1. Review of Algebra

2. Equations & Inequalities

3. Functions & Graphs

4. Polynomial Functions

5. Rational Functions

6. Exponential and Logarithmic Functions

7. Systems of Equations and Inequalities

8. Matrices and Determinants

9. Conic Sections

10. Sequences, Induction, and Probability

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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

In Exercises 1–22, factor the greatest common factor from each polynomial.

30x²y³ − 10xy²

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

x³y − 16xy³

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

(x − y)³ − y³

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﻿<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">(</mo><mi>x</mi><mo>−</mo><mn>5</mn><mo fence="true">)</mo></mrow><mrow><mo fence="true">(</mo><mi>x</mi><mo>−</mo><mn>8</mn><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\left(x-5\right)\left(x-8\right)</annotation></semantics></math>(x−5)(x−8)﻿

﻿<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">(</mo><mi>x</mi><mo>−</mo><mn>4</mn><mo fence="true">)</mo></mrow><mrow><mo fence="true">(</mo><mi>x</mi><mo>−</mo><mn>10</mn><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\left(x-4\right)\left(x-10\right)</annotation></semantics></math>(x−4)(x−10)﻿

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Factor the polynomial. ﻿<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>13</mn><mi>x</mi><mo>+</mo><mn>40</mn></mrow><annotation encoding="application/x-tex">x^2-13x+40</annotation></semantics></math>x2−13x+40﻿

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

x² − 9xy + 14y²

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

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

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

x² − 4a² + 12x + 36

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

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

In Exercises 1–22, factor the greatest common factor from each polynomial.

15x²ⁿ − 25xⁿ

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

a²(x − y) + 4(y − x)

Factoring Polynomials

Factoring Perfect Square Trinomials - Ex1

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

64y² − 16y + 1

In Exercises 83–92, factor by introducing an appropriate substitution.

5x⁴ + 2x² − 3

In Exercises 23–34, factor out the negative of the greatest common factor.

−x² − 7x + 5

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

In Exercises 93–100, factor completely.

acx² − bcx − adx + bd

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

8x² + 8y²

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

Factoring Trinomials : Factor by Grouping - ex 3

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

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

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Factor the polynomial. ﻿<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">3x^2-2x-5</annotation></semantics></math>3x2−2x−5﻿

In Exercises 1–22, factor the greatest common factor from each polynomial.

32x⁴ + 2x³ + 8x²

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

Factoring by Grouping - Ex 1

Factoring Perfect Square Trinomials - Ex3

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

In Exercises 69–80, factor completely.

(x + y)² + 6(x + y) + 9

Factoring by Grouping

In Exercises 69–78, factor each polynomial.

ab − c − ac + b

Lesson: Factoring a Sum or Difference of Cubes

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

x³ + 64

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

x² − 10x + 25

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

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

a² + 5a − 14

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

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

In Exercises 23–34, factor out the negative of the greatest common factor.

−2x² + 6x − 14

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

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

16y² − 4y − 2

In Exercises 16–17, factor completely.

xy − 6x + 2y − 12

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

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

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

8x² - 8y²

In Exercises 65–74, factor by grouping to obtain the difference of two squares. 

x⁴ − x² − 2x − 1

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

﻿<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><mi>x</mi><mrow><mo fence="true">(</mo><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>6</mn><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">3x\left(-x^2+4x-6\right)</annotation></semantics></math>3x(−x2+4x−6)﻿

﻿<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mrow><mo fence="true">(</mo><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>6</mn><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">3x^2\left(-x^2+4x-6\right)</annotation></semantics></math>3x2(−x2+4x−6)﻿

﻿<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><mrow><mo fence="true">(</mo><mo>−</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">3\left(-x^3+4x^2-6x\right)</annotation></semantics></math>3(−x3+4x2−6x)﻿

﻿<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mrow><mo fence="true">(</mo><mo>−</mo><mn>3</mn><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>12</mn><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>18</mn><msup><mi>x</mi><mn>2</mn></msup><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">3x^2\left(-3x^4+12x^3-18x^2\right)</annotation></semantics></math>3x2(−3x4+12x3−18x2)﻿

Factor out the Greatest Common Factor in the polynomial.﻿<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>3</mn><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>12</mn><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>18</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">-3x^4+12x^3-18x^2</annotation></semantics></math>−3x4+12x3−18x2﻿

In Exercises 83–92, factor by introducing an appropriate substitution.

3(x+1)² − 5(x+1) + 2 (Let u = x+1)

Factor Using the AC Method When a Is 1

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

6bx² + 6by²

In Exercises 93–100, factor completely.

−5x⁴y³ + 7x³y⁴ − 2x²y⁵

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

x²y − 16y + 32 − 2x²

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

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

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

x³ − 27

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

x² − x + 7

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

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

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

a² − 18ab + 45b²

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

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

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

x² − 12x + 36 − 49y²

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

9x² + 48xy + 64y²

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

x² + 4x + 4

Factor by Grouping

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

Introduction to Factoring Polynomials

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

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

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

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

Factor an expression: taking out the greatest common factor

In Exercises 45–68, factor by grouping.

4x³ − x² + 12x - 3

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

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

x⁸ − y⁸

Factor Using the AC Method When a Is Not 1

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

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

(x - 3)² - y²

In Exercises 45–68, factor by grouping.

10x² − 12xy + 35xy − 42y²

In Exercises 1–22, factor the greatest common factor from each polynomial.

12x⁴ − 8x²

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

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

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

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

In Exercises 95–104, factor completely.

8x⁴ − x/8

Factor an expression using the diamond and box method

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

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

Factoring Sum and Difference of Cubes

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

In Exercises 69–82, factor completely.

8a²b + 34ab – 84b

In Exercises 95–104, factor completely.

0.04x² + 0.12x + 0.09

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. 

(c + d)⁴ − 1

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

x⁶y⁶ − x³y³

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

In Exercises 69–78, factor each polynomial.

2y⁷(3x−1)⁵ − 7y⁶(3x−1)⁴

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

x⁴ - 16

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

x² − 8x + 15

Factor: using factor sums

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

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

Factoring the Difference of Two Squares - Ex 3

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

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

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

In Exercises 35–44, factor the greatest common binomial factor from each polynomial.

3x(x+y) − (x+y)

Factoring the Difference of Two Squares - Ex 1

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 Example 7. m^4-3m^2-10

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

12x³y − 12xy³

Factoring a sum or difference of two cubes

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

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

x² − xy − 30y²

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

a² - (b - 2)²

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

In Exercises 35–44, factor the greatest common binomial factor from each polynomial.

4x²(3x−1) + 3x − 1

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

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

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

x² + 25y²

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

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

x² − 12xy + 36y²

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

x² + 10x − y² + 25

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

2x⁵ + 54x²

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

In Exercises 69–82, factor completely.

12x² + 10xy – 8y²

In Exercises 1–22, factor the greatest common factor from each polynomial.

4x²y³ + 6xy

Factoring the Difference of Two Squares - Ex 2

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

x² + 8x + 12

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

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

In Exercises 69–80, factor completely.

x³ − y³ − x + y

Factor an expression

Rewrite a trinomial by factoring using grouping

In Exercises 69–80, factor completely.

(x − y)⁴ − 4(x − y)²

In Exercises 45–68, factor by grouping.

x² − ax − bx + ab

In Exercises 65–74, factor by grouping to obtain the difference of two squares. 

9x^2 − 30x + 25 − 36x^4

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

How to factor a perfect square trinomial

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

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

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)

﻿<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">(</mo><mo>−</mo><mi>x</mi><mo>+</mo><mn>7</mn><mo fence="true">)</mo></mrow><mrow><mo fence="true">(</mo><mi>x</mi><mo>+</mo><mn>5</mn><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\left(-x+7\right)\left(x+5\right)</annotation></semantics></math>(−x+7)(x+5)﻿

﻿<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">(</mo><mi>x</mi><mo>+</mo><mn>7</mn><mo fence="true">)</mo></mrow><mrow><mo fence="true">(</mo><mi>x</mi><mo>+</mo><mn>5</mn><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\left(x+7\right)\left(x+5\right)</annotation></semantics></math>(x+7)(x+5)﻿

﻿<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">(</mo><mi>x</mi><mo>−</mo><mn>7</mn><mo fence="true">)</mo></mrow><mrow><mo fence="true">(</mo><mi>x</mi><mo>+</mo><mn>5</mn><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\left(x-7\right)\left(x+5\right)</annotation></semantics></math>(x−7)(x+5)﻿

﻿<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">(</mo><mi>x</mi><mo>+</mo><mn>7</mn><mo fence="true">)</mo></mrow><mrow><mo fence="true">(</mo><mo>−</mo><mi>x</mi><mo>+</mo><mn>5</mn><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\left(x+7\right)\left(-x+5\right)</annotation></semantics></math>(x+7)(−x+5)﻿

Factor the polynomial by grouping	﻿<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>7</mn><mi>x</mi><mo>+</mo><mn>35</mn></mrow><annotation encoding="application/x-tex">-x^2-5x+7x+35</annotation></semantics></math>−x2−5x+7x+35﻿

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

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

In Exercises 95–104, factor completely.

x⁶ − 9x³ + 8

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)

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

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

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

In Exercises 23–34, factor out the negative of the greatest common factor.

−4x + 12

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

Finding the Greatest Common Factor, GCF

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

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

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

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

In Exercises 69–82, factor completely.

10y⁵ – 17y⁴ + 3y³

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

In Exercises 65–74, factor by grouping to obtain the difference of two squares. 

z² − x² + 4xy − 4y²

In Exercises 1–22, factor the greatest common factor from each polynomial.

x³ + 5x²

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

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

2x³ - 8x

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

In Exercises 69–82, factor completely.

12x²y – 46xy² + 14y³

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

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

y² + 10y + 16

In Exercises 45–68, factor by grouping.

x² + 7x − 4x − 28

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

9x⁴ - 25y⁶