# Factoring Polynomials - Video Tutorials & Practice Problems

## Introduction to Factoring Polynomials

Factor out the Greatest Common Factor in the polynomial. $4x^2y-100y$

$y\left(4x^2-100\right)$

$4\left(x^2y-25y\right)$

$4y$($x^2-25$)

$4y\left(x^2-100\right)$

Factor out the Greatest Common Factor in the polynomial.

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

$3x\left(-x^2+4x-6\right)$

$3x^2\left(-x^2+4x-6\right)$

$3\left(-x^3+4x^2-6x\right)$

$3x^2\left(-3x^4+12x^3-18x^2\right)$

## Factor by Grouping

Factor the polynomial by grouping $-x^2-5x+7x+35$

$\left(-x+7\right)\left(x+5\right)$

$\left(x+7\right)\left(x+5\right)$

$\left(x-7\right)\left(x+5\right)$

$\left(x+7\right)\left(-x+5\right)$

Factor the polynomial by grouping. $6x^3-2x^2+3x-1$

$2x^2\left(3x-1\right)$

$\left(2x^2+x\right)\left(3x-1\right)$

$\left(2x+1\right)\left(3x^2-1\right)$

$\left(2x^2+1\right)\left(3x-1\right)$

## Factor Using Special Product Formulas

Factor the polynomial using special product formulas. $25x^2-110x+121$

$\left(5x-10\right)^2$

$\left(5x+11\right)\left(5x-11\right)$

$\left(5x+11\right)^2$

$\left(5x-11\right)^2$

Factor the polynomial using special product formulas. $\frac{x^2}{49}-9$

$\left(\frac{x}{7}-3\right)^2$

$\left(\frac{x}{7}+3\right)\left(\frac{x}{7}-3\right)$

$\left(7x-3\right)^2$

$\left(7x+3\right)\left(7x-3\right)$

## Factor Using the AC Method When a Is 1

Factor the polynomial. $x^2-13x+40$

$\left(x+5\right)\left(x+8\right)$

$\left(x-5\right)\left(x-8\right)$

$\left(x-4\right)\left(x-10\right)$

$\left(x+4\right)\left(x+10\right)$

Factor the polynomial. $x^2-2x-15$

$\left(x-3\right)\left(x-5\right)$

$\left(x-3\right)\left(x+5\right)$

$\left(x+3\right)\left(x-5\right)$

$\left(x+3\right)\left(x+5\right)$

## Factor Using the AC Method When a Is Not 1

Factor the polynomial. $4x^2-19x+12$

$\left(4x-3\right)\left(x-4\right)$

$\left(4x-6\right)\left(x-2\right)$

$\left(2x-3\right)\left(2x-4\right)$

$\left(2x-6\right)\left(2x-2\right)$

Factor the polynomial. $3x^2-2x-5$

$\left(x+3\right)\left(x-5\right)$

$\left(x+1\right)\left(x-5\right)$

$\left(3x+1\right)\left(x-5\right)$

$\left(x+1\right)\left(3x-5\right)$

## Do you want more practice?

- In Exercises 1–22, factor each difference of two squares. Assume that any variable exponents represent whole n...
- In Exercises 1–10, factor out the greatest common factor. 18x+27
- In Exercises 1–68, factor completely, or state that the polynomial is prime. x³ − 16x
- In Exercises 1–10, factor out the greatest common factor. 3x^2+6x
- In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using...
- In Exercises 1–10, factor out the greatest common factor. 9x^4−18x^3+27x^2
- In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using...
- In Exercises 1–22, factor the greatest common factor from each polynomial. x³ + 5x²
- Work each problem. Match each polynomial in Column I with its factored form in Column II. a. x^2 + 10xy +25y^...
- In Exercises 1–68, factor completely, or state that the polynomial is prime. x²y − 16y + 32 − 2x²
- In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using...
- In Exercises 1–22, factor the greatest common factor from each polynomial. 12x⁴ − 8x²
- In Exercises 1–22, factor each difference of two squares. Assume that any variable exponents represent whole n...
- In Exercises 1–68, factor completely, or state that the polynomial is prime. 4a²b − 2ab − 30b
- In Exercises 1–22, factor the greatest common factor from each polynomial. 32x⁴ + 2x³ + 8x²
- In Exercises 1–22, factor each difference of two squares. Assume that any variable exponents represent whole n...
- Work each problem. Which of the following is the correct complete factorization of x^4-1? A. (x^2-1)(x^2+1) B....
- 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)...
- In Exercises 1–22, factor the greatest common factor from each polynomial. 4x²y³ + 6xy
- Factor out the greatest common factor from each polynomial. See Example 1. 12m+60
- Express the distance between the numbers -17 and 4 using absolute value. Then evaluate the absolute value.
- In Exercises 11–16, factor by grouping. x^3−3x^2+4x−12
- In Exercises 1–22, factor the greatest common factor from each polynomial. 30x²y³ − 10xy²
- Factor out the greatest common factor from each polynomial. See Example 1. 8k^3+24k
- In Exercises 1–68, factor completely, or state that the polynomial is prime. 11x⁵ − 11xy²
- In Exercises 11–16, factor by grouping. x^3+6x^2−2x−12
- In Exercises 1–22, factor the greatest common factor from each polynomial. 12xy − 6xz + 4xw
- Factor out the greatest common factor from each polynomial. See Example 1. xy-5xy^2
- In Exercises 1–68, factor completely, or state that the polynomial is prime. 4x⁵ − 64x
- Factor out the greatest common factor from each polynomial. See Example 1. 5h^2j+hj
- In Exercises 16–17, factor completely. xy − 6x + 2y − 12
- Factor out the greatest common factor from each polynomial. See Example 1. -4p^3q^4-2p^2q^5
- In Exercises 1–68, factor completely, or state that the polynomial is prime. x³ − 4x² − 9x + 36
- In Exercises 16–17, factor completely. 24x³y + 16x²y − 30xy
- In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. x^2+8x+15
- In Exercises 1–68, factor completely, or state that the polynomial is prime. 2x⁵ + 54x²
- Simplify the algebraic expression. 5(2x - 3) + 7x
- Simplify the algebraic expression. (1/5)(5x) + [(3y) + (- 3y)] - (-x)
- In Exercises 1–22, factor the greatest common factor from each polynomial. 15x²ⁿ − 25xⁿ
- Factor out the greatest common factor from each polynomial. See Example 1. 2(a+b)+4m(a+b)
- In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. x^2−14x+45
- Factor out the greatest common factor from each polynomial. See Example 1. 6x(a+b)-4y(a+b)
- In Exercises 23–48, factor completely, or state that the polynomial is prime. 2x³ - 8x
- Factor out the greatest common factor from each polynomial. See Example 1. (5r-6)(r+3)-(2r-1)(r+3)
- In Exercises 23–34, factor out the negative of the greatest common factor. −4x + 12
- Factor out the greatest common factor from each polynomial. See Example 1. (4z-5)(3z-2)-(3z-9)(3z-2)
- In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. 2x^2+5x−3
- In Exercises 23–34, determine the constant that should be added to the binomial so that it becomes a perfect s...
- Factor out the greatest common factor from each polynomial. See Example 1. 2(m-1)-3(m-1)^2+2(m-1)^3
- In Exercises 23–48, factor completely, or state that the polynomial is prime. 50 - 2y²
- In Exercises 23–34, factor out the negative of the greatest common factor. −8x − 48
- In Exercises 1–68, factor completely, or state that the polynomial is prime. x² − 12x + 36 − 49y²
- Factor out the greatest common factor from each polynomial. See Example 1. 5(a+3)^3-2(a+3)+(a+3)^2
- In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. 6x^2−11x+4
- In Exercises 23–48, factor completely, or state that the polynomial is prime. 8x² - 8y²
- In Exercises 23–34, factor out the negative of the greatest common factor. −2x² + 6x − 14
- Concept Check When directed to completely factor the polynomial 4x^2y^5-8xy^3,a student wrote 2xy^3(2xy^2-4). ...
- In Exercises 1–68, factor completely, or state that the polynomial is prime. 4x² + 25y²
- Concept Check Kurt factored 16a^2-40a-6a+15 by grouping and obtained (8a-3)(2a-5). Callie factored the same po...
- In Exercises 23–34, factor out the negative of the greatest common factor. −5y² + 40x
- In Exercises 1–68, factor completely, or state that the polynomial is prime. 12x³y − 12xy³
- Factor each polynomial by grouping. See Example 2. 10ab-6b+35a-21
- In Exercises 23–34, determine the constant that should be added to the binomial so that it becomes a perfect s...
- In Exercises 23–34, factor out the negative of the greatest common factor. −4x³ + 32x² − 20x
- In Exercises 1–68, factor completely, or state that the polynomial is prime. 6bx² + 6by²
- Factor each polynomial by grouping. See Example 2. 4x^6+36-x^6y-9y
- In Exercises 23–34, determine the constant that should be added to the binomial so that it becomes a perfect s...
- In Exercises 23–34, factor out the negative of the greatest common factor. −x² − 7x + 5
- In Exercises 1–68, factor completely, or state that the polynomial is prime. x⁴ − xy³ + x³y − y⁴
- Factor each polynomial by grouping. See Example 2. 20z^2-8x+5pz^2-2px
- In Exercises 1–68, factor completely, or state that the polynomial is prime. x² − 4a² + 12x + 36
- In Exercises 23–48, factor completely, or state that the polynomial is prime. 8x² + 8y²
- Factor each trinomial, if possible. See Examples 3 and 4. 8h^2-2h-21
- In Exercises 31–38, factor completely. 4y³ + 12y² − 72y
- In Exercises 23–48, factor completely, or state that the polynomial is prime. x² + 25y²
- In Exercises 31–38, factor completely. 3x⁴ + 54x³ + 135x²
- In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. 6x^2−7xy−5y^2
- In Exercises 23–48, factor completely, or state that the polynomial is prime. x⁴ - 16
- In Exercises 35–44, factor the greatest common binomial factor from each polynomial. 3x(x+y) − (x+y)
- Factor each trinomial, if possible. See Examples 3 and 4. 9x^2+4x-2
- In Exercises 39–48, factor the difference of two squares. x^2−144
- In Exercises 35–44, factor the greatest common binomial factor from each polynomial. 4x²(3x−1) + 3x − 1
- A rigid bar (negligible mass) of length 80 cm connects a large sphere with a mass (m1) of 25 g to a small sphe...
- Factor each trinomial, if possible. See Examples 3 and 4. 36x^3+18x^2-4x
- In Exercises 39–48, factor the difference of two squares. 64x^2−81
- In Exercises 39–44, factor by introducing an appropriate substitution. x⁴ − 4x² − 5
- In Exercises 35–44, factor the greatest common binomial factor from each polynomial. (x + 2)(x + 3) + (x − 1)...
- In Exercises 1–68, factor completely, or state that the polynomial is prime. x² + 10x − y² + 25
- Factor each trinomial, if possible. See Examples 3 and 4. 14m^2+11mr-15r^2
- In Exercises 39–48, factor the difference of two squares. 36x^2−49y^2
- In Exercises 39–44, factor by introducing an appropriate substitution. (x + 1)² + 8(x + 1) + 7 (Let u = x+1.)
- Factor each trinomial, if possible. See Examples 3 and 4. 5a^2-7ab-6b^2
- In Exercises 45–68, factor by grouping. x² + 3x + 5x + 15
- In Exercises 1–68, factor completely, or state that the polynomial is prime. x⁸ − y⁸
- In Exercises 23–48, factor completely, or state that the polynomial is prime. x³ + 3x² - 4x - 12
- In Exercises 45–68, use the method of your choice to factor each trinomial, or state that the trinomial is pri...
- In Exercises 39–48, factor the difference of two squares. 16x^4−81
- In Exercises 45–68, factor by grouping. x² + 7x − 4x − 28
- In Exercises 1–68, factor completely, or state that the polynomial is prime. x³y − 16xy³
- In Exercises 23–48, factor completely, or state that the polynomial is prime. x³ - 7x² - x + 7
- Factor each trinomial, if possible. See Examples 3 and 4. 12x^2-xy-y^2
- In Exercises 45–68, use the method of your choice to factor each trinomial, or state that the trinomial is pri...
- In Exercises 49–56, factor each perfect square trinomial. x^2+2x+1
- In Exercises 45–68, factor by grouping. x³− 3x² + 4x − 12
- In Exercises 1–68, factor completely, or state that the polynomial is prime. x + 8x⁴
- In Exercises 49–64, factor any perfect square trinomials, or state that the polynomial is prime. x² + 4x + 4
- Factor each trinomial, if possible. See Examples 3 and 4. 24a^4+10a^3b-4a^2b^2
- In Exercises 45–68, use the method of your choice to factor each trinomial, or state that the trinomial is pri...
- In Exercises 1–68, factor completely, or state that the polynomial is prime. 16y² − 4y − 2
- In Exercises 49–64, factor any perfect square trinomials, or state that the polynomial is prime. x² − 10x + 2...
- Factor each trinomial, if possible. See Examples 3 and 4. 9m^2 -12m+4
- In Exercises 49–64, factor any perfect square trinomials, or state that the polynomial is prime. x⁴ − 4x² + 4
- Factor each trinomial, if possible. See Examples 3 and 4. 32a^2+48ab+18b^2
- Factor each trinomial, if possible. See Examples 3 and 4. 4x^2y^2+28xy+49
- In Exercises 49–64, factor any perfect square trinomials, or state that the polynomial is prime. 9y² + 6y + 1
- In Exercises 45–68, factor by grouping. 10x² − 12xy + 35xy − 42y²
- Factor each trinomial, if possible. See Examples 3 and 4. 9m^2n^2+12mn+4
- In Exercises 49–56, factor each perfect square trinomial. 64x^2−16x+1
- Factor each trinomial, if possible. See Examples 3 and 4. (a-3b)^2-6(a-3b)+9
- In Exercises 49–64, factor any perfect square trinomials, or state that the polynomial is prime. 64y² − 16y +...
- In Exercises 45–68, factor by grouping. 4x³ − x² + 12x - 3
- In Exercises 1–68, factor completely, or state that the polynomial is prime. 12x³ + 3xy²
- In Exercises 57–64, factor using the formula for the sum or difference of two cubes. x^3+64
- In Exercises 49–64, factor any perfect square trinomials, or state that the polynomial is prime. x² − 12xy + ...
- In Exercises 45–68, factor by grouping. x² − ax − bx + ab
- In Exercises 1–68, factor completely, or state that the polynomial is prime. x⁶y⁶ − x³y³
- Factor each polynomial. See Examples 5 and 6. 9a^2-16
- In Exercises 57–64, factor using the formula for the sum or difference of two cubes. x^3−27
- In Exercises 45–68, factor by grouping. x³ − 12 − 3x² + 4x
- In Exercises 1–68, factor completely, or state that the polynomial is prime. (x + 5)(x − 3) + (x + 5)(x − 7)
- Factor each polynomial. See Examples 5 and 6. x^4-16
- Factor each polynomial. See Examples 5 and 6. y^4-81
- In Exercises 49–64, factor any perfect square trinomials, or state that the polynomial is prime. 9x² + 48xy +...
- In Exercises 45–68, factor by grouping. ay − by + bx − ax
- In Exercises 1–68, factor completely, or state that the polynomial is prime. a²(x − y) + 4(y − x)
- Factor each polynomial. See Examples 5 and 6. 25s^4-9t^2
- Factor each polynomial. See Examples 5 and 6. 36z^2-81y^4
- In Exercises 57–64, factor using the formula for the sum or difference of two cubes. 8x^3+125
- Factor each polynomial. See Examples 5 and 6. (a+b)^2-16
- In Exercises 1–68, factor completely, or state that the polynomial is prime. (c + d)⁴ − 1
- Factor each polynomial. See Examples 5 and 6. (p-2q)^2-100
- In Exercises 65–92, factor completely, or state that the polynomial is prime. 5x^3−45x
- In Exercises 1–68, factor completely, or state that the polynomial is prime. p³ − pq² + p²q − q³
- In Exercises 65–92, factor completely, or state that the polynomial is prime. 2x^4−162
- In Exercises 69–80, factor completely. x⁴ − 5x²y² + y⁴
- In Exercises 65–74, factor by grouping to obtain the difference of two squares. 9x^2 − 30x + 25 − 36x^4
- In Exercises 69–78, factor each polynomial. ab − c − ac + b
- In Exercises 69–80, factor completely. (x + y)² + 6(x + y) + 9
- In Exercises 65–74, factor by grouping to obtain the difference of two squares. x⁴ − x² − 2x − 1
- In Exercises 69–78, factor each polynomial. x³ − 5 + 4x³y − 20y
- Factor each polynomial. See Examples 5 and 6. y^2-x^2+12x-36
- Factor each polynomial. See Examples 5 and 6. 9m^2-n^2-2n-1
- In Exercises 69–80, factor completely. (x − y)⁴ − 4(x − y)²
- In Exercises 65–74, factor by grouping to obtain the difference of two squares. z² − x² + 4xy − 4y²
- In Exercises 69–78, factor each polynomial. 2y⁷(3x−1)⁵ − 7y⁶(3x−1)⁴
- Factor each polynomial. See Examples 5 and 6. 8-a^3
- Factor each polynomial. See Examples 5 and 6. 27-r^3
- In Exercises 69–82, factor completely. 10y⁵ – 17y⁴ + 3y³
- Factor each polynomial. See Examples 5 and 6. 125x^3-27
- In Exercises 75–94, factor using the formula for the sum or difference of two cubes. x³ + 64
- In Exercises 69–82, factor completely. 12x² + 10xy – 8y²
- In Exercises 65–92, factor completely, or state that the polynomial is prime. 9x^3−9x
- In Exercises 69–80, factor completely. x³ − y³ − x + y
- In Exercises 75–94, factor using the formula for the sum or difference of two cubes. x³ − 27
- Factor each polynomial. See Examples 5 and 6. 27z^9+64y^12
- In Exercises 69–82, factor completely. 8a²b + 34ab – 84b
- Factor: x² + 4x + 4 − 9y². (Section 5.6, Example 4)
- In Exercises 75–94, factor using the formula for the sum or difference of two cubes. 8y³ + 1
- In Exercises 65–92, factor completely, or state that the polynomial is prime. x^3+2x^2−4x−8
- Factor each polynomial. See Examples 5 and 6. (b+3)^3-27
- In Exercises 69–82, factor completely. 12x²y – 46xy² + 14y³
- In Exercises 69–82, factor completely. 4x³y⁵ + 24x²y⁵ – 64xy⁵
- Factor each polynomial. See Examples 5 and 6. 125-(4a-b)^3
- In Exercises 65–92, factor completely, or state that the polynomial is prime. 20y^4−45y^2
- In Exercises 83–92, factor by introducing an appropriate substitution. 5x⁴ + 2x² − 3
- Factor each polynomial. See Example 7. 6(4z-3)^2+7(4z-3)-3
- Factor each polynomial. See Example 7. 9(a-4)^2+30(a-4)+25
- In Exercises 65–92, factor completely, or state that the polynomial is prime. x^2−10x+25−36y^2
- Factor each polynomial. See Example 7. 4(5x+7)^2+12(5x+7)+9
- Factor each polynomial. See Example 7. (a+1)^3+27
- Factor each polynomial. See Example 7. (x-4)^3+64
- Factor each polynomial. See Example 7. (3x+4)^3-1
- In Exercises 65–92, factor completely, or state that the polynomial is prime. x^2 y−16y+32−2x^2
- Factor completely, or state that the polynomial is prime. 15x^3+3x^2
- Factor completely, or state that the polynomial is prime. x^2-11x+28
- In Exercises 83–92, factor by introducing an appropriate substitution. 3(x+1)² − 5(x+1) + 2 (Let u = x+1)
- In Exercises 65–92, factor completely, or state that the polynomial is prime. 2x^3−8a^2 x+24x^2+72x
- Factor each polynomial. See Example 7. m^4-3m^2-10
- Factor completely, or state that the polynomial is prime. 64-x^2
- In Exercises 83–92, factor by introducing an appropriate substitution. 3(x−2)² − 5(x−2) − 2
- Factor each polynomial. See Example 7. a^4-2a^2-48
- In Exercises 75–94, factor using the formula for the sum or difference of two cubes. (x − y)³ − y³
- In Exercises 93–100, factor completely. x² + 0.3x − 0.04
- Factor each polynomial. See Example 7. 10m^4+43m^2-9
- In Exercises 95–104, factor completely. 0.04x² + 0.12x + 0.09
- In Exercises 93–100, factor completely. x² − 6/25 + 1/5 x
- Factor by any method. See Examples 1–7. (2y-1)^2-4(2y-1)+4
- In Exercises 95–104, factor completely. 8x⁴ − x/8
- Factor by any method. See Examples 1–7. x^2+xy-5x-5y
- In Exercises 93–100, factor completely. acx² − bcx − adx + bd
- In Exercises 95–104, factor completely. x⁶ − 9x³ + 8
- Factor by any method. See Examples 1–7. p^4(m-2n)+q(m-2n)
- In Exercises 93–102, factor and simplify each algebraic expression. (x+5)^−1/2−(x+5)^−3/2
- In Exercises 93–100, factor completely. −5x⁴y³ + 7x³y⁴ − 2x²y⁵
- Factor by any method. See Examples 1–7. 4z^2+28z+49
- In Exercises 93–102, factor and simplify each algebraic expression. −8(4x+3)^−2+10(5x+1)(4x+3)^−1
- In Exercises 103–114, factor completely. 6x^4+35x^2−6
- Factor and simplify each algebraic expression. 16x^(-3/4)+32x(1/4)
- Factor completely: 50x³ − 18x. (Section 5.5, Example 2)
- Factor by any method. See Examples 1–7. q^2+6q+9-p^2
- Factor by any method. See Examples 1–7. 64+(3x+2)^3
- Factor and simplify each algebraic expression. [12x(-1/2)+6x^(-3/2)]
- In Exercises 103–114, factor completely. x^4−5x^2 y^2+4y^4
- Factor by any method. See Examples 1–7. (x+y)^3-(x-y)^3
- In Exercises 103–114, factor completely. (x+y)^4−100(x+y)^2
- Factor by any method. See Examples 1–7. (3a+5)^2-18(3a+5)+81
- Factor by any method. See Examples 1–7. (x+y)^2-(x-y)^2
- Factor by any method. See Examples 1–7. 4z^4-7z^2-15
- Factor out the least power of the variable or variable expression. Assume all variables represent positive rea...
- Factor out the least power of the variable or variable expression. Assume all variables represent positive rea...
- Factor out the least power of the variable or variable expression. Assume all variables represent positive rea...
- Factor each polynomial over the set of rational number coefficients. 49x^2-1/25
- In Exercises 133–136, factor each polynomial completely. Assume that any variable exponents represent whole nu...
- In Exercises 133–136, factor each polynomial completely. Assume that any variable exponents represent whole nu...
- Factor each polynomial over the set of rational number coefficients. (25/9)x^4-(9y^2)
- Find all values of b or c that will make the polynomial a perfect square trinomial. 4z^2+bz+81
- Find all values of b or c that will make the polynomial a perfect square trinomial. 100r^2-60r+c
- Find all values of b or c that will make the polynomial a perfect square trinomial. 49x^2+70x+c
- Exercises 143–145 will help you prepare for the material covered in the next section. In each exercise, factor...
- Exercises 177–179 will help you prepare for the material covered in the next section. Factor completely: x^3 ...