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0. Review of Algebra

Factoring Polynomials

0. Review of Algebra

Factoring Polynomials

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Introduction to Factoring Polynomials

Introduction to Factoring Polynomials

Factor by Grouping

Factor by Grouping

Factor Using Special Product Formulas

Factor Using Special Product Formulas

Factor Using the AC Method When a Is 1

Factor Using the AC Method When a Is 1

Factor Using the AC Method When a Is Not 1

Factor Using the AC Method When a Is Not 1

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Whiteboard Math: Factoring by Grouping

Whiteboard Math: Factoring by Grouping

Whiteboard Math

How to factor a perfect square trinomial

How to factor a perfect square trinomial

Factoring Sum and Difference of Cubes

Factoring Sum and Difference of Cubes

Mario's Math Tutoring

Lesson: Factoring a Sum or Difference of Cubes

Lesson: Factoring a Sum or Difference of Cubes

Factoring a sum or difference of two cubes

Factoring a sum or difference of two cubes

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Practice this topic

Multiple Choice

Factor out the Greatest Common Factor in the polynomial.

$4x^2y-100y$

Multiple Choice

Factor out the Greatest Common Factor in the polynomial.

$-3x^4+12x^3-18x^2$

Multiple Choice

Factor the polynomial by grouping

$-x^2-5x+7x+35$

Multiple Choice

Factor the polynomial by grouping.

$6x^3-2x^2+3x-1$

Multiple Choice

Factor the polynomial using special product formulas.

$25x^2-110x+121$

Multiple Choice

Factor the polynomial using special product formulas.

$\frac{x^2}{49}-9$

Multiple Choice

Factor the polynomial.

$x^2-13x+40$

Multiple Choice

Factor the polynomial.

$x^2-2x-15$

Multiple Choice

Factor the polynomial.

$4x^2-19x+12$

Multiple Choice

Factor the polynomial.

$3x^2-2x-5$

Textbook Question

In Exercises 1–22, factor each difference of two squares. Assume that any variable exponents represent whole numbers. x² - 4

Textbook Question

In Exercises 1–10, factor out the greatest common factor. 18x+27

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. x³ − 16x

Textbook Question

In Exercises 1–10, factor out the greatest common factor. 3x^2+6x

Textbook Question

In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. x² + 8x + 12

Textbook Question

In Exercises 1–10, factor out the greatest common factor. 9x^4−18x^3+27x^2

Textbook Question

In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. x² + 9x + 20

Textbook Question

In Exercises 1–22, factor the greatest common factor from each polynomial. x³ + 5x²

Textbook Question

Work each problem. Match each polynomial in Column I with its factored form in Column II. a. x^2 + 10xy +25y^2 b. x^2-10xy +25y^2 c.x^2-25y^2 d. 25y^2-x^2 A. (x+5y ) (x-5y) B. (x+5y)^2 C. (x-5y)^2 D. (5y+x)(5y-x)

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. x²y − 16y + 32 − 2x²

Textbook Question

In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. y² + 10y + 16

Textbook Question

In Exercises 1–22, factor the greatest common factor from each polynomial. 12x⁴ − 8x²

Textbook Question

In Exercises 1–22, factor each difference of two squares. Assume that any variable exponents represent whole numbers. 36x² - 49y²

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. 4a²b − 2ab − 30b

Textbook Question

In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. x² − 8x + 15

Textbook Question

In Exercises 1–22, factor the greatest common factor from each polynomial. 32x⁴ + 2x³ + 8x²

Textbook Question

In Exercises 1–22, factor each difference of two squares. Assume that any variable exponents represent whole numbers. x²y² - 1

Textbook Question

Work each problem. Which of the following is the correct complete factorization of x^4-1? A. (x^2-1)(x^2+1) B.(x^2+1)(x+1)(x-1) C. (x^2-1)^2 D. (x-1)^2(x+1)^2

Textbook Question

Work each problem. Which of the following is the correct factorization of x^3+8? A. (x+2)^3 B. (x+2)(x^2+2x+4) C. (x+2)(x^2-2x+4) D. (x+2)(x^2-4x+4)

Textbook Question

In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. y² − 12y + 20

Textbook Question

In Exercises 1–22, factor the greatest common factor from each polynomial. 4x²y³ + 6xy

Textbook Question

In Exercises 1–22, factor each difference of two squares. Assume that any variable exponents represent whole numbers. 9x⁴ - 25y⁶

Textbook Question

Factor out the greatest common factor from each polynomial. See Example 1. 12m+60

Textbook Question

Express the distance between the numbers -17 and 4 using absolute value. Then evaluate the absolute value.

Textbook Question

In Exercises 11–16, factor by grouping. x^3−3x^2+4x−12

Textbook Question

In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. a² + 5a − 14

Textbook Question

In Exercises 1–22, factor the greatest common factor from each polynomial. 30x²y³ − 10xy²

Textbook Question

In Exercises 1–22, factor each difference of two squares. Assume that any variable exponents represent whole numbers. x¹⁴ - y⁴

Textbook Question

Factor out the greatest common factor from each polynomial. See Example 1. 8k^3+24k

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. 11x⁵ − 11xy²

Textbook Question

In Exercises 11–16, factor by grouping. x^3+6x^2−2x−12

Textbook Question

In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. x² + x − 30

Textbook Question

In Exercises 1–22, factor the greatest common factor from each polynomial. 12xy − 6xz + 4xw

Textbook Question

In Exercises 1–22, factor each difference of two squares. Assume that any variable exponents represent whole numbers. (x - 3)² - y²

Textbook Question

Factor out the greatest common factor from each polynomial. See Example 1. xy-5xy^2

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. 4x⁵ − 64x

Textbook Question

Factor out the greatest common factor from each polynomial. See Example 1. 5h^2j+hj

Textbook Question

In Exercises 16–17, factor completely. xy − 6x + 2y − 12

Textbook Question

Factor out the greatest common factor from each polynomial. See Example 1. -4p^3q^4-2p^2q^5

Textbook Question

In Exercises 1–22, factor each difference of two squares. Assume that any variable exponents represent whole numbers. a² - (b - 2)²

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. x³ − 4x² − 9x + 36

Textbook Question

In Exercises 16–17, factor completely. 24x³y + 16x²y − 30xy

Textbook Question

In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. x^2+8x+15

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. 2x⁵ + 54x²

Textbook Question

Simplify the algebraic expression. 5(2x - 3) + 7x

Textbook Question

Simplify the algebraic expression. (1/5)(5x) + [(3y) + (- 3y)] - (-x)

Textbook Question

In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. x² − x + 7

Textbook Question

In Exercises 1–22, factor the greatest common factor from each polynomial. 15x²ⁿ − 25xⁿ

Textbook Question

Factor out the greatest common factor from each polynomial. See Example 1. 2(a+b)+4m(a+b)

Textbook Question

In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. x^2−14x+45

Textbook Question

Factor out the greatest common factor from each polynomial. See Example 1. 6x(a+b)-4y(a+b)

Textbook Question

In Exercises 23–48, factor completely, or state that the polynomial is prime. 2x³ - 8x

Textbook Question

Factor out the greatest common factor from each polynomial. See Example 1. (5r-6)(r+3)-(2r-1)(r+3)

Textbook Question

In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. x² − 9xy + 14y²

Textbook Question

In Exercises 23–34, factor out the negative of the greatest common factor. −4x + 12

Textbook Question

Factor out the greatest common factor from each polynomial. See Example 1. (4z-5)(3z-2)-(3z-9)(3z-2)

Textbook Question

In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. 2x^2+5x−3

Textbook Question

In Exercises 23–34, determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial. 24. x² + 4x

Textbook Question

Factor out the greatest common factor from each polynomial. See Example 1. 2(m-1)-3(m-1)^2+2(m-1)^3

Textbook Question

In Exercises 23–48, factor completely, or state that the polynomial is prime. 50 - 2y²

Textbook Question

In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. x² − xy − 30y²

Textbook Question

In Exercises 23–34, factor out the negative of the greatest common factor. −8x − 48

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. x² − 12x + 36 − 49y²

Textbook Question

Factor out the greatest common factor from each polynomial. See Example 1. 5(a+3)^3-2(a+3)+(a+3)^2

Textbook Question

In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. 6x^2−11x+4

Textbook Question

In Exercises 23–48, factor completely, or state that the polynomial is prime. 8x² - 8y²

Textbook Question

In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. x² + xy + y²

Textbook Question

In Exercises 23–34, factor out the negative of the greatest common factor. −2x² + 6x − 14

Textbook Question

Concept Check When directed to completely factor the polynomial 4x^2y^5-8xy^3,a student wrote 2xy^3(2xy^2-4). When the teacher did not give him full credit, he complained because when his answer is multiplied out, the result is the original polynomial. Give the correct answer.

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. 4x² + 25y²

Textbook Question

In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. x² − xy + y²

Textbook Question

Concept Check Kurt factored 16a^2-40a-6a+15 by grouping and obtained (8a-3)(2a-5). Callie factored the same polynomial and gave an answer of (3-8a)(5-2a). Which answer is correct?

Textbook Question

In Exercises 23–34, factor out the negative of the greatest common factor. −5y² + 40x

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. 12x³y − 12xy³

Textbook Question

In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. a² − 18ab + 45b²

Textbook Question

Factor each polynomial by grouping. See Example 2. 10ab-6b+35a-21

Textbook Question

In Exercises 23–34, determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial. 30. x² - 1/3x

Textbook Question

In Exercises 23–34, factor out the negative of the greatest common factor. −4x³ + 32x² − 20x

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. 6bx² + 6by²

Textbook Question

Factor each polynomial by grouping. See Example 2. 4x^6+36-x^6y-9y

Textbook Question

In Exercises 23–34, determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial. 32. x²+4/5x

Textbook Question

In Exercises 23–34, factor out the negative of the greatest common factor. −x² − 7x + 5

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. x⁴ − xy³ + x³y − y⁴

Textbook Question

Factor each polynomial by grouping. See Example 2. 20z^2-8x+5pz^2-2px

Textbook Question

In Exercises 23–34, determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial. 34. x²-9/5x

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. x² − 4a² + 12x + 36

Textbook Question

In Exercises 23–48, factor completely, or state that the polynomial is prime. 8x² + 8y²

Textbook Question

Factor each trinomial, if possible. See Examples 3 and 4. 8h^2-2h-21

Textbook Question

In Exercises 31–38, factor completely. 4y³ + 12y² − 72y

Textbook Question

In Exercises 23–48, factor completely, or state that the polynomial is prime. x² + 25y²

Textbook Question

In Exercises 31–38, factor completely. 3x⁴ + 54x³ + 135x²

Textbook Question

In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. 6x^2−7xy−5y^2

Textbook Question

In Exercises 23–48, factor completely, or state that the polynomial is prime. x⁴ - 16

Textbook Question

In Exercises 35–44, factor the greatest common binomial factor from each polynomial. 3x(x+y) − (x+y)

Textbook Question

Factor each trinomial, if possible. See Examples 3 and 4. 9x^2+4x-2

Textbook Question

In Exercises 39–48, factor the difference of two squares. x^2−144

Textbook Question

In Exercises 35–44, factor the greatest common binomial factor from each polynomial. 4x²(3x−1) + 3x − 1

Textbook Question

A rigid bar (negligible mass) of length 80 cm connects a large sphere with a mass (m₁) of 25 g to a small sphere with an unknown mass (m₂). The large sphere is located at one end of the bar, with the center of mass of the bar located 22 cm away from this sphere. Determine the mass of the sphere (in grams) at the other end of the bar.

Textbook Question

Factor each trinomial, if possible. See Examples 3 and 4. 36x^3+18x^2-4x

Textbook Question

In Exercises 39–48, factor the difference of two squares. 64x^2−81

Textbook Question

In Exercises 39–44, factor by introducing an appropriate substitution. x⁴ − 4x² − 5

Textbook Question

In Exercises 35–44, factor the greatest common binomial factor from each polynomial. (x + 2)(x + 3) + (x − 1)(x + 3)

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. x² + 10x − y² + 25

Textbook Question

Factor each trinomial, if possible. See Examples 3 and 4. 14m^2+11mr-15r^2

Textbook Question

In Exercises 39–48, factor the difference of two squares. 36x^2−49y^2

Textbook Question

In Exercises 39–44, factor by introducing an appropriate substitution. (x + 1)² + 8(x + 1) + 7 (Let u = x+1.)

Textbook Question

Factor each trinomial, if possible. See Examples 3 and 4. 5a^2-7ab-6b^2

Textbook Question

In Exercises 45–68, factor by grouping. x² + 3x + 5x + 15

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. x⁸ − y⁸

Textbook Question

In Exercises 23–48, factor completely, or state that the polynomial is prime. x³ + 3x² - 4x - 12

Textbook Question

In Exercises 45–68, use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. 2x² + 9x + 7

Textbook Question

In Exercises 39–48, factor the difference of two squares. 16x^4−81

Textbook Question

In Exercises 45–68, factor by grouping. x² + 7x − 4x − 28

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. x³y − 16xy³

Textbook Question

In Exercises 23–48, factor completely, or state that the polynomial is prime. x³ - 7x² - x + 7

Textbook Question

Factor each trinomial, if possible. See Examples 3 and 4. 12x^2-xy-y^2

Textbook Question

In Exercises 45–68, use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. 5x² − 16x + 3

Textbook Question

In Exercises 49–56, factor each perfect square trinomial. x^2+2x+1

Textbook Question

In Exercises 45–68, factor by grouping. x³− 3x² + 4x − 12

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. x + 8x⁴

Textbook Question

In Exercises 49–64, factor any perfect square trinomials, or state that the polynomial is prime. x² + 4x + 4

Textbook Question

Factor each trinomial, if possible. See Examples 3 and 4. 24a^4+10a^3b-4a^2b^2

Textbook Question

In Exercises 45–68, use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. 5y² + 33y − 14

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. 16y² − 4y − 2

Textbook Question

In Exercises 49–64, factor any perfect square trinomials, or state that the polynomial is prime. x² − 10x + 25

Textbook Question

Factor each trinomial, if possible. See Examples 3 and 4. 9m^2 -12m+4

Textbook Question

In Exercises 45–68, use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. 8y² + 10y + 3

Textbook Question

In Exercises 49–64, factor any perfect square trinomials, or state that the polynomial is prime. x⁴ − 4x² + 4

Textbook Question

Factor each trinomial, if possible. See Examples 3 and 4. 32a^2+48ab+18b^2

Textbook Question

In Exercises 45–68, use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. 6x² + 19x + 15

Textbook Question

Factor each trinomial, if possible. See Examples 3 and 4. 4x^2y^2+28xy+49

Textbook Question

In Exercises 49–64, factor any perfect square trinomials, or state that the polynomial is prime. 9y² + 6y + 1

Textbook Question

In Exercises 45–68, factor by grouping. 10x² − 12xy + 35xy − 42y²

Textbook Question

Factor each trinomial, if possible. See Examples 3 and 4. 9m^2n^2+12mn+4

Textbook Question

In Exercises 45–68, use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. 4x² − 27x + 18

Textbook Question

In Exercises 49–56, factor each perfect square trinomial. 64x^2−16x+1

Textbook Question

Factor each trinomial, if possible. See Examples 3 and 4. (a-3b)^2-6(a-3b)+9

Textbook Question

In Exercises 49–64, factor any perfect square trinomials, or state that the polynomial is prime. 64y² − 16y + 1

Textbook Question

In Exercises 45–68, factor by grouping. 4x³ − x² + 12x - 3

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. 12x³ + 3xy²

Textbook Question

In Exercises 45–68, use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. 16y² − 6y − 27

Textbook Question

In Exercises 57–64, factor using the formula for the sum or difference of two cubes. x^3+64

Textbook Question

In Exercises 49–64, factor any perfect square trinomials, or state that the polynomial is prime. x² − 12xy + 36y²

Textbook Question

In Exercises 45–68, factor by grouping. x² − ax − bx + ab

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. x⁶y⁶ − x³y³

Textbook Question

Factor each polynomial. See Examples 5 and 6. 9a^2-16

Textbook Question

In Exercises 57–64, factor using the formula for the sum or difference of two cubes. x^3−27

Textbook Question

In Exercises 45–68, factor by grouping. x³ − 12 − 3x² + 4x

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. (x + 5)(x − 3) + (x + 5)(x − 7)

Textbook Question

Factor each polynomial. See Examples 5 and 6. x^4-16

Textbook Question

Factor each polynomial. See Examples 5 and 6. y^4-81

Textbook Question

In Exercises 49–64, factor any perfect square trinomials, or state that the polynomial is prime. 9x² + 48xy + 64y²

Textbook Question

In Exercises 45–68, factor by grouping. ay − by + bx − ax

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. a²(x − y) + 4(y − x)

Textbook Question

Factor each polynomial. See Examples 5 and 6. 25s^4-9t^2

Textbook Question

Factor each polynomial. See Examples 5 and 6. 36z^2-81y^4

Textbook Question

In Exercises 57–64, factor using the formula for the sum or difference of two cubes. 8x^3+125

Textbook Question

Factor each polynomial. See Examples 5 and 6. (a+b)^2-16

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. (c + d)⁴ − 1

Textbook Question

Factor each polynomial. See Examples 5 and 6. (p-2q)^2-100

Textbook Question

In Exercises 65–92, factor completely, or state that the polynomial is prime. 5x^3−45x

Textbook Question

In Exercises 45–68, use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. 15x² + 111xy − 14y²

Textbook Question

In Exercises 1–68, factor completely, or state that the polynomial is prime. p³ − pq² + p²q − q³

Textbook Question

In Exercises 45–68, use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. 15a² − ab − 6b²

Textbook Question

In Exercises 65–92, factor completely, or state that the polynomial is prime. 2x^4−162

Textbook Question

In Exercises 69–80, factor completely. x⁴ − 5x²y² + y⁴

Textbook Question

In Exercises 65–74, factor by grouping to obtain the difference of two squares. 9x^2 − 30x + 25 − 36x^4

Textbook Question

In Exercises 69–78, factor each polynomial. ab − c − ac + b

Textbook Question

In Exercises 69–80, factor completely. (x + y)² + 6(x + y) + 9

Textbook Question

In Exercises 65–74, factor by grouping to obtain the difference of two squares. x⁴ − x² − 2x − 1

Textbook Question

In Exercises 69–78, factor each polynomial. x³ − 5 + 4x³y − 20y

Textbook Question

Factor each polynomial. See Examples 5 and 6. y^2-x^2+12x-36

Textbook Question

Factor each polynomial. See Examples 5 and 6. 9m^2-n^2-2n-1

Textbook Question

In Exercises 69–80, factor completely. (x − y)⁴ − 4(x − y)²

Textbook Question

In Exercises 65–74, factor by grouping to obtain the difference of two squares. z² − x² + 4xy − 4y²

Textbook Question

In Exercises 69–78, factor each polynomial. 2y⁷(3x−1)⁵ − 7y⁶(3x−1)⁴

Textbook Question

Factor each polynomial. See Examples 5 and 6. 8-a^3

Textbook Question

Factor each polynomial. See Examples 5 and 6. 27-r^3

Textbook Question

In Exercises 69–82, factor completely. 10y⁵ – 17y⁴ + 3y³

Textbook Question

Factor each polynomial. See Examples 5 and 6. 125x^3-27

Textbook Question

In Exercises 75–94, factor using the formula for the sum or difference of two cubes. x³ + 64

Textbook Question

In Exercises 69–82, factor completely. 12x² + 10xy – 8y²

Textbook Question

In Exercises 65–92, factor completely, or state that the polynomial is prime. 9x^3−9x

Textbook Question

In Exercises 69–80, factor completely. x³ − y³ − x + y

Textbook Question

In Exercises 75–94, factor using the formula for the sum or difference of two cubes. x³ − 27

Textbook Question

Factor each polynomial. See Examples 5 and 6. 27z^9+64y^12

Textbook Question

In Exercises 69–82, factor completely. 8a²b + 34ab – 84b

Textbook Question

Factor: x² + 4x + 4 − 9y². (Section 5.6, Example 4)

Textbook Question

In Exercises 75–94, factor using the formula for the sum or difference of two cubes. 8y³ + 1

Textbook Question

In Exercises 65–92, factor completely, or state that the polynomial is prime. x^3+2x^2−4x−8

Textbook Question

Factor each polynomial. See Examples 5 and 6. (b+3)^3-27

Textbook Question

In Exercises 69–82, factor completely. 12x²y – 46xy² + 14y³

Textbook Question

In Exercises 69–82, factor completely. 4x³y⁵ + 24x²y⁵ – 64xy⁵

Textbook Question

Factor each polynomial. See Examples 5 and 6. 125-(4a-b)^3

Textbook Question

In Exercises 65–92, factor completely, or state that the polynomial is prime. 20y^4−45y^2

Textbook Question

In Exercises 83–92, factor by introducing an appropriate substitution. 5x⁴ + 2x² − 3

Textbook Question

Factor each polynomial. See Example 7. 6(4z-3)^2+7(4z-3)-3

Textbook Question

Factor each polynomial. See Example 7. 9(a-4)^2+30(a-4)+25

Textbook Question

In Exercises 65–92, factor completely, or state that the polynomial is prime. x^2−10x+25−36y^2

Textbook Question

Factor each polynomial. See Example 7. 4(5x+7)^2+12(5x+7)+9

Textbook Question

Factor each polynomial. See Example 7. (a+1)^3+27

Textbook Question

Factor each polynomial. See Example 7. (x-4)^3+64

Textbook Question

Factor each polynomial. See Example 7. (3x+4)^3-1

Textbook Question

In Exercises 65–92, factor completely, or state that the polynomial is prime. x^2 y−16y+32−2x^2

Textbook Question

Factor completely, or state that the polynomial is prime. 15x^3+3x^2

Textbook Question

Factor completely, or state that the polynomial is prime. x^2-11x+28

Textbook Question

In Exercises 83–92, factor by introducing an appropriate substitution. 3(x+1)² − 5(x+1) + 2 (Let u = x+1)

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In Exercises 65–92, factor completely, or state that the polynomial is prime. 2x^3−8a^2 x+24x^2+72x

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Factor each polynomial. See Example 7. m^4-3m^2-10

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Factor completely, or state that the polynomial is prime. 64-x^2

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In Exercises 83–92, factor by introducing an appropriate substitution. 3(x−2)² − 5(x−2) − 2

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Factor each polynomial. See Example 7. a^4-2a^2-48

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In Exercises 75–94, factor using the formula for the sum or difference of two cubes. (x − y)³ − y³

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In Exercises 93–100, factor completely. x² + 0.3x − 0.04

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Factor each polynomial. See Example 7. 10m^4+43m^2-9

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In Exercises 95–104, factor completely. 0.04x² + 0.12x + 0.09

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In Exercises 93–100, factor completely. x² − 6/25 + 1/5 x

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Factor by any method. See Examples 1–7. (2y-1)^2-4(2y-1)+4

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In Exercises 95–104, factor completely. 8x⁴ − x/8

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Factor by any method. See Examples 1–7. x^2+xy-5x-5y

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In Exercises 93–100, factor completely. acx² − bcx − adx + bd

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In Exercises 95–104, factor completely. x⁶ − 9x³ + 8

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Factor by any method. See Examples 1–7. p^4(m-2n)+q(m-2n)

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In Exercises 93–102, factor and simplify each algebraic expression. (x+5)^−1/2−(x+5)^−3/2

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In Exercises 93–100, factor completely. −5x⁴y³ + 7x³y⁴ − 2x²y⁵

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Factor by any method. See Examples 1–7. 4z^2+28z+49

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In Exercises 93–102, factor and simplify each algebraic expression. −8(4x+3)^−2+10(5x+1)(4x+3)^−1

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In Exercises 103–114, factor completely. 6x^4+35x^2−6

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Factor and simplify each algebraic expression. 16x^(-3/4)+32x(1/4)

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Factor completely: 50x³ − 18x. (Section 5.5, Example 2)

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Factor by any method. See Examples 1–7. q^2+6q+9-p^2

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Factor by any method. See Examples 1–7. 64+(3x+2)^3

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Factor and simplify each algebraic expression. [12x(-1/2)+6x^(-3/2)]

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In Exercises 103–114, factor completely. x^4−5x^2 y^2+4y^4

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Factor by any method. See Examples 1–7. (x+y)^3-(x-y)^3

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In Exercises 103–114, factor completely. (x+y)^4−100(x+y)^2

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Factor by any method. See Examples 1–7. (3a+5)^2-18(3a+5)+81

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Factor by any method. See Examples 1–7. (x+y)^2-(x-y)^2

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Factor by any method. See Examples 1–7. 4z^4-7z^2-15

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Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8. 4k^-1+k^-2

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Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8. 9z^-1/2+2z^1/2

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Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8. 3m^2/3-4m^-1/3

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Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8. 6r^-2/3-5r^-5/3

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Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8. -4a^-2/5+16a^-7/5

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Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8. (3r+1)^-2/3+(3r+1)^1/3+(3r+1)^4/3

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Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8. 2(3x+1)^-3/2+4(3x+1)^-1/2+6(3x+1)^1/2

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Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8. 7(5t+3)^-5/3+14(5t+3)^-2/3-21(5t+3)^1/3

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Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8. 6y^3(4y-1)^-3/7-8y^2(4y-1)^4/7+16y(4y-1)^11/7

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Factor each polynomial over the set of rational number coefficients. 49x^2-1/25

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In Exercises 133–136, factor each polynomial completely. Assume that any variable exponents represent whole numbers. 36x²ⁿ − y²ⁿ

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In Exercises 133–136, factor each polynomial completely. Assume that any variable exponents represent whole numbers. x³ⁿ + y¹²ⁿ

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Factor each polynomial over the set of rational number coefficients. (25/9)x^4-(9y^2)

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Find all values of b or c that will make the polynomial a perfect square trinomial. 4z^2+bz+81

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Find all values of b or c that will make the polynomial a perfect square trinomial. 100r^2-60r+c

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Find all values of b or c that will make the polynomial a perfect square trinomial. 49x^2+70x+c

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Exercises 143–145 will help you prepare for the material covered in the next section. In each exercise, factor completely. 9b²x + 9b²y − 16x − 16y

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Exercises 177–179 will help you prepare for the material covered in the next section. Factor completely: x^3 + x^2 - 4x - 4.

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Write each rational expression in lowest terms. 8x^2 + 16 / 4x^2

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Write each rational expression in lowest terms. 3(3 - t) / (t + 5)(t - 3)

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Write each rational expression in lowest terms. 8k + 16 / 9k + 18

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Write each rational expression in lowest terms. m^2 - 4m + 4 / m^2 + m - 6

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Write each rational expression in lowest terms. 8m^2 + 6m - 9 / 16m^2 - 9

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Write each rational expression in lowest terms. x^3 + 64 / x + 4

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Multiply or divide, as indicated. (2k + 8)/6 ÷ (3k + 12)/2

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Multiply or divide, as indicated. (y^3 + y^2)/7 * 49/(y^4 + y^3)

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Multiply or divide, as indicated. (m^2 + 3m + 2)/(m^2 + 5m + 4) ÷ (m^2 + 5m + 6)/(m^2 + 10m + 24)

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Multiply or divide, as indicated. (x^2 - y^2)/(x - y)^2 * (x^2 - xy + y^2)/(x^2 - 2xy + y^2) ÷ (x^3 + y^3)/(x - y)^4

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Multiply or divide, as indicated. (xz - xw + 2yz - 2yw)/(z^2 - w^2) * (4z + 4w + xz + wx)/(16 - x^2)

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Multiply or divide, as indicated. ac + ad + bc + bd/a^2 - b^2 * a^3 - b^3/2a^2 + 2ab + 2b^2

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Add or subtract, as indicated. 5/x + 2 + 2/x^2 - 2x + 4 - 60/x^3 + 8

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Add or subtract, as indicated. 4/x + 1 + 1/x^2 - x + 1 - 12/x^3 + 1

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Simplify each complex fraction. [1 + (1/x)] / (1-1/x)

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Simplify each complex fraction. (2 - 2/y) / (2 + 2/y)

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Simplify each complex fraction. [ 1/(x+1) - 1/x ] / (1/x)

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Simplify each complex fraction. [ 1/(y+3) - 1/y ] / (1/y)

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Simplify each complex fraction. [ 2 + 2/(1+x) ] / [ 2 - 2/(1-x) ]

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Simplify each complex fraction. [ 1/(a^3+b^3) ] / [ 1/(a^2 + 2ab + b^2) ]

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Simplify each complex fraction. [ 1/(x^3-y^3) ] / [ 1/(x^2 -y^2) ]

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Simplify each complex fraction. [ m - 1/(m^2-4) ] / [ 1/(m + 2) ]

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Simplify each complex fraction. [ y + 1/(y^2-9) ] / [ 1/(y + 3) ]

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Simplify each complex fraction. [ 3/(p^2-16) + p ] / [ 1/(p - 4) ]

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Simplify each complex fraction. [ 6/(x^2-25) + x ] / [ 1/(x - 5) ]

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Simplify each complex fraction. [ (-2)/(x+h) - (-2)/x ] / h

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Perform all indicated operations, and write each answer with positive integer exponents. [ (x^-2 + y^-2)/ (x^-2 - y^-2) ] * [ (x+y)/(x-y) ]

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Simplify each rational expression. Assume all variable expressions represent positive real numbers. (Hint: Use factoring and divide out any common factors as a first step.) [(y^2 +2)^5(3y) - y^3(6)(y^2+2)^4(3y)] / [(y^2+2)^7]

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Simplify each rational expression. Assume all variable expressions represent positive real numbers. (Hint: Use factoring and divide out any common factors as a first step.) [(x^2 +1)^4(2x) - x^2(4)(x^2+1)^3(2x)] / [(x^2+1)^8]

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Simplify each rational expression. Assume all variable expressions represent positive real numbers. (Hint: Use factoring and divide out any common factors as a first step.) [4(x^2- 1)^3 + 8x(x^2-1)^4] / [16(x^2-1)^3]

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Simplify each rational expression. Assume all variable expressions represent positive real numbers. (Hint: Use factoring and divide out any common factors as a first step.) [10(4x^2-9)^2 - 25x(4x^2-9)^3] / [15(4x^2-9)^6]

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In Exercises 91–100, find all values of x satisfying the given conditions. y = x^3 + 4x^2 - x + 6 and y = 10

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Solve each polynomial equation in Exercises 1–10 by factoring and then using the zero-product principle. 4x^3 - 12x^2 = 9x - 27

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Solve each polynomial equation in Exercises 1–10 by factoring and then using the zero-product principle. 4y^3 - 2 = y - 8y^2

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Solve each polynomial equation in Exercises 1–10 by factoring and then using the zero-product principle. 5x^4 - 20x^2 = 0

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Solve each equation in Exercises 96–102 by the method of your choice. x^3 + 2x^2 = 9x + 18

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