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GED Math : Simplifying, Distributing, and Factoring

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Example Questions

Example Question #31 : Single Variable Algebra

Factor completely:

$36x^{2} - 49y^{2}$

Possible Answers:

$\left (3x + 7y \right )\left (12x - 7y \right )$

$\left (6x - 7y \right ) ^{2}$

$\left (12x + 7y \right )\left (3x - 7y \right )$

$\left (6x + 7y \right )\left (6x - 7y \right )$

Correct answer:

$\left (6x + 7y \right )\left (6x - 7y \right )$

Explanation:

The polynomial is the difference of squares and can be factored using the pattern

$A^{2} - B ^{2} = (A+B)(A-B)$

where

A = 6x, B = 7y

as seen here:

$36x^{2} - 49y^{2}$

$= 6^{2}x^{2} - 7^{2}y^{2}$

$= \left (6 x \right )^{2} - \left (7 y \right )^{2}$

$= \left (6x + 7y \right )\left (6x - 7y \right )$

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Example Question #31 : Algebra

Which of the following is a factor of the polynomial $8x ^{3}y - 20x ^{2}y ^{2}$ ?

Possible Answers:

$2x ^{2}- 5y ^{2}$

2xy - 5

2x - 5y

2- 5xy

Correct answer:

2x - 5y

Explanation:

The greatest common factor of the two terms is the monomial term $4x ^{2}y$ , so factor it out:

$8x ^{3}y - 20x ^{2}y ^{2}$

$= 4x ^{2}y \cdot 2x - 4x ^{2}y \cdot 5y$

$= 4x ^{2}y \left (2x - 5y \right )$

Of the four choices, 2x - 5y is correct.

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Example Question #31 : Single Variable Algebra

Which of the following is a factor of the polynomial $8x ^{3}y - 20x ^{2}y ^{2}$ ?

Possible Answers:

$2x ^{2}- 5y ^{2}$

2xy - 5

2- 5xy

2x - 5y

Correct answer:

2x - 5y

Explanation:

The greatest common factor of the two terms is the monomial term $4x ^{2}y$ , so factor it out:

$8x ^{3}y - 20x ^{2}y ^{2}$

$= 4x ^{2}y \cdot 2x - 4x ^{2}y \cdot 5y$

$= 4x ^{2}y \left (2x - 5y \right )$

Of the four choices, 2x - 5y is correct.

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Example Question #31 : Simplifying, Distributing, And Factoring

Simplify:

$\left (\frac{5}{t} \right )^{-3}$

Possible Answers:

$-\frac{1 5}{t^{3}}$

$\frac{t^{3}}{125}$

$-\frac{125}{t^{3}}$

$\frac{t^{3}}{1 5}$

Correct answer:

$\frac{t^{3}}{125}$

Explanation:

Raise a fraction to a negative power by raising its reciprocal to the power of the absolute value of the exponent. Then apply the power of a quotient rule:

$\left (\frac{5}{t} \right )^{-3} = \left (\frac{t}{5} \right )^{ 3} = \frac{t^{ 3}}{5^{ 3}} = \frac{t^{ 3}}{125}$

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Example Question #32 : Simplifying, Distributing, And Factoring

Simplify:

$\left (4 t^{2} \right )^{-3}$

Possible Answers:

$-\frac{12}{ t^{6}}$

$\frac{1}{64t^{6}}$

$\frac{4}{ t^{6}}$

$\frac{1}{ t^{6}}$

Correct answer:

$\frac{1}{64t^{6}}$

Explanation:

To raise a number to a negative exponent, raise it to the absolute value of that exponent, then take its reciprocal. We do this, then apply the various properties of exponents:

$\left (4 t^{2} \right )^{-3} = \frac{1}{\left (4 t^{2} \right )^{3} } = \frac{1}{4^{3} \left ( t^{2} \right )^{3} } =\frac{1}{64 t^{2 \cdot 3} } =\frac{1}{64 t^{6} }$

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Example Question #33 : Algebra

Factor completely:

$6t ^{2} + t - 15$

Possible Answers:

$\left ( 3t - 5 \right )\left ( 2t + 3 \right )$

$\left ( 3t + 5 \right )\left ( 2t - 3 \right )$

$\left ( 6t - 5 \right )\left ( t + 3 \right )$

$\left ( 6t + 5 \right )\left ( t - 3 \right )$

Correct answer:

$\left ( 3t + 5 \right )\left ( 2t - 3 \right )$

Explanation:

For a quadratic trinomial with a quadratic coefficient other than 1, use the factoring by grouping method.

First, find two integers whose product is $6 \times (-15) = -90$ (the product of the quadratic and constant coefficients) and whose sum is 1 (the implied coefficient of ). By trial and error, we find that these are -9, 10 .

Split the linear term accordingly, then factor by grouping, as follows.

$6t ^{2} + t - 15$

$= 6t ^{2} - 9t +10 t - 15$

$=\left ( 6t ^{2} - 9t \right )+\left (10 t - 15 \right )$

$= \left ( 3t \cdot 2t - 3t \cdot 3 \right )+\left (5 \cdot 2t - 5 \cdot 3 \right )$

$= 3t \left ( 2t - 3 \right )+ 5 \left ( 2t - 3 \right )$

$=\left ( 3t + 5 \right )\left ( 2t - 3 \right )$

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Example Question #35 : Algebra

Factor:

$4y^{3} - 12y ^{2}- 72y$

Possible Answers:

y (y+2)(4y-9)

4y (y-6)(y+3)

y (y-6)(4y+3)

4y (y+2)(y-9)

Correct answer:

4y (y-6)(y+3)

Explanation:

The greatest common factor of the terms is , so factor it out:

$4y^{3} - 12y ^{2}- 72y$

$=4y \cdot y^{2} - 4y \cdot 3y - 4y \cdot 18$

$=4y \left ( y^{2} - 3y - 18 \right )$

The trinomial might be able to be factored as

(y+A)(y + B) ,

where AB = -18 and A + B = -3 .

By trial and error, we find that

A = -6 , B = 3 ,

so the factorization becomes

4y (y-6)(y+3) .

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Example Question #33 : Simplifying, Distributing, And Factoring

Decrease 8t + 72 by 40%. Which of the following will this be equal to?

Possible Answers:

3.2t + 28.8

4.8t + 43.2

4.8t + 72

3.2t + 72

Correct answer:

4.8t + 43.2

Explanation:

A number decreased by 40% is equivalent to 100% of the number minus 40% of the number. This is taking 60% of the number, or, equivalently, multiplying it by 0.6.

Therefore, 8t + 72 decreased by 40% is 0.6 times this, or

$0.6 \left (8t + 72 \right ) =4.8t + 43.2$ .

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Example Question #34 : Simplifying, Distributing, And Factoring

Increase 8x + 9y by 20%. Which of the following will this be equal to?

Possible Answers:

9.2x+10 y

9.6x + 10.8 y

8.2x+9.2y

9x + 10y

Correct answer:

9.6x + 10.8 y

Explanation:

A number increased by 20% is equivalent to 100% of the number plus 20% of the number. This is taking 120% of the number, or, equivalently, multiplying it by 1.2.

Therefore, 8x + 9y increased by 20% is 1.2 times this, or

$1.2 \left (8x + 9y \right ) = 1.2 \cdot 8x + 1.2 \cdot 9y = 9.6x + 10.8 y$ .

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Example Question #35 : Simplifying, Distributing, And Factoring

Which of the following is a prime factor of $n^{4} - 16n^{2} - 225$ ?

Possible Answers:

$n^{2}+9$

$n^{2}+3$

$n^{2}+25$

$n^{2}+75$

Correct answer:

$n^{2}+9$

Explanation:

This can be most easily solved by first substituting for $n^{2}$ , and, subsequently, $t^{2}$ for $n^{4}$ :

$n^{4} - 16n^{2} - 225$

$= t^{2} - 16t - 225$

This becomes quadratic in the new variable, and can be factored as

$\left (t +\; \; \right )\left (t +\; \; \right )$ ,

filling out the blanks with two numbers whose sum is and whose product is . Through some trial and error, the numbers can be seen to be 9, -25 .

Therefore, after factoring and substituting back,

$n^{4} - 16n^{2} - 225$

$= t^{2} - 16t - 225$

= (t+9)(t-25)

$= (n^{2}+9)(n^{2}-25)$

The first factor, the sum of squares, is prime. The second factors as the difference of squares, so the final factorization is

$(n^{2}+9)(n+5)(n-5)$ .

Of the choices given, $n^{2}+9$ is correct.

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GED Math : Simplifying, Distributing, and Factoring

Study concepts, example questions & explanations for GED Math

All GED Math Resources

Example Questions

Example Question #31 : Single Variable Algebra

Example Question #31 : Algebra

Example Question #31 : Single Variable Algebra

Example Question #31 : Simplifying, Distributing, And Factoring

Example Question #32 : Simplifying, Distributing, And Factoring

Example Question #33 : Algebra

Example Question #35 : Algebra

Example Question #33 : Simplifying, Distributing, And Factoring

Example Question #34 : Simplifying, Distributing, And Factoring

Example Question #35 : Simplifying, Distributing, And Factoring

All GED Math Resources

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