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Algebra II : Factoring Polynomials

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Example Question #11 : Factoring Polynomials

Factor 3u^4 - 24uv^3 .

Possible Answers:

3u(u - 2v)(u^2 - 2uv -4v^2)

3u[u^3 - (2v)^3]

3u(u - 2v)(u^2 + 2uv + 4v^2)

3u(u^3 - 8v^3)

3u(u - 2v)(u + 2v)

Correct answer:

3u(u - 2v)(u^2 + 2uv + 4v^2)

Explanation:

First pull out 3u from both terms.

3u⁴ – 24uv³= 3u(u³ – 8v³) = 3u[u³ – (2v)³]

This is a difference of cubes. You will see this type of factoring if you get to the challenging questions on the GRE. They can be a pain to remember, but pat yourself on the back for getting to such hard questions! The difference of cubes formula is a³ – b³ = (a – b)(a² + ab + b²). In our problem, a = u and b = 2v:

3u⁴ – 24uv³= 3u(u³ – 8v³) = 3u[u³ – (2v)³]

= 3u(u – 2v)(u² + 2uv + 4v²)

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Example Question #11 : Factoring Polynomials

Factor 36x^2 - 49y^2 .

Possible Answers:

(6x + 7y)(6x -7y)

Cannot be factored any further.

6x^2 - 7y^2

(6x - 7y)(6x - 7y)

(6x + 7y)(6x + 7y)

Correct answer:

(6x + 7y)(6x -7y)

Explanation:

This is a difference of squares. The difference of squares formula is a² – b² = (a + b)(a – b).

In this problem, a = 6x and b = 7y:

36x² – 49y²= (6x + 7y)(6x – 7y)

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Example Question #2 : Factoring Polynomials

Factor:

$2x^{2}-7x+6$

Possible Answers:

(x-1)(2x-6)

(x+2)(2x+3)

(x+1)(2x+6)

(2x-2)(x-3)

(x-2)(2x-3)

Correct answer:

(x-2)(2x-3)

Explanation:

$2x^{2}-7x+6$

$=2x^{2}-3x-4x+6$

$=2x^{2}-3x-(4x-6)$

=x(2x-3)-2(2x-3)

=(x-2)(2x-3)

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Example Question #3 : Factoring Polynomials

Factor the following expression:

x^2 -10x - 24

Possible Answers:

(x-6)(x+4)

(x-24)(x-1)

(x-12)(x+2)

(x+3)(x-8)

(x-12)(x-2)

Correct answer:

(x-12)(x+2)

Explanation:

x^2 -10x - 24

To factor, we are looking for two terms that multiply to give and add together to get .

Possible factors of :

$(-1,24),\ (1,-24),\ (-2,12),\ (2,-12),\ (-3,8),\ (3,-8),\ (-4,6),\ (4,-6)$

Based on these options, it is clear our factors are and .

$(2)(-12)=-24\ \text{and}\ 2+(-12)=-10$

Our final answer will be:

(x-12)(x+2)

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Example Question #11 : Factoring Polynomials

Factor the following expression:

x^2 +7x - 8

Possible Answers:

(x+8)(x-1)

(x-4)(x-2)

(x+4)(x-2)

(x-8)(x+1)

(x+4)(x-4)

Correct answer:

(x+8)(x-1)

Explanation:

x^2 +7x - 8

To factor, we are looking for two terms that multiply to give and add together to get .

Possible factors of :

$(-1,8),\ (1,-8),\ (-2,4),\ (2,-4)$

Based on these options, it is clear our factors are and .

$(-1)(8)=-8\ \text{and}\ (-1)+8=7$

Our final answer will be:

(x+8)(x-1)

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Example Question #12 : Factoring Polynomials

Factor the following expression:

x^2 - 11x + 10

Possible Answers:

(x-2)(x-5)

(x-5)(x+2)

(x-5)(x-5)

(x-10)(x+1)

(x-10)(x-1)

Correct answer:

(x-10)(x-1)

Explanation:

x^2 - 11x + 10

To factor, we are looking for two terms that multiply to give and add together to get .

Possible factors of :

$(1,10),\ (-1,-10),\ (2,5),\ (-2,-5)$

Based on these options, it is clear our factors are and .

$(-1)(-10)=10\ \text{and}\ (-1)+(-10)=-11$

Our final answer will be:

(x-10)(x-1)

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Example Question #4 : Factoring Polynomials

Factor the following expression:

x^2 + 20x + 96

Possible Answers:

(x - 12)(x-8)

(x + 6)(x+16)

(x + 12)(x+8)

(x + 2)(x-48)

(x+24)(x+4)

Correct answer:

(x + 12)(x+8)

Explanation:

x^2 + 20x + 96

To factor, we are looking for two terms that multiply to give and add together to get . There are numerous factors of , so we will only list a few.

Possible factors of :

$(3,32),\ (-3,-32),\ (4,24),\ (-4,-24),\ (6,16),\ (8,12)$

Based on these options, it is clear our factors are and .

$(8)(12)=96\ \text{and}\ 8+12=20$

Our final answer will be:

(x + 12)(x+8)

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Example Question #13 : Factoring Polynomials

Factor the following expression:

x^2 -38x -600

Possible Answers:

(x-50)(x-12)

(x-300)(x+2)

(x-50)(x+12)

(x-100)(x+6)

(x-60)(x+10)

Correct answer:

(x-50)(x+12)

Explanation:

x^2 -38x -600

To factor, we are looking for two terms that multiply to give and add together to get . There are numerous factors of , so we will only list a few.

Possible factors of :

$(2,-300),\ (3,-200),\ (5,-120),\ (6,-100),\ (10,-60),\ (12,-50),\ (15,-40)$

Based on these options, it is clear our factors are and .

$(12)(-50)=-600\ \text{and}\ 12+(-50)=-38$

Our final answer will be:

(x-50)(x+12)

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Example Question #14 : Factoring Polynomials

Factor the following expression:

x^2 - 26x + 105

Possible Answers:

(x + 7)(x-15)

(x + 7)(x-21)

(x - 7)(x-15)

(x - 5)(x-21)

(x - 5)(x+21)

Correct answer:

(x - 5)(x-21)

Explanation:

x^2 - 26x + 105

To factor, we are looking for two terms that multiply to give 105 and add together to get .

Possible factors of 105 :

$(3,35),\ (-3,-35),\ (5,21),\ (-5,-21),\ (7,15),\ (-7,-15)$

Based on these options, it is clear our factors are and .

$(-5)(-21)=105\ \text{and}\ (-5)+(-21)=-26$

Our final answer will be:

(x - 5)(x-21)

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Example Question #15 : Factoring Polynomials

Simplify:

$\frac{x^2+6x+9}{x+3}$

Possible Answers:

x+3

$\frac{(x+3)}{(x-3)}$

x+15

x+6x+3

Correct answer:

x+3

Explanation:

Factor the numerator:

$x^2+6x+9\rightarrow (x+3)(x+3)$

Simplify the fraction:

$\frac{(x+3)(x+3)}{(x+3)}\rightarrow (x+3)$

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Algebra II : Factoring Polynomials

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #11 : Factoring Polynomials

Example Question #11 : Factoring Polynomials

Example Question #2 : Factoring Polynomials

Example Question #3 : Factoring Polynomials

Example Question #11 : Factoring Polynomials

Example Question #12 : Factoring Polynomials

Example Question #4 : Factoring Polynomials

Example Question #13 : Factoring Polynomials

Example Question #14 : Factoring Polynomials

Example Question #15 : Factoring Polynomials

All Algebra II Resources

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