remaining paper rectangle in terms of w, the width of the original sheet. If there are 5in of paper remaining, what were the dimensions of the cut-out hole? 9. A company manufactures windows that are 0 inches taller than they are wide. The window comes with an aluminum frame that is 6 inches wide on three sides, and 10 inches wide at the bottom. Epress the area of the aluminum frame in terms of the window s width (w). If the area of the frame is 1,9in, what is the height of the window?

22 Standard Form and Factoring Algebra 9.5 A Quadratic Equation written as a function looks like this: y A B C We will call this Standard Form. Eamples: List values for A, B, and C: y 5 y 5 When you multiply a pair of (1 st degree) binomials, you get a quadratic epression. ( )( 5) 15 Think! In the equation above, what are the A, B, and C values? How did we get the values for B and C? Factoring: Easy ones. Today we will learn to factor simple quadratics by reversing the FOIL method. Review: Multiply ( )( 4) 6 8 Factoring: Find two numbers which can be added to get 6 and multiplied to get 8. More Eamples: Factor

23 Standard Form and Factoring Algebra 9.5 Practice: Factor. Write Prime for any that cannot be factored Practice: Factor. Write Prime for any that cannot be factored

24 Factoring: GCF with Easy Ones Algebra 9.5 Eamples: Factor completely. Begin by factoring out the GCF. Finish by using reverse FOIL y 14y 4y Practice: Factor Completely a 10a 4a y 1y 0y Practice: Factor. Write Prime for any that cannot be factored y 4y48y

25 Name Period Factoring Easy Ones with GCFs Algebra 9.5 Factor eact epression by first factoring the GCF and then using reverse FOIL. Write Prime for any that cannot be factored y 18 y 17 y y 1y 48y y y 7y a a 1a

26 Factoring: Difference of Squares Algebra 9.7 Eamples: Factor completely Practice: Factor Completely You can factor out the GCF first. Eamples: Factor completely Practice: Factor Completely y y 4y

27 Factoring: Perfect Squares Algebra 9.7 Eamples: Factor completely Practice: Factor Completely Recognizing Perfect Squares: Don t be fooled by these imposters! Only one is a perfect square. Can you find it? Factor and Check! Easy Ones, Perfect Squares, and Difference of Squares: Put it all together. Try to recognize how to factor each.. a

28 Factoring and FOIL Review Name Period Algebra Multiply (FOIL or Grid method) 1. ( 1)( 5). ( 4a ) 1.. ( )( y) 4. (1 7)(1 7).. 4. Factor Each Completely (Look for GCFs, Perfect Squares, and Difference of Squares. Write PRIME for any that cannot be factored.)

29 Factoring Practice Challenge 1: Factor Completely. Name Period Algebra y y Ch. 1. Challenge : The number 65,55 is equal to Use what you know about a difference of squares to find the four prime factors of 65,55 without a calculator (be ready to eplain how this can be done). Ch..

30 Name Period Factoring and FOIL Practice Quiz Algebra Multiply (FOIL or Grid method) 1. ( a b)( a b ) 1.. ( 5).. ( y)( y). 4. ( )( ) 4. Factor Each Completely (Look for GCFs, Perfect Squares, and Difference of Squares. Write PRIME for any that cannot be factored.) a b

31 Name Period Factoring and FOIL Practice Quiz Algebra Factor Each Completely (Look for GCFs, Perfect Squares, and Difference of Squares. Write PRIME for any that cannot be factored.) 8. y 8y 15y

32 Name Period Factoring and FOIL Self-Check Algebra 9.7 Factor each (Look for perfect squares and difference of squares, GCF, and easy ones).. y 5y 6y Name Period Factoring and FOIL Self-Check Algebra 9.7 Factor each (Look for perfect squares and difference of squares, GCF, and easy ones).. y 5y 6y

33 Factoring Review. Algebra 9.7 Easy Ones: Factor completely. Write PRIME for any that cannot be factored. E.: y 1y 15y Difference of Squares: Factor completely. Write PRIME where applicable. E.: a ay Perfect Squares: Factor completely. Write PRIME where applicable. E.: y 6y

34 Hard Ones Magic Number Algebra 9.7 Look at the trinomial below. Is there a GCF to be factored? Is it an Easy One, a Perfect Square, or a Difference of Squares? The answer to all of these questions is No. We will call this type of factoring the Magic Number Method. Eample: Factor Find/Factor the Magic Number.. Rewrite the middle term.. Regroup. 4. Factor out the GCFs. 5. Finish ( )( ) Two more eamples. Watch Carefully! Practice: Factor each completely Practice: Factor each completely

35 Hard Ones Magic Number Algebra 9.7 Look at each trinomial below. DO NOT TRY TO FACTOR THEM. Label each with: EASY ONE DIFFERENCE OF SQUARES PERFECT SQUARE HARD ONE (MAGIC NUMBER) (hint: there are two of each) y Now, try to factor them y

36 Factoring Review Algebra 9.7 Factor each: Write Prime for any that cannot be factored Name Period y

37 Factoring Review Algebra 9.7 Factor each: Write Prime for any that cannot be factored Name Period y y y 5y y y

38 Factoring Quiz Review Algebra 9.7 Factor each: Write Prime for any that cannot be factored y 9y Factor each: Write Prime for any that cannot be factored y 9y 9 Factor each: Write Prime for any that cannot be factored

39 Name Period Factoring and FOIL Practice Quiz Algebra 9.8 Multiply each (Look for perfect squares and difference of squares, order the terms by descending powers of ): 1. ( ) 1.. ( a 1)(a 1).. ( y)( y ). Factor each COMPLETELY NONE OF THE PROBLEMS BELOW ARE PRIME. (Look for perfect squares and difference of squares, easy ones and hard ones) y a 9a

40 Name Period Factoring and FOIL Practice Quiz Algebra 9.8 Factor each COMPLETELY Write PRIME for any that cannot be factored. (Look for perfect squares and difference of squares, easy ones and hard ones) y y y 1.

41 Simplifying Epressions Algebra Practice: Factor each Now, try to simplify the following: Practice: Simplify each epression. ( 7) 1. ( 7)( ) Practice: Simplify each epression Practice: Simplify each epression. ( )( ) 1. ( )( )

42 Simplify by Factoring Algebra 9.7 Factor each and simplify where possible. Name Period 1. ( 7) ( 7)

43 Solving Equations by Factoring Algebra 9.8 Practice: Solve each ( 5) 15. ( ) 0 If 0 If ( )( 5) 0 ab then either a 0 or 0 b. then either ( ) 0 or ( 5) 0 Eamples: Solve each for. Each will have two solutions.. ( 9)( 5) 0 1. ( 7) Practice: Solve for. Each will have two solutions Tricky Eamples: Solve each for Tricky Practice: Solve each for

44 Solving Quadratics by Factoring Algebra 9.7 Factor each and simplify where possible. 1. ( )( 5) 0 Name Period Challenge: (4 solutions)

45 Factoring Problems Algebra For the problems below, you must know the Pythagorean Theorem: a c In any right triangle: a + b = c Eample: Find the lengths of the sides of the right triangle below. b + + Practice: Find the lengths of the sides of the right triangle below Practice: 1. In a right triangle, the hypotenuse is 9 inches longer than the shortest side. The length of the medium side is just one inch longer than the length of the shortest side. What is the perimeter in inches of the triangle?. The hypotenuse of a right triangle is 1cm longer than the long leg. The short leg is 1cm shorter than half the long leg. What is the triangle s area?

46 Solving Quadratics by Factoring Factor each and simplify where possible Name Period Algebra ( solutions) 9. The equation k 6 0 has only one solution for positive integer k. What is k? 10. Find the perimeter of the triangle below

47 Clever Factoring: Some tricks and more difficult problems: Algebra Eample: One of the solutions to the equation a. What is the value of a? b. What is the other solution? Practice: 1. The equation a 8 a. What is the value of a? b. What is the other solution?. The equation 5 a. What is the value of a? b. What is the other solution? a 6 has 4 a has 1 is 5. as a solution. as a solution. Eample: How can the polynomial ( y) 9 can be factored into the product of two trinomials? Practice: 1. Factor the following into a product of trinomials: ( y ).. Factor the following into a product of trinomials: y Solving Trickier Equations Practice: 1. Solve for : ( 4) ( 4) 0. Hint: Where have you seen something similar to this before? 10. Solve for : 0. Hint: use a common denominator.. Solve for :

48 Factoring Test Review Algebra 9.8 Perfect Squares and Difference of Squares: Factor each y 1y Easy Ones and Magic Number: Write Prime for any that cannot be factored y y Solve each: Write Prime for any that cannot be factored

49 Name Period Factoring and FOIL Practice Test Algebra 9.8 Factor each COMPLETELY Write PRIME for any that cannot be factored. (Look for perfect squares and difference of squares, easy ones and hard ones, and GCF problems) y a 4a 4a

50 Name Period Factoring & FOIL Practice Test (4) Algebra 9.8 Solve for : Some problems may have more than one solution. List all solutions in the blank provided. 8. ( 7) 0 8. = = = = Multiply: 1. ( 5 4)( 5) ( 5 ) ( )( )( ) 14.

51 Name Period Factoring and FOIL Practice Test Algebra 9.8 Solve for : Some problems may have more than one solution. List all solutions in the blank provided. 8. ( 7) 0 8. = = = = Simplify each: