1. Fundamental Concepts of Algebra

Factor by Grouping

Factoring Polynomials

Factoring by Grouping

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

Factor Expressions with Fractional Exponents

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

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

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

Factoring Perfect Square Trinomials - Ex1

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

In Exercises 49–56, factor each perfect square trinomial. x^2−14x+49

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

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

Finding all the Zeros of a Polynomial - Example 1

Factoring by Grouping - Ex 1

Factoring Perfect Square Trinomials - Ex3

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

Lesson: Factoring a Sum or Difference of Cubes

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 65–92, factor completely, or state that the polynomial is prime. x^3+2x^2−9x−18

Factor each polynomial. See Examples 5 and 6. x^2-8x+16-y^2

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

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

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

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

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

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

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

In Exercises 65–92, factor completely, or state that the polynomial is prime. 6x^2−6x−12

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

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

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

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

Factor each polynomial. See Example 7. (5x-2)^3-8

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

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

Factoring Expressions with Rational Exponents

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

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

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

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

Factoring Trinomials by Trial and Error - Ex 2

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

Factoring the Difference of Two Squares - Ex 1

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

Factoring a sum or difference of two cubes

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

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

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

Factor each trinomial, if possible. See Examples 3 and 4. 12s^2+11st-5t^2

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

Factoring the Difference of Two Squares - Ex 2

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

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 17–38, factor each trinomial, or state that the trinomial is prime. 9x^2+5x−4

Factor an expression

Rewrite a trinomial by factoring using grouping

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)

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)

Factor by any method. See Examples 1–7. 1000x^3+343y^3

Factor completely, or state that the polynomial is prime. 16x^2-40x+25

Use a general strategy for factoring polynomials

Factoring with Negative Exponents

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

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

Finding the Greatest Common Factor, GCF

In Exercises 1–10, factor out the greatest common factor. x^2(2x+5)+17(2x+5)

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

Factor completely, or state that the polynomial is prime. 3x^4-12x^2

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

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

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

Factoring Trinomials (A quadratic Trinomial) by Trial and Error

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

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

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

Whiteboard Math: Factoring by Grouping

Factoring Polynomials By Grouping

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

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

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

Exercises 177–179 will help you prepare for the material covered in the next section.

Factor completely: x^3 + x^2 - 4x - 4.

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

Factor expressions by grouping diagrams

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. x^4-16

Factoring with fractional and negative exponents.

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

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

Factoring Sums and Differences of Cubes

Factoring By Grouping - Ex 2

Finding all the Zeros of a Polynomial - Example 2

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

Factoring Sums and Differences of Cubes - Ex 3

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

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

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

In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. 3x^2+4xy+y^2

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

In Exercises 93–102, factor and simplify each algebraic expression. x^3/4−x^1/4

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

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

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

In Exercises 1–10, factor out the greatest common factor. x(2x+1)+4(2x+1)

Factoring Perfect Square Trinomials - Ex 2

Factor each polynomial. See Examples 5 and 6. 27-(m+2n)^3

Factor using multiple strategies

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

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

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

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

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.

Factor perfect square trinomials

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

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

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

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?

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

❖ Finding all the Zeros of a Polynomial - Example 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 out the greatest common factor from each polynomial. See Example 1. 4k^2m^3+8k^4m^3-12k^2m^4

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

In Exercises 49–56, factor each perfect square trinomial. 4x^2+4x+1

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

Factoring Using the Great Common Factor, GCF - Example 1

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

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

Rewrite an expression using the difference of two squares

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

Factoring Trinomials: Factor by Grouping - ex 1

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

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

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

Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. See Example 8. 4x(2x+3)^-5/9+6x^2(2x+3)^4/9-8x^3(2x+3)^13/9

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. 24a^4+10a^3b-4a^2b^2

Factor a quadratic equation: difference between 2 squares

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

Factor each trinomial, if possible. See Examples 3 and 4. (2p+q)^2-10(2p+q)+25

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

Factoring Trinomials: Factor by Grouping - ex 2

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

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

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

A General Factoring Strategy

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

Factoring Using the Great Common Factor, GCF - Example 2 - Factoring Out Binomials

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

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

Work each problem. Match each polynomial in Column I with its factored form in Column II.  a. 8x^3-27 b.8x^3+27 c. 27-8x^3  A. (3-2x)(9+6x+4x^2) B. (2x-3)(4x^2+6x+9) C. (2x+3)(4x^2-6x+9)

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

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

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

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

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