m81347 Introduction to Factoring Polynomials By the end of this section, you will be able to: Find the greatest common factor of two or more expressions Factor the greatest common factor from a polynomial 29daf534-a9bb-4e3e-90d2-065c7cdf9027 Before you get started, take this readiness quiz. Factor 56 into primes. If you missed this problem, review . 2⋅2⋅2⋅7 Multiply: −3(6a+11). If you missed this problem, review . −18a−33 Multiply: 4x2(x2+3x−1). If you missed this problem, review . 4x4+12x3−4x2

Earlier we multiplied factors together to get a product. Now, we will be reversing this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring. In The Language of Algebra we factored numbers to find the least common multiple (LCM) of two or more numbers. Now we will factor expressions and find the greatest common factor of two or more expressions. The method we use is similar to what we used to find the LCM. The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions. First we will find the greatest common factor of two numbers. Find the greatest common factor of 24 and 36. Step 1: Factor each coefficient into primes. Write all variables with exponents in expanded form. Factor 24 and 36. Step 2: List all factors--matching common factors in a column. In each column, circle the common factors. Circle the 2, 2, and 3 that are shared by both numbers. Step 3: Bring down the common factors that all expressions share. Bring down the 2, 2, 3 and then multiply. Step 4: Multiply the factors. The GCF of 24 and 36 is 12. Notice that since the GCF is a factor of both numbers, 24 and 36 can be written as multiples of 12. 24=12·236=12·3 Find the greatest common factor: 54,36. 18 Find the greatest common factor: 48,80. 16 In the previous example, we found the greatest common factor of constants. The greatest common factor of an algebraic expression can contain variables raised to powers along with coefficients. We summarize the steps we use to find the greatest common factor. Factor each coefficient into primes. Write all variables with exponents in expanded form. List all factors—matching common factors in a column. In each column, circle the common factors. Bring down the common factors that all expressions share. Multiply the factors. Find the greatest common factor of 5xand15. Factor each number into primes. Circle the common factors in each column. Bring down the common factors. The GCF of 5x and 15 is 5. Find the greatest common factor: 7y,14. 7 Find the greatest common factor: 22,11m. 11 In the examples so far, the greatest common factor was a constant. In the next two examples we will get variables in the greatest common factor. Find the greatest common factor of 12x2 and 18x3. Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column. Bring down the common factors. Multiply the factors. The GCF of12x2and18x3is6x2 Find the greatest common factor: 16x2,24x3. 8x² Find the greatest common factor: 27y3,18y4. 9y³ Find the greatest common factor of 14x3,8x2,10x. Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column. Bring down the common factors. Multiply the factors. The GCF of14x3and8x2, and10xis2x Find the greatest common factor: 21x3,9x2,15x. 3x Find the greatest common factor: 25m4,35m3,20m2. 5m²

Just like in arithmetic, where it is sometimes useful to represent a number in factored form (for example, 12 as 2·6or3·4), in algebra it can be useful to represent a polynomial in factored form. One way to do this is by finding the greatest common factor of all the terms. Remember that you can multiply a polynomial by a monomial as follows: 2(x + 7)factors 2·x + 2·7 2x + 14product Here, we will start with a product, like 2x+14, and end with its factors, 2(x+7). To do this we apply the Distributive Property “in reverse”. If a,b,c are real numbers, then a(b+c)=ab+acandab+ac=a(b+c) The form on the left is used to multiply. The form on the right is used to factor. So how do we use the Distributive Property to factor a polynomial? We find the GCF of all the terms and write the polynomial as a product! Factor: 2x+14. Step 1: Find the GCF of all the terms of the polynomial. Find the GCF of 2x and 14. Step 2: Rewrite each term as a product using the GCF. Rewrite 2x and 14 as products of their GCF, 2.2x=2⋅x14=2⋅7 Step 3: Use the Distributive Property 'in reverse' to factor the expression. 2(x+7) Step 4: Check by multiplying the factors. Check: Factor: 4x+12. 4(x + 3) Factor: 6a+24. 6(a + 4) Notice that in , we used the word factor as both a noun and a verb: Noun7is a factor of14Verbfactor2from2x+14 Find the GCF of all the terms of the polynomial. Rewrite each term as a product using the GCF. Use the Distributive Property ‘in reverse’ to factor the expression. Check by multiplying the factors. Factor: 3a+3. Rewrite each term as a product using the GCF. Use the Distributive Property 'in reverse' to factor the GCF. Check by multiplying the factors to get the original polynomial. Factor: 9a+9. 9(a + 1) Factor: 11x+11. 11(x + 1) The expressions in the next example have several factors in common. Remember to write the GCF as the product of all the common factors. Factor: 12x−60. Rewrite each term as a product using the GCF. Factor the GCF. Check by multiplying the factors. Factor: 11x−44. 11(x − 4) Factor: 13y−52. 13(y − 4) Now we’ll factor the greatest common factor from a trinomial. We start by finding the GCF of all three terms. Factor: 3y2+6y+9. Rewrite each term as a product using the GCF. Factor the GCF. Check by multiplying. Factor: 4y2+8y+12. 4(y² + 2y + 3) Factor: 6x2+42x−12. 6(x² + 7x − 2) In the next example, we factor a variable from a binomial. Factor: 6x2+5x. 6x2+5x Find the GCF of 6x2 and 5x and the math that goes with it. Rewrite each term as a product. Factor the GCF. x(6x+5) Check by multiplying. x(6x+5)x⋅6x+x⋅56x2+5x✓ Factor: 9x2+7x. x(9x + 7) Factor: 5a2−12a. a(5a − 12) When there are several common factors, as we’ll see in the next two examples, good organization and neat work helps! Factor: 4x3−20x2. Rewrite each term. Factor the GCF. Check. Factor: 2x3+12x2. 2x²(x + 6) Factor: 6y3−15y2. 3y²(2y − 5) Factor: 21y2+35y. Find the GCF of 21y2 and 35y Rewrite each term. Factor the GCF. Factor: 18y2+63y. 9y(2y + 7) Factor: 32k2+56k. 8k(4k + 7) Factor: 14x3+8x2−10x. Previously, we found the GCF of 14x3,8x2,and10x to be 2x. 14x3+8x2−10x Rewrite each term using the GCF, 2x. Factor the GCF. 2x(7x2+4x−5) Factor: 18y3−6y2−24y. 6y(3y² − y − 4) Factor: 16x3+8x2−12x. 4x(4x² + 2x − 3) When the leading coefficient, the coefficient of the first term, is negative, we factor the negative out as part of the GCF. Factor: −9y−27. When the leading coefficient is negative, the GCF will be negative. Ignoring the signs of the terms, we first find the GCF of 9y and 27 is 9. Since the expression −9y−27 has a negative leading coefficient, we use −9 as the GCF. −9y − 27 Rewrite each term using the GCF. Factor the GCF. −9(y+3) Factor: −5y−35. −5(y + 7) Factor: −16z−56. −8(2z + 7) Pay close attention to the signs of the terms in the next example. Factor: −4a2+16a. The leading coefficient is negative, so the GCF will be negative. Since the leading coefficient is negative, the GCF is negative, −4a. −4a2+16a Rewrite each term. Factor the GCF. −4a(a−4) Check on your own by multiplying. Factor: −7a2+21a. −7a(a − 3) Factor: −6x2+x. −x(6x − 1) Factor GCF Factor a Binomial Identify GCF

Find the greatest common factor. Factor each coefficient into primes. Write all variables with exponents in expanded form. List all factors—matching common factors in a column. In each column, circle the common factors. Bring down the common factors that all expressions share. Multiply the factors. Distributive Property If a, b, c are real numbers, then a(b+c)=ab+ac and ab+ac=a(b+c) Factor the greatest common factor from a polynomial. Find the GCF of all the terms of the polynomial. Rewrite each term as a product using the GCF. Use the Distributive Property ‘in reverse’ to factor the expression. Check by multiplying the factors.

Find the Greatest Common Factor of Two or More Expressions In the following exercises, find the greatest common factor. 40,56 45,75 15 72,162 150,275 25 3x,12 4y,28 4 10a,50 5b,30 5 16y,24y2 9x,15x2 3x 18m3,36m2 12p4,48p3 12p³ 10x,25x2,15x3 18a,6a2,22a3 2a 24u,6u2,30u3 40y,10y2,90y3 10y 15a4,9a5,21a6 35x3,10x4,5x5 5x³ 27y2,45y3,9y4 14b2,35b3,63b4 7b² Factor the Greatest Common Factor from a Polynomial In the following exercises, factor the greatest common factor from each polynomial. 2x+8 5y+15 5(y + 3) 3a−24 4b−20 4(b − 5) 9y−9 7x−7 7(x − 1) 5m2+20m+35 3n2+21n+12 3(n² + 7n + 4) 8p2+32p+48 6q2+30q+42 6(q² + 5q + 7) 8q2+15q 9c2+22c c(9c + 22) 13k2+5k 17x2+7x x(17x + 7) 5c2+9c 4q2+7q q(4q + 7) 5p2+25p 3r2+27r 3r(r + 9) 24q2−12q 30u2−10u 10u(3u − 1) yz+4z ab+8b b(a + 8) 60x−6x3 55y−11y4 11y(5 − y³) 48r4−12r3 45c3−15c2 15c²(3c − 1) 4a3−4ab2 6c3−6cd2 6c(c² − d²) 30u3+80u2 48x3+72x2 24x²(2x + 3) 120y6+48y4 144a6+90a3 18a³(8a³ + 5) 4q2+24q+28 10y2+50y+40 10(y² + 5y + 4) 15z2−30z−90 12u2−36u−108 12(u² − 3u − 9) 3a4−24a3+18a2 5p4−20p3−15p2 5p²(p² − 4p − 3) 11x6+44x5−121x4 8c5+40c4−56c3 8c³(c² + 5c − 7) −3n−24 −7p−84 −7(p + 12) −15a2−40a −18b2−66b −6b(3b + 11) −10y3+60y2 −8a3+32a2 −8a²(a − 4) −4u5+56u3 −9b5+63b3 −9b³(b² − 7)

Revenue A manufacturer of microwave ovens has found that the revenue received from selling microwaves a cost of p dollars each is given by the polynomial −5p2+150p. Factor the greatest common factor from this polynomial. Height of a baseball The height of a baseball hit with velocity 80 feet/second at 4 feet above ground level is −16t2+80t+4, with t= the number of seconds since it was hit. Factor the greatest common factor from this polynomial. −4(4t² − 20t − 1)

The greatest common factor of 36 and 60 is 12. Explain what this means. What is the GCF of y4, y5, and y10? Write a general rule that tells how to find the GCF of ya, yb, and yc. Answers will vary.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

Identify Polynomials, Monomials, Binomials and Trinomials In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial. y2+8y−20 trinomial −6a4 9x3−1 binomial n3−3n2+3n−1 Determine the Degree of Polynomials In the following exercises, determine the degree of each polynomial. 16x2−40x−25 2 5m+9 −15 0 y2+6y3+9y4 Add and Subtract Monomials In the following exercises, add or subtract the monomials. 4p+11p 15p −8y3−5y3 Add 4n5,−n5,−6n5 −3n⁵ Subtract 10x2 from 3x2 Add and Subtract Polynomials In the following exercises, add or subtract the polynomials. (4a2+9a−11)+(6a2−5a+10) 10a² + 4a − 1 (8m2+12m−5)−(2m2−7m−1) (y2−3y+12)+(5y2−9) 6y² − 3y + 3 (5u2+8u)−(4u−7) Find the sum of 8q3−27 and q2+6q−2 8q³ + q² + 6q − 29 Find the difference of x2+6x+8 and x2−8x+15 Evaluate a Polynomial for a Given Value of the Variable In the following exercises, evaluate each polynomial for the given value. 200x−15x2 when x=5 995 200x−15x2 when x=0 200x−15x2 when x=15 2,955 5+40x−12x2 when x=10 5+40x−12x2 when x=−4 −163 5+40x−12x2 when x=0 A pair of glasses is dropped off a bridge 640 feet above a river. The polynomial −16t2+640 gives the height of the glasses t seconds after they were dropped. Find the height of the glasses when t=6. 64 feet The fuel efficiency (in miles per gallon) of a bus going at a speed of x miles per hour is given by the polynomial −1160x2+12x. Find the fuel efficiency when x=20 mph.

Simplify Expressions with Exponents In the following exercises, simplify. 63 216 (12)4 (−0.5)2 0.25 −32 Simplify Expressions Using the Product Property of Exponents In the following exercises, simplify each expression. p3·p10 p¹³ 2·26 a·a2·a3 a⁶ x·x8 Simplify Expressions Using the Power Property of Exponents In the following exercises, simplify each expression. (y4)3 y¹² (r3)2 (32)5 3¹⁰ (a10)y Simplify Expressions Using the Product to a Power Property In the following exercises, simplify each expression. (8n)2 64n² (−5x)3 (2ab)8 256a⁸b⁸ (−10mnp)4 Simplify Expressions by Applying Several Properties In the following exercises, simplify each expression. (3a5)3 27a¹⁵ (4y)2(8y) (x3)5(x2)3 x²¹ (5st2)3(2s3t4)2 Multiply Monomials In the following exercises, multiply the monomials. (−6p4)(9p) −54p⁵ (13c2)(30c8) (8x2y5)(7xy6) 56x³y¹¹ (23m3n6)(16m4n4)

Multiply a Polynomial by a Monomial In the following exercises, multiply. 7(10−x) 70 − 7x a2(a2−9a−36) −5y(125y3−1) −625y⁴ + 5y (4n−5)(2n3) Multiply a Binomial by a Binomial In the following exercises, multiply the binomials using various methods. (a+5)(a+2) a² + 7a + 10 (y−4)(y+12) (3x+1)(2x−7) 6x² − 19x − 7 (6p−11)(3p−10) (n+8)(n+1) n² + 9n + 8 (k+6)(k−9) (5u−3)(u+8) 5u² + 37u − 24 (2y−9)(5y−7) (p+4)(p+7) p² + 11p + 28 (x−8)(x+9) (3c+1)(9c−4) 27c² − 3c − 4 (10a−1)(3a−3) Multiply a Trinomial by a Binomial In the following exercises, multiply using any method. (x+1)(x2−3x−21) x³ − 2x² − 24x − 21 (5b−2)(3b2+b−9) (m+6)(m2−7m−30) m³ − m² − 72m − 180 (4y−1)(6y2−12y+5)

Simplify Expressions Using the Quotient Property of Exponents In the following exercises, simplify. 2822 2⁶ or 64 a6a n3n12 1n9 xx5 Simplify Expressions with Zero Exponents In the following exercises, simplify. 30 1 y0 (14t)0 1 12a0−15b0 Simplify Expressions Using the Quotient to a Power Property In the following exercises, simplify. (35)2 925 (x2)5 (5mn)3 125m3n3 (s10t)2 Simplify Expressions by Applying Several Properties In the following exercises, simplify. (a3)2a4 a² u3u2·u4 (xx9)5 1x40 (p4·p5p3)2 (n5)3(n2)8 1n (5s24t)3 Divide Monomials In the following exercises, divide the monomials. 72p12÷8p3 9p⁹ −26a8÷(2a2) 45y6−15y10 −3y4 −30x8−36x9 28a9b7a4b3 4a5b2 11u6v355u2v8 (5m9n3)(8m3n2)(10mn4)(m2n5) 4m9n4 42r2s46rs3−54rs29s

Use the Definition of a Negative Exponent In the following exercises, simplify. 6−2 136 (−10)−3 5·2−4 516 (8n)−1 Simplify Expressions with Integer Exponents In the following exercises, simplify. x−3·x9 x⁶ r−5·r−4 (uv−3)(u−4v−2) 1u3v5 (m5)−1 (k−2)−3 k⁶ q4q20 b8b−2 b¹⁰ n−3n−5 Convert from Decimal Notation to Scientific Notation In the following exercises, write each number in scientific notation. 5,300,000 5.3 × 10⁶ 0.00814 The thickness of a piece of paper is about 0.097 millimeter. 9.7 × 10⁻² millimeter According to www.cleanair.com, U.S. businesses use about 21,000,000 tons of paper per year. Convert Scientific Notation to Decimal Form In the following exercises, convert each number to decimal form. 2.9×104 29,000 1.5×108 3.75×10−1 0.375 9.413×10−5 Multiply and Divide Using Scientific Notation In the following exercises, multiply and write your answer in decimal form. (3×107)(2×10−4) 6,000 (1.5×10−3)(4.8×10−1) 6×1092×10−1 30,000,000,000 9×10−31×10−6

Find the Greatest Common Factor of Two or More Expressions In the following exercises, find the greatest common factor. 5n,45 5 8a,72 12x2,20x3,36x4 4x² 9y4,21y5,15y6 Factor the Greatest Common Factor from a Polynomial In the following exercises, factor the greatest common factor from each polynomial. 16u−24 8(2u − 3) 15r+35 6p2+6p 6p(p + 1) 10c2−10c −9a5−9a3 −9a³(a² + 1) −7x8−28x3 5y2−55y+45 5(y² − 11y + 9) 2q5−16q3+30q2

For the polynomial 8y4−3y2+1 ⓐ Is it a monomial, binomial, or trinomial? ⓑ What is its degree? ⓐ trinomialⓑ 4 In the following exercises, simplify each expression. (5a2+2a−12)+(9a2+8a−4) (10x2−3x+5)−(4x2−6) 6x² − 3x + 11 (−34)3 n·n4 n⁵ (10p3q5)2 (8xy3)(−6x4y6) −48x⁵y⁹ 4u(u2−9u+1) (s+8)(s+9) s² + 17s + 72 (m+3)(7m−2) (11a−6)(5a−1) 55a² − 41a + 6 (n−8)(n2−4n+11) (4a+9b)(6a−5b) 24a² + 34ab − 45b² 5658 (x3·x9x5)2 x¹⁴ (47a18b23c5)0 24r3s6r2s7 4rs6 8y2−16y+204y (15xy3−35x2y)÷5xy 3y² − 7x 4−1 (2y)−3 18y3 p−3·p−8 x4x−5 x⁹ In the following exercises, factor the greatest common factor from each polynomial. 80a3+120a2+40a −6x2−30x −6x(x + 5) According to www.cleanair.org, the amount of trash generated in the US in one year averages out to 112,000 pounds of trash per person. Write this number in scientific notation. Convert 5.25×10−4 to decimal form. 0.000525 In the following exercises, simplify, and write your answer in decimal form. (2.4×108)(2×10−5) 9×1043×10−1 300,000 A hiker drops a pebble from a bridge 240 feet above a canyon. The polynomial −16t2+240 gives the height of the pebble t seconds a after it was dropped. Find the height when t=3.

greatest common factor The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.