# F.3 Special Factoring and a General Strategy of Factoring

Size: px
Start display at page:

Download "F.3 Special Factoring and a General Strategy of Factoring"

Transcription

1 F.3 Special Factoring and a General Strategy of Factoring Difference of Squares section F4 233 Recall that in Section P2, we considered formulas that provide a shortcut for finding special products, such as a product of two conjugate binomials, or the perfect square of a binomial, ( + bb)( bb) = 22 bb 22, ( ± bb) 2 = 22 ± bb 22. Since factoring reverses the multiplication process, these formulas can be used as shortcuts in factoring binomials of the form 22 bb 22 (difference of squares), and trinomials of the form 22 ± bb 22 (perfect square). In this section, we will also introduce a formula for factoring binomials of the form 33 ± bb 33 (sum or difference of cubes). These special product factoring techniques are very useful in simplifying expressions or solving equations, as they allow for more efficient algebraic manipulations. At the end of this section, we give a summary of all the factoring strategies shown in this chapter. 2 bb 2 + bb bb Out of the special factoring formulas, the easiest one to use is the difference of squares, 22 bb 22 = ( + bb)( bb) Figure 3.1 shows a geometric interpretation of this formula. The area of the yellow square, 2, diminished by the area of the blue square, bb 2, can be rearranged to a rectangle with the length of ( + bb) and the width of ( bb). bb bb Figure 3.1 bb To factor a difference of squares 22 bb 22, first, identify and bb, which are the expressions being squared, and then, form two factors, the sum ( + bb), and the difference ( bb), as illustrated in the example below. Solution Factoring Differences of Squares Factor each polynomial completely. a. 25xx 2 1 b. 3.6xx 4 0.9yy 6 c. xx 4 81 d. 16 ( 2) 2 a. First, we rewrite each term of 25xx 2 1 as a perfect square of an expression. bb 25xx 2 1 = (5xx) Then, treating 5xx as the and 1 as the bb in the difference of squares formula 2 bb 2 = ( + bb)( bb), we factor: Solving Polynomial Equations and Applications of Factoring

2 section F bb 2 = ( + bb)( bb) 25xx 2 1 = (5xx) = ( )( ) b. First, we factor out 0.9 to leave the coefficients in a perfect square form. So, 3.6xx 4 0.9yy 6 = 0.9(4xx 4 yy 6 ) Then, after writing the terms of 4xx 4 yy 6 as perfect squares of expressions that correspond to and bb in the difference of squares formula 2 bb 2 = ( + bb)( bb), we factor bb 0.9(4xx 4 yy 6 ) = 0.9[(2xx 2 ) 2 (yy 3 ) 2 ] = xx 22 + yy 33 22xx 22 yy 33 c. Similarly as in the previous two examples, xx 4 81 can be factored by following the difference of squares pattern. So, xx 4 81 = (xx 2 ) 2 (9) 2 = (xx 2 + 9)(xx 2 9) However, this factorization is not complete yet. Notice that xx 2 9 is also a difference of squares, so the original polynomial can be factored further. Thus, xx 4 81 = (xx 2 + 9)(xx 2 9) = xx (xx + 33)(xx 33) Attention: The sum of squares, xx 2 + 9, cannot be factored using real coefficients. Generally, except for a common factor, a quadratic binomial of the form 22 + bb 22 is not factorable over the real numbers. d. Following the difference of squares formula, we have 16 ( 2) 2 = 4 2 ( 2) 2 = [4 + ( 2)][4 ( 2)] Remember to use brackets after the negative sign! = (4 + 2)(4 + 2) work out the inner brackets = (22 + )(66 ) combine like terms Perfect Squares + bb 2 bb + bb bb bb 2 Another frequently used special factoring formula is the perfect square of a sum or a difference bb + bb 22 = ( + bb) 22 or 22 22bb + bb 22 = ( bb) 22 Figure 3.2 Figure 3.2 shows the geometric interpretation of the perfect square of a sum. We encourage the reader to come up with a similar interpretation of the perfect square of a difference. Factoring

3 section F4 235 To factor a perfect square trinomial 22 ± 22bb + bb 22, we find and bb, which are the expressions being squared. Then, depending on the middle sign, we use and bb to form the perfect square of the sum ( + bb) 22, or the perfect square of the difference ( bb) 22. Identifying Perfect Square Trinomials Decide whether the given polynomial is a perfect square. a. 9xx 2 + 6xx + 4 b. 9xx 2 + 4yy 2 12xxxx c. 25pp pp 2 16 d. 49yy xxxx xx 2 Solution a. Observe that the outside terms of the trinomial 9xx 2 + 6xx + 4 are perfect squares, as 9xx 2 = (3xx) 2 and 4 = 2 2. So, the trinomial would be a perfect square if the middle terms would equal 2 3xx 2 = 12xx. Since this is not the case, our trinomial is not a perfect square. Attention: Except for a common factor, trinomials of the type 22 ± bb + bb 22 are not factorable over the real numbers! b. First, we arrange the trinomial in decreasing order of the powers of xx. So, we obtain 9xx 2 12xxxx + 4yy 2. Then, since 9xx 2 = (3xx) 2, 4yy 2 = (2yy) 2, and the middle term (except for the sign) equals 2 3xx 2 = 12xx, we claim that the trinomial is a perfect square. Since the middle term is negative, this is the perfect square of a difference. So, the trinomial 9xx 2 12xxxx + 4yy 2 can be seen as 2 2 bb + bb 2 = ( bb ) 2 (3xx) 2 2 3xx 2yy + (2yy) 2 = (3xx 2yy) 2 c. Even though the coefficients of the trinomial 25pp pp 2 16 and the distribution of powers seem to follow the pattern of a perfect square, the last term is negative, which makes it not a perfect square. d. Since 49yy 6 = (7yy 3 ) 2, 36xx 2 = (6xx) 2, and the middle term equals 2 7yy 3 6xx = 84xxxx 3, we claim that the trinomial is a perfect square. Since the middle term is positive, this is the perfect square of a sum. So, the trinomial 9xx 2 12xxxx + 4yy 2 can be seen as bb + bb 2 = ( bb ) 2 (7yy 3 ) yy 3 6xx + (6xx) 2 = (7yy 3 6xx) 2 Factoring Perfect Square Trinomials Factor each polynomial completely. a. 25xx xx + 1 b bb 2 c. mm 2 8mm nn 2 d. 4yy 2 144yy yy 5 Solving Polynomial Equations and Applications of Factoring

4 section F4 236 Solution a. The outside terms of the trinomial 25xx xx + 1 are perfect squares of 5 and 1, and the middle term equals 2 5xx 1 = 10xx, so we can follow the perfect square formula. Therefore, 25xx xx + 1 = ( ) 22 b. The outside terms of the trinomial bb 2 are perfect squares of and 6bb, and the middle term (disregarding the sign) equals 2 6bb = 12, so we can follow the perfect square formula. Therefore, bb 2 = ( 6666) 22 c. Observe that the first three terms of the polynomial mm 2 8mm nn 2 form a perfect square of mm 6 and the last term is a perfect square of 7nn. So, we can write mm 2 8mm nn 2 = (mm 6) 2 (7nn) 2 Notice that this way we have formed a difference of squares. So we can factor it by following the difference of squares formula (mm 6) 2 (7nn) 2 = (mm )(mm ) d. As in any factoring problem, first we check the polynomial 4yy 2 144yy yy 5 for a common factor, which is 4yy 2. To leave the leading term of this polynomial positive, we factor out 4yy 2. So, we obtain 4yy 2 144yy yy 5 = 4yy 2 (1 + 36yy 6 12yy 3 ) = 4yy 2 (36yy 6 12yy 3 + 1) = 44yy 22 66yy This is not in factored form yet! arrange the polynomial in decreasing powers fold to the perfect square form Sum or Difference of Cubes bb bb The last special factoring formula to discuss in this section is the sum or difference of cubes. 3 bb 3 bb bb bb bb or 33 + bb 33 = ( + bb) 22 bb + bb bb 33 = ( bb) 22 + bb + bb 22 3 bb 3 = 2 ( bb)+( bb) + bb 2 ( bb) Figure 3.3 = ( bb)( bb 2 ) The reader is encouraged to confirm these formulas by multiplying the factors in the right-hand side of each equation. In addition, we offer a geometric visualization of one of these formulas, the difference of cubes, as shown in Figure 3.3. Factoring

5 section F4 237 Hints for memorization of the sum or difference of cubes formulas: The binomial factor is a copy of the sum or difference of the terms that were originally cubed. The trinomial factor follows the pattern of a perfect square, except that the middle term is single, not doubled. The signs in the factored form follow the pattern Same-Opposite-Positive (SOP). Factoring Sums or Differences of Cubes Factor each polynomial completely. a. 8xx b. 27xx 7 yy 125xxxx 4 c. 2nn d. (pp 2) 3 + qq 3 Solution a. First, we rewrite each term of 8xx as a perfect cube of an expression. bb 8xx = (2xx) Then, treating 2xx as the and 1 as the bb in the sum of cubes formula 3 + bb 3 = ( + bb)( 2 + bb 2 ), we factor: 33 + bb 33 = ( + bb) 22 bb + bb 22 Quadratic trinomials of the form 22 ± + bb 22 are not factorable! 8xx = (2xx) = (2xx + 1)((2xx) 2 2xx ) = ( ) 44xx Notice that the trinomial 4xx 2 2xx + 1 in not factorable anymore. b. Since the two terms of the polynomial 27xx 7 yy 125xxxx 4 contain the common factor xxxx, we factor it out and obtain 27xx 7 yy 125xxxx 4 = xxxx(27xx 6 125yy 3 ) Observe that the remaining polynomial is a difference of cubes, (3xx 2 ) 3 (5yy) 3. So, we factor, 27xx 7 yy 125xxxx 4 = xxxx[(3xx 2 ) 3 (5yy) 3 ] c. After factoring out the common factor 2, we obtain ( + bb ) 22 bb + bb 22 = xxxx(3xx 2 5yy)[(3xx 2 ) 2 + 3xx 2 5yy + (5yy) 2 ] = xxxx 33xx xx xx 22 yy yy 22 2nn = 2(nn 6 64) Difference of squares or difference of cubes? Solving Polynomial Equations and Applications of Factoring

6 section F4 238 Notice that nn 6 64 can be seen either as a difference of squares, (nn 3 ) 2 8 2, or as a difference of cubes, (nn 2 ) It turns out that applying the difference of squares formula first leads us to a complete factorization while starting with the difference of cubes does not work so well here. See the two approaches below. (nn 3 ) (nn 2 ) = (nn 3 + 8)(nn 3 8) = (nn 2 4)(nn 4 + 4nn ) = (nn + 2)(nn 2 2nn + 4)(nn 2)(nn 2 + 2nn + 4) = (nn + 2)(nn 2)(nn 4 + 4nn ) 4 prime factors, so the factorization is complete There is no easy way of factoring this trinomial! Therefore, the original polynomial should be factored as follows: 2nn = 2(nn 6 64) = 2[(nn 3 ) ] = 2(nn 3 + 8)(nn 3 8) = 22(nn + 22) nn (nn 22) nn d. To factor (pp 2) 3 + qq 3, we follow the sum of cubes formula ( + bb)( 2 + bb 2 ) by assuming = pp 2 and bb = qq. So, we have (pp 2) 3 + qq 3 = (pp 2 + qq) [(pp 2) 2 (pp 2)qq + qq 2 ] = (pp 2 + qq) [pp 2 2pppp + 4 pppp + 2qq + qq 2 ] = (pp 22 + qq) pp qq 22 General Strategy of Factoring Recall that a polynomial with integral coefficients is factored completely if all of its factors are prime over the integers. How to Factorize Polynomials Completely? 1. Factor out all common factors. Leave the remaining polynomial with a positive leading term and integral coefficients, if possible. 2. Check the number of terms. If the polynomial has - more than three terms, try to factor by grouping; a four term polynomial may require 2-2, 3-1, or 1-3 types of grouping. - three terms, factor by guessing, decomposition, or follow the perfect square formula, if applicable. - two terms, follow the difference of squares, or sum or difference of cubes formula, if applicable. Remember that sum of squares, 22 + bb 22, is not factorable over the real numbers, except for possibly a common factor. Factoring

7 section F Keep in mind the special factoring formulas: Difference of Squares Perfect Square of a Sum Perfect Square of a Difference Sum of Cubes Difference of Cubes 22 bb 22 = ( + bb)( bb) bb 22 = ( + bb) bb 22 = ( bb) bb 33 = ( + bb) 22 + bb bb 33 = ( bb) bb Keep factoring each of the obtained factors until all of them are prime over the integers. Multiple-step Factorization Factor each polynomial completely. a. 80xx 5 5xx b bb 2 c. (5rr + 8) 2 6(5rr + 8) + 9 d. (pp 2qq) 3 + (pp + 2qq) 3 Solution a. First, we factor out the GCF of 80xx 5 and 5xx, which equals to 5xx. So, we obtain 80xx 5 5xx = 5xx(16xx 4 1) repeated difference of squares Then, we notice that 16xx 4 1 can be seen as the difference of squares (4xx 2 ) So, we factor further 80xx 5 5xx = 5xx(4xx 2 + 1)(4xx 2 1) The first binomial factor, 4xx 2 + 1, cannot be factored any further using integral coefficients as it is the sum of squares, (2xx) However, the second binomial factor, 4xx 2 1, is still factorable as a difference of squares, (2xx) Therefore, 80xx 5 5xx = xx ( )( ) This is a complete factorization as all the factors are prime over the integers. 3-1 type of grouping b. The polynomial bb 2 consists of four terms, so we might be able to factor it by grouping. Observe that the 2-2 type of grouping has no chance to succeed, as the first two terms involve only the variable while the second two terms involve only the variable bb. This means that after factoring out the common factor in each group, the remaining binomials would not be the same. So, the 2-2 grouping would not lead us to a factorization. However, the 3-1 type of grouping should help. This is because the first three terms form the perfect square, (2 1) 2, and there is a subtraction before the last term bb 2, which is also a perfect square. So, in the end, we can follow the difference of squares formula to complete the factoring process bb 2 = (2 1) 2 bb 2 = ( bb)( bb) Solving Polynomial Equations and Applications of Factoring

8 section F4 240 factoring by substitution c. To factor (5rr + 8) 2 6(5rr + 8) + 9, it is convenient to substitute a new variable, say, for the expression 5rr + 8. Then, (5rr + 8) 2 6(5rr + 8) + 9 = perfect square! Remember to represent the new variable by a different letter than the original variable! = ( 3) 2 = (5rr + 8 3) 2 = (5rr + 5) 2 go back to the original variable Notice that 5rr + 5 can still be factored by taking the 5 out. So, for a complete factorization, we factor further (5rr + 5) 2 = 5(rr + 1) 2 = 2222(rr + 11) 22 d. To factor (pp 2qq) 3 + (pp + 2qq) 3, we follow the sum of cubes formula ( + bb)( 2 + bb 2 ) by assuming = pp 2qq and bb = pp + 2qq. So, we have multiple special formulas and simplifying (pp 2qq) 3 + (pp + 2qq) 3 = (pp 2qq + pp + 2qq) [(pp 2qq) 2 (pp 2qq)(pp + 2qq) + (pp + 2qq) 2 ] = 2pp [pp 2 4pppp + 4qq 2 (pp 2 4qq 2 ) + pp 2 + 4pppp + 4qq 2 ] = 2pp (2pp 2 + 8qq 2 pp 2 + 4qq 2 ) = 2222 pp qq 22 F.3 Exercises Vocabulary Check Complete each blank with one of the suggested words, or the most appropriate term or phrase from the given list: difference of cubes, difference of squares, perfect square, sum of cubes, sum of squares. 1. If a binomial is a its factorization has the form ( + bb)( bb). 2. Trinomials of the form 2 ± 2 + bb 2 are trinomials. 3. The product ( + bb)( 2 + bb 2 ) is the factorization of the. 4. The product ( bb)( bb 2 ) is the factorization of the. 5. A is not factorable. 6. Quadratic trinomials of the form 2 ± + bb 2 / nnnnnn factorable. Factoring

9 section F4 241 Concept Check Determine whether each polynomial is a perfect square, a difference of squares, a sum or difference of cubes, or neither xx yy 2 8. xx 2 14xx xx 4 + 4xx xx 2 (xx + 4) xx yy xx yy xx xx 3 + 9xx xx 6 10xx 3 yy 2 + yy xx 6 yy xx yy xx 2 16xx Concept Check 19. The binomial 4xx is an example of a sum of two squares that can be factored. Under what conditions can the sum of two squares be factored? 20. Insert the correct signs into the blanks. a = (2 )(4 2 2 ) b. bb 3 1 = (bb 1)(bb 2 bb 1) Factor each polynomial completely, if possible. 21. xx 2 yy xx 2 + 2xxxx + yy xx 3 yy xx zz 2 4zz xx zz yy yy yy xx 2 64yy nn nnnn + 100mm bb bb xx + 16xx pp 6 64qq xx 2 zz 2 100yy pp pp 38. xx yy rr 4 9rr bb xx yy xx 4 + 8xx tt xx 6 121xx 2 yy pppp + 100pp 2 qq bb 3 125cc nn 2tt bb + 64bb xx (xx + 1) uu2 uuuu + vv tt 4 128tt (nn + 3) xx 2nn + 6xx nn ( + 2) zz cc cc cc 59. (xx + 5) 3 xx bb zz 2 0.7zz (xx 1) 3 + (xx + 1) (xx 2yy) 2 (xx + yy) pp 8 + 9pp (xx + 2) 3 (xx 2) 3 Solving Polynomial Equations and Applications of Factoring

10 section F4 242 Factor each polynomial completely yy 3 12xx 2 yy 67. 2xx xx 3 xxyy 2 + xx 2 yy yy yy yy uu mm 3 + mm nn 2 + 2nn 3 + 4nn bb yy 9 yy 75. (xx 2 2) 2 4(xx 2 2) (pp 3) 2 64(pp 3) bb 2 6bb (2 bb) xx 2 yy 2 zz + 25xxxxzz zz xx 8 yy xx 6 2xx 5 + xx 4 xx 2 + 2xx xx 2 yy 4 9yy 4 4xx 2 zz 4 + 9zz cc 2ww+1 + 2cc ww+1 + cc Factoring

### F.1 Greatest Common Factor and Factoring by Grouping

section F1 214 is the reverse process of multiplication. polynomials in algebra has similar role as factoring numbers in arithmetic. Any number can be expressed as a product of prime numbers. For example,

### F.1 Greatest Common Factor and Factoring by Grouping

1 Factoring Factoring is the reverse process of multiplication. Factoring polynomials in algebra has similar role as factoring numbers in arithmetic. Any number can be expressed as a product of prime numbers.

### 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

### Lesson 1: Successive Differences in Polynomials

Lesson 1 Lesson 1: Successive Differences in Polynomials Classwork Opening Exercise John noticed patterns in the arrangement of numbers in the table below. 2.4 3.4 4.4 5.4 6.4 5.76 11.56 19.36 29.16 40.96

### P.2 Multiplication of Polynomials

1 P.2 Multiplication of Polynomials aa + bb aa + bb As shown in the previous section, addition and subtraction of polynomials results in another polynomial. This means that the set of polynomials is closed

### F.4 Solving Polynomial Equations and Applications of Factoring

section F4 243 F.4 Zero-Product Property Many application problems involve solving polynomial equations. In Chapter L, we studied methods for solving linear, or first-degree, equations. Solving higher

### Quadratic Equations and Functions

50 Quadratic Equations and Functions In this chapter, we discuss various ways of solving quadratic equations, aaxx 2 + bbbb + cc 0, including equations quadratic in form, such as xx 2 + xx 1 20 0, and

### R.3 Properties and Order of Operations on Real Numbers

1 R.3 Properties and Order of Operations on Real Numbers In algebra, we are often in need of changing an expression to a different but equivalent form. This can be observed when simplifying expressions

### Algebra 1: Hutschenreuter Chapter 10 Notes Adding and Subtracting Polynomials

Algebra 1: Hutschenreuter Chapter 10 Notes Name 10.1 Adding and Subtracting Polynomials Polynomial- an expression where terms are being either added and/or subtracted together Ex: 6x 4 + 3x 3 + 5x 2 +

### P.3 Division of Polynomials

00 section P3 P.3 Division of Polynomials In this section we will discuss dividing polynomials. The result of division of polynomials is not always a polynomial. For example, xx + 1 divided by xx becomes

### Algebra 2. Factoring Polynomials

Algebra 2 Factoring Polynomials Algebra 2 Bell Ringer Martin-Gay, Developmental Mathematics 2 Algebra 2 Bell Ringer Answer: A Martin-Gay, Developmental Mathematics 3 Daily Learning Target (DLT) Tuesday

### Systems of Linear Equations

Systems of Linear Equations As stated in Section G, Definition., a linear equation in two variables is an equation of the form AAAA + BBBB = CC, where AA and BB are not both zero. Such an equation has

### Polynomials and Polynomial Functions

1 Polynomials and Polynomial Functions One of the simplest types of algebraic expressions are polynomials. They are formed only by addition and multiplication of variables and constants. Since both addition

### = The algebraic expression is 3x 2 x The algebraic expression is x 2 + x. 3. The algebraic expression is x 2 2x.

Chapter 7 Maintaining Mathematical Proficiency (p. 335) 1. 3x 7 x = 3x x 7 = (3 )x 7 = 5x 7. 4r 6 9r 1 = 4r 9r 6 1 = (4 9)r 6 1 = 5r 5 3. 5t 3 t 4 8t = 5t t 8t 3 4 = ( 5 1 8)t 3 4 = ()t ( 1) = t 1 4. 3(s

### 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

### Section 6.5 A General Factoring Strategy

Difference of Two Squares: a 2 b 2 = (a + b)(a b) NOTE: Sum of Two Squares, a 2 b 2, is not factorable Sum and Differences of Two Cubes: a 3 + b 3 = (a + b)(a 2 ab + b 2 ) a 3 b 3 = (a b)(a 2 + ab + b

### 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

### Rational Expressions and Functions

Rational Expressions and Functions In the previous two chapters we discussed algebraic expressions, equations, and functions related to polynomials. In this chapter, we will examine a broader category

### 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

### Math 10 - Unit 5 Final Review - Polynomials

Class: Date: Math 10 - Unit 5 Final Review - Polynomials Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Factor the binomial 44a + 99a 2. a. a(44 + 99a)

### Find two positive factors of 24 whose sum is 10. Make an organized list.

9.5 Study Guide For use with pages 582 589 GOAL Factor trinomials of the form x 2 1 bx 1 c. EXAMPLE 1 Factor when b and c are positive Factor x 2 1 10x 1 24. Find two positive factors of 24 whose sum is

### 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

### KEY CONCEPTS. Factoring is the opposite of expanding.

KEY CONCEPTS Factoring is the opposite of expanding. To factor simple trinomials in the form x 2 + bx + c, find two numbers such that When you multiply them, their product (P) is equal to c When you add

### 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

### 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

### Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2

Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2 April 11, 2016 Chapter 10 Section 1: Addition and Subtraction of Polynomials A monomial is

### 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

### 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

### Eureka Math. Algebra II Module 1 Student File_A. Student Workbook. This file contains Alg II-M1 Classwork Alg II-M1 Problem Sets

Eureka Math Algebra II Module 1 Student File_A Student Workbook This file contains Alg II- Classwork Alg II- Problem Sets Published by the non-profit GREAT MINDS. Copyright 2015 Great Minds. No part of

### Solving Quadratic Equations

Solving Quadratic Equations MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to: solve quadratic equations by factoring, solve quadratic

### Math 0320 Final Exam Review

Math 0320 Final Exam Review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Factor out the GCF using the Distributive Property. 1) 6x 3 + 9x 1) Objective:

### Rational Expressions and Functions

1 Rational Expressions and Functions In the previous two chapters we discussed algebraic expressions, equations, and functions related to polynomials. In this chapter, we will examine a broader category

0 Radicals and Radical Functions So far we have discussed polynomial and rational expressions and functions. In this chapter, we study algebraic expressions that contain radicals. For example, + 2, xx,

### Controlling the Population

Lesson.1 Skills Practice Name Date Controlling the Population Adding and Subtracting Polynomials Vocabulary Match each definition with its corresponding term. 1. polynomial a. a polynomial with only 1

### Can there be more than one correct factorization of a polynomial? There can be depending on the sign: -2x 3 + 4x 2 6x can factor to either

MTH95 Day 9 Sections 5.5 & 5.6 Section 5.5: Greatest Common Factor and Factoring by Grouping Review: The difference between factors and terms Identify and factor out the Greatest Common Factor (GCF) Factoring

### Topic 7: Polynomials. Introduction to Polynomials. Table of Contents. Vocab. Degree of a Polynomial. Vocab. A. 11x 7 + 3x 3

Topic 7: Polynomials Table of Contents 1. Introduction to Polynomials. Adding & Subtracting Polynomials 3. Multiplying Polynomials 4. Special Products of Binomials 5. Factoring Polynomials 6. Factoring

### 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..........

### Some examples of radical equations are. Unfortunately, the reverse implication does not hold for even numbers nn. We cannot

40 RD.5 Radical Equations In this section, we discuss techniques for solving radical equations. These are equations containing at least one radical expression with a variable, such as xx 2 = xx, or a variable

### Chapter 5: Exponents and Polynomials

Chapter 5: Exponents and Polynomials 5.1 Multiplication with Exponents and Scientific Notation 5.2 Division with Exponents 5.3 Operations with Monomials 5.4 Addition and Subtraction of Polynomials 5.5

### Review of Operations on the Set of Real Numbers

1 Review of Operations on the Set of Real Numbers Before we start our journey through algebra, let us review the structure of the real number system, properties of four operations, order of operations,

### Basic ALGEBRA 2 SUMMER PACKET

Name Basic ALGEBRA SUMMER PACKET This packet contains Algebra I topics that you have learned before and should be familiar with coming into Algebra II. We will use these concepts on a regular basis throughout

### 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

### Lesson 18: Recognizing Equations of Circles

Student Outcomes Students complete the square in order to write the equation of a circle in center-radius form. Students recognize when a quadratic in xx and yy is the equation for a circle. Lesson Notes

### Chapter One: Pre-Geometry

Chapter One: Pre-Geometry Index: A: Solving Equations B: Factoring (GCF/DOTS) C: Factoring (Case Two leading into Case One) D: Factoring (Case One) E: Solving Quadratics F: Parallel and Perpendicular Lines

### MULTIPLYING TRINOMIALS

Name: Date: 1 Math 2 Variable Manipulation Part 4 Polynomials B MULTIPLYING TRINOMIALS Multiplying trinomials is the same process as multiplying binomials except for there are more terms to multiply than

### We say that a polynomial is in the standard form if it is written in the order of decreasing exponents of x. Operations on polynomials:

R.4 Polynomials in one variable A monomial: an algebraic expression of the form ax n, where a is a real number, x is a variable and n is a nonnegative integer. : x,, 7 A binomial is the sum (or difference)

### Review for Mastery. Integer Exponents. Zero Exponents Negative Exponents Negative Exponents in the Denominator. Definition.

LESSON 6- Review for Mastery Integer Exponents Remember that means 8. The base is, the exponent is positive. Exponents can also be 0 or negative. Zero Exponents Negative Exponents Negative Exponents in

### 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

### How to write polynomials in standard form How to add, subtract, and multiply polynomials How to use special products to multiply polynomials

PRC Ch P_3.notebook How to write polynomials in standard form How to add, subtract, and multiply polynomials How to use special products to multiply polynomials How to remove common factors from polynomials

### Maintaining Mathematical Proficiency

Chapter 7 Maintaining Mathematical Proficiency Simplify the expression. 1. 5x 6 + 3x. 3t + 7 3t 4 3. 8s 4 + 4s 6 5s 4. 9m + 3 + m 3 + 5m 5. 4 3p 7 3p 4 1 z 1 + 4 6. ( ) 7. 6( x + ) 4 8. 3( h + 4) 3( h

### Student Instruction Sheet: Unit 1 Lesson 3. Polynomials

Student Instruction Sheet: Unit 1 Lesson 3 Suggested time: 150 min Polynomials What s important in this lesson: You will use algebra tiles to learn how to add/subtract polynomials. Problems are provided

### Mathwithsheppard.weebly.com

Unit #: Powers and Polynomials Unit Outline: Date Lesson Title Assignment Completed.1 Introduction to Algebra. Discovering the Exponent Laws Part 1. Discovering the Exponent Laws Part. Multiplying and

### Algebra I Unit Report Summary

Algebra I Unit Report Summary No. Objective Code NCTM Standards Objective Title Real Numbers and Variables Unit - ( Ascend Default unit) 1. A01_01_01 H-A-B.1 Word Phrases As Algebraic Expressions 2. A01_01_02

### 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

### MathB65 Ch 4 VII, VIII, IX.notebook. November 06, 2017

Chapter 4: Polynomials I. Exponents & Their Properties II. Negative Exponents III. Scientific Notation IV. Polynomials V. Addition & Subtraction of Polynomials VI. Multiplication of Polynomials VII. Greatest

### Chapter y. 8. n cd (x y) 14. (2a b) 15. (a) 3(x 2y) = 3x 3(2y) = 3x 6y. 16. (a)

Chapter 6 Chapter 6 opener A. B. C. D. 6 E. 5 F. 8 G. H. I. J.. 7. 8 5. 6 6. 7. y 8. n 9. w z. 5cd.. xy z 5r s t. (x y). (a b) 5. (a) (x y) = x (y) = x 6y x 6y = x (y) = (x y) 6. (a) a (5 a+ b) = a (5

### I CAN classify polynomials by degree and by the number of terms.

13-1 Polynomials I CAN classify polynomials by degree and by the number of terms. 13-1 Polynomials Insert Lesson Title Here Vocabulary monomial polynomial binomial trinomial degree of a polynomial 13-1

### Polynomials and Polynomial Functions

1 Polynomials and Polynomial Functions One of the simplest types of algebraic expressions are polynomials. They are formed only by addition and multiplication of variables and constants. Since both addition

### 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

### Worksheets for GCSE Mathematics. Algebraic Expressions. Mr Black 's Maths Resources for Teachers GCSE 1-9. Algebra

Worksheets for GCSE Mathematics Algebraic Expressions Mr Black 's Maths Resources for Teachers GCSE 1-9 Algebra Algebraic Expressions Worksheets Contents Differentiated Independent Learning Worksheets

### Factoring Polynomials. Review and extend factoring skills. LEARN ABOUT the Math. Mai claims that, for any natural number n, the function

Factoring Polynomials GOAL Review and extend factoring skills. LEARN ABOUT the Math Mai claims that, for any natural number n, the function f (n) 5 n 3 1 3n 2 1 2n 1 6 always generates values that are

### Review Unit Multiple Choice Identify the choice that best completes the statement or answers the question.

Review Unit 3 1201 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which of the following numbers is not both a perfect square and a perfect cube? a. 531

### Q.3 Properties and Graphs of Quadratic Functions

374 Q.3 Properties and Graphs of Quadratic Functions In this section, we explore an alternative way of graphing quadratic functions. It turns out that if a quadratic function is given in form, ff() = aa(

### { independent variable some property or restriction about independent variable } where the vertical line is read such that.

Page 1 of 5 Introduction to Review Materials One key to Algebra success is identifying the type of work necessary to answer a specific question. First you need to identify whether you are dealing with

### Polynomials and Polynomial Functions

Unit 5: Polynomials and Polynomial Functions Evaluating Polynomial Functions Objectives: SWBAT identify polynomial functions SWBAT evaluate polynomial functions. SWBAT find the end behaviors of polynomial

### Math 3 Variable Manipulation Part 3 Polynomials A

Math 3 Variable Manipulation Part 3 Polynomials A 1 MATH 1 & 2 REVIEW: VOCABULARY Constant: A term that does not have a variable is called a constant. Example: the number 5 is a constant because it does

### MathB65 Ch 4 IV, V, VI.notebook. October 31, 2017

Part 4: Polynomials I. Exponents & Their Properties II. Negative Exponents III. Scientific Notation IV. Polynomials V. Addition & Subtraction of Polynomials VI. Multiplication of Polynomials VII. Greatest

### Lesson 3 Algebraic expression: - the result obtained by applying operations (+, -,, ) to a collection of numbers and/or variables o

Lesson 3 Algebraic expression: - the result obtained by applying operations (+, -,, ) to a collection of numbers and/or variables o o ( 1)(9) 3 ( 1) 3 9 1 Evaluate the second expression at the left, if

### Just DOS Difference of Perfect Squares. Now the directions say solve or find the real number solutions :

5.4 FACTORING AND SOLVING POLYNOMIAL EQUATIONS To help you with #1-1 THESE BINOMIALS ARE EITHER GCF, DOS, OR BOTH!!!! Just GCF Just DOS Difference of Perfect Squares Both 1. Break each piece down.. Pull

### THE DISTRIBUTIVE LAW. Note: To avoid mistakes, include arrows above or below the terms that are being multiplied.

THE DISTRIBUTIVE LAW ( ) When an equation of the form a b c is epanded, every term inside the bracket is multiplied by the number or pronumeral (letter), and the sign that is located outside the brackets.

### Quadratic Functions. Key Terms. Slide 1 / 200. Slide 2 / 200. Slide 3 / 200. Table of Contents

Slide 1 / 200 Quadratic Functions Table of Contents Key Terms Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic Equations

### Quadratic Functions. Key Terms. Slide 2 / 200. Slide 1 / 200. Slide 3 / 200. Slide 4 / 200. Slide 6 / 200. Slide 5 / 200.

Slide 1 / 200 Quadratic Functions Slide 2 / 200 Table of Contents Key Terms Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic

### Slide 1 / 200. Quadratic Functions

Slide 1 / 200 Quadratic Functions Key Terms Slide 2 / 200 Table of Contents Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic

### Chapter Six. Polynomials. Properties of Exponents Algebraic Expressions Addition, Subtraction, and Multiplication Factoring Solving by Factoring

Chapter Six Polynomials Properties of Exponents Algebraic Expressions Addition, Subtraction, and Multiplication Factoring Solving by Factoring Properties of Exponents The properties below form the basis

### Collecting Like Terms

MPM1D Unit 2: Algebra Lesson 5 Learning goal: how to simplify algebraic expressions by collecting like terms. Date: Collecting Like Terms WARM-UP Example 1: Simplify each expression using exponent laws.

### Algebra 1 FINAL EXAM REVIEW Spring Semester Material (by chapter)

Name: Per.: Date: Algebra 1 FINAL EXAM REVIEW Spring Semester Material (by chapter) Your Algebra 1 Final will be on at. You will need to bring your tetbook and number 2 pencils with you to the final eam.

### Factors of Polynomials Factoring For Experts

Factors of Polynomials SUGGESTED LEARNING STRATEGIES: Shared Reading, Activating Prior Knowledge, Discussion Group, Note-taking When you factor a polynomial, you rewrite the original polynomial as a product

### Section 5: Quadratic Equations and Functions Part 1

Section 5: Quadratic Equations and Functions Part 1 Topic 1: Real-World Examples of Quadratic Functions... 121 Topic 2: Factoring Quadratic Expressions... 125 Topic 3: Solving Quadratic Equations by Factoring...

### 5.1 Modelling Polynomials

5.1 Modelling Polynomials FOCUS Model, write, and classify polynomials. In arithmetic, we use Base Ten Blocks to model whole numbers. How would you model the number 234? In algebra, we use algebra tiles

### LESSON 9.1 ROOTS AND RADICALS

LESSON 9.1 ROOTS AND RADICALS LESSON 9.1 ROOTS AND RADICALS 67 OVERVIEW Here s what you ll learn in this lesson: Square Roots and Cube Roots a. Definition of square root and cube root b. Radicand, radical

### Section 2.4: Add and Subtract Rational Expressions

CHAPTER Section.: Add and Subtract Rational Expressions Section.: Add and Subtract Rational Expressions Objective: Add and subtract rational expressions with like and different denominators. You will recall

### Chapter 3: Section 3.1: Factors & Multiples of Whole Numbers

Chapter 3: Section 3.1: Factors & Multiples of Whole Numbers Prime Factor: a prime number that is a factor of a number. The first 15 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43,

### UNIT 2 FACTORING. M2 Ch 11 all

UNIT 2 FACTORING M2 Ch 11 all 2.1 Polynomials Objective I will be able to put polynomials in standard form and identify their degree and type. I will be able to add and subtract polynomials. Vocabulary

### Adding and Subtracting Polynomials

Adding and Subtracting Polynomials When you add polynomials, simply combine all like terms. When subtracting polynomials, do not forget to use parentheses when needed! Recall the distributive property:

### TECHNIQUES IN FACTORISATION

TECHNIQUES IN FACTORISATION The process where brackets are inserted into an equation is referred to as factorisation. Factorisation is the opposite process to epansion. METHOD: Epansion ( + )( 5) 15 Factorisation

### 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

### Spring 2018 Math Week Week 1 Task List

Spring 2018 Math 143 - Week 1 25 Week 1 Task List This week we will cover Sections 1.1 1.4 in your e-book. Work through each of the following tasks, carefully filling in the following pages in your notebook.

### Unit # 4 : Polynomials

Name: Block: Teacher: Miss Zukowski Date Submitted: / / 2018 Unit # 4 : Polynomials Submission Checklist: (make sure you have included all components for full marks) Cover page & Assignment Log Class Notes

### Algebra 2 Segment 1 Lesson Summary Notes

Algebra 2 Segment 1 Lesson Summary Notes For each lesson: Read through the LESSON SUMMARY which is located. Read and work through every page in the LESSON. Try each PRACTICE problem and write down the

### 27 Wyner Math 2 Spring 2019

27 Wyner Math 2 Spring 2019 CHAPTER SIX: POLYNOMIALS Review January 25 Test February 8 Thorough understanding and fluency of the concepts and methods in this chapter is a cornerstone to success in the

### When you square a binomial, you can apply the FOIL method to find the product. You can also apply the following rules as a short cut.

Squaring a Binomial When you square a binomial, you can apply the FOIL method to find the product. You can also apply the following rules as a short cut. Solve. (x 3) 2 Step 1 Square the first term. Rules

### 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

### Warm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2

6-5 Warm Up Lesson Presentation Lesson Quiz 2 Warm Up Factor completely. 1. 2y 3 + 4y 2 30y 2y(y 3)(y + 5) 2. 3x 4 6x 2 24 Solve each equation. 3(x 2)(x + 2)(x 2 + 2) 3. x 2 9 = 0 x = 3, 3 4. x 3 + 3x

### 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

### 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

### 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: