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GED Math : Simplifying, Distributing, and Factoring

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Example Question #11 : Simplifying, Distributing, And Factoring

Express 286 in base five.

Possible Answers:

$2201_{\textrm{five}}$

$2101_{\textrm{five}}$

$2221_{\textrm{five}}$

$2121_{\textrm{five}}$

Correct answer:

$2121_{\textrm{five}}$

Explanation:

To convert a base ten number to base five, divide the number by five, with the remainder being the digit in the units place; continue, dividing each successive quotient by five and putting the remainder in the next position to the left until the final quotient is less than five.

$286\div 5 = 57 \textrm{ R }\underline{1}$ - 1 is the last digit.

$57 \div 5 = 11 \textrm{ R }\underline{2}$ - 2 is the second-to-last digit.

$11 \div 5 = \underline{2} \textrm{ R }\underline{1}$ - 1 is the third-to-last digit; 2 is the first digit.

286 is equal to $2121_{\textrm{five}}$ .

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Example Question #12 : Simplifying, Distributing, And Factoring

Simplify the following: 2x^2 - 3y - 4x^2 + 3y^2 - 3x

Possible Answers:

-2x^2 - 3y^2 - 3x - 3y

2x^2 + 3y^2 - 3x + 3y

-5x^2 - 3x

-2x^2 + 3y^2 - 3x - 3y

2x^2 - 3x - 3y

Correct answer:

-2x^2 + 3y^2 - 3x - 3y

Explanation:

Group all like terms by their order:

2x^2 - 3y - 4x^2 + 3y^2 - 3x

2x^2 - 4x^2 + 3y^2 - 3x- 3y

Simplify:

-2x^2 + 3y^2 - 3x - 3y

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Example Question #13 : Simplifying, Distributing, And Factoring

Simplify the following: -(2-x)(6-x)

Possible Answers:

-x^2 +8x-12

-x^2 -4x-12

-x^2 +4x-12

-x^2 -8x-12

-x^2 +8x+12

Correct answer:

-x^2 +8x-12

Explanation:

This can be solved using the FOIL method. The steps are shown below.

-(2-x)(6-x)

-((2)(6) + (2)(-x) + (-x)(6)+(-x)(-x))

-(12-2x-6x+x^2)

-(12-8x+x^2)

-12+8x-x^2

Therefore, after reordering, the answer is: -x^2+8x-12

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Example Question #14 : Simplifying, Distributing, And Factoring

Simplify:

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

Possible Answers:

$\frac{1}{256x^{8}}$

$\frac{x^{8}}{4}$

$\frac{1}{4x^{8}}$

$\frac{x^{8}}{256}$

Correct answer:

$\frac{x^{8}}{4}$

Explanation:

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

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

$=\left ( \frac{x^{4}}{2} \right )^{2}$

$= \frac{\left (x^{4}\right )^{2}}{2^{2}}$

$= \frac{ x^{4 \cdot 2} }{2^{2}}$

$= \frac{ x^{8} }{4}$

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Example Question #11 : Simplifying, Distributing, And Factoring

Factor completely:

$8t^{3} -56t^{2} +3t -21$

Possible Answers:

$(8t^{2}-3) (t+7)$

$(8t^{2}+3) (t-7)$

$(8t^{2}+7) (t-3)$

$(8t^{2}-7) (t+3)$

Correct answer:

$(8t^{2}+3) (t-7)$

Explanation:

Use the grouping technique, then distribute out the greatest common factor of each group as follows:

$8t^{3} -56t^{2} +3t -21$

$=\left ( 8t^{3} -56t^{2} \right ) +\left (3t -21 \right )$

$=\left ( 8t^{2} \cdot t -8t^{2} \cdot 7 \right ) +\left (3 \cdot t -3 \cdot 7\right )$

$=8t^{2} \left ( t - 7 \right ) + 3\left ( t - 7 \right )$

$=\left (8t^{2} + 3 \right ) \left ( t - 7 \right )$

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Example Question #16 : Simplifying, Distributing, And Factoring

Factor completely:

$9t^{2} - 81t$

Possible Answers:

9(t-3)(t+27)

$9 (t-3)^{2}$

9t (t-9)

$9 \left ( t+3\right )(t-3)$

Correct answer:

9t (t-9)

Explanation:

The common factor of the terms $9t^{2}$ and 81t can be found as follows:

GCF (9,81) = 9 , and the lesser of the two powers of is ; therefore, $GCF (9t^{2},81t) = 9t$ , their product. Distribute this out:

$9t^{2} - 81t$

$= 9t \cdot t - 9t \cdot 9$

$= 9t \left (t - 9 \right )$

This is as far as we can go with the factoring.

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Example Question #17 : Simplifying, Distributing, And Factoring

Factor completely:

30xt + 15x -22t - 11

Possible Answers:

$\left (15x - 11 \right ) \left ( 2t + 1 \right )$

$\left (6x+11 \right ) \left ( 5t - 1 \right )$

$\left (15x + 11 \right ) \left ( 2t -1 \right )$

$\left (6x- 11 \right ) \left ( 5t + 1 \right )$

Correct answer:

$\left (15x - 11 \right ) \left ( 2t + 1 \right )$

Explanation:

Use the grouping technique - group the terms into pairs, then factor out the greatest common factor of each pair.

30xt + 15x -22t - 11

$= \left (30xt + 15x \right ) -\left (22t + 11 \right )$

$= \left (15x \cdot 2t + 15x \cdot 1 \right ) -\left ( 11 \cdot 2t + 11 \cdot 1 \right )$

$= 15x \left ( 2t + 1 \right ) - 11 \left ( 2t + 1 \right )$

$= \left (15x - 11 \right ) \left ( 2t + 1 \right )$

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Example Question #11 : Algebra

Factor completely:

$2x^{2} + 21x - 50$

Possible Answers:

$\left (2x+ 25 \right ) \left (x-2 \right )$

$2 \left (x+5 \right ) \left (x-5 \right )$

$\left (2x- 1\right ) \left (x+25 \right )$

$\left (2x+ 5\right ) \left (x-2 \right )$

Correct answer:

$\left (2x+ 25 \right ) \left (x-2 \right )$

Explanation:

A polynomial of the form $Ax^{2} + Bx + C$ can be factored by first splitting the term into two terms whose coefficients add up to and have product , then factoring out by the grouping technique.

We are looking for two integers whose sum is B = 21 and whose product is AC = 2 (-50) = -100 . Through some trial and error, we can see that the integers are $\left \{ 25, -4 \right \}$ , so:

$2x^{2} + 21x - 50$

$=2x^{2} - 4x + 25x - 50$

$=\left (2x^{2} - 4x \right )+\left ( 25x - 50 \right )$

$=\left (2x \cdot x - 2x \cdot 2 \right )+\left ( 25 \cdot x - 25 \cdot 2 \right )$

$= 2x \left (x-2 \right )+ 25 \left (x-2 \right )$

$= \left (2x+ 25 \right ) \left (x-2 \right )$

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Example Question #11 : Simplifying, Distributing, And Factoring

Factor completely:

$4x ^{3}y - 100xy$

Possible Answers:

$4x y \left ( x + 5 \right ) \left ( x - 5 \right )$

$2x y \left ( 2x - 5 \right ) \left ( x + 10 \right )$

$2x y \left ( 2x + 5 \right ) \left ( x - 10 \right )$

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

Correct answer:

$4x y \left ( x + 5 \right ) \left ( x - 5 \right )$

Explanation:

First, factor out the greatest common factor of the terms, which is 4x y :

$4x ^{3}y - 100xy$

$= 4x y \cdot x^{2} - 4x y \cdot 25$

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

$x^{2} - 25$ is the difference of squares, so we can factor further:

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

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

$4x y \left ( x + 5 \right ) \left ( x - 5 \right )$

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Example Question #12 : Algebra

Simplify:

7 (2x + 8)- 5 (7x - 3)

Possible Answers:

-49x+ 41

-21x+ 41

-49x+ 71

-21x+ 71

Correct answer:

-21x+ 71

Explanation:

7 (2x + 8)- 5 (7x - 3)

$= 7 \cdot 2x +7 \cdot 8- 5 \cdot 7x + 5 \cdot 3$

= 14x +56- 35x + 15

= 14x- 35x +56+ 15

$= \left (14 - 35 \right )x +\left (56+ 15 \right )$

= -21x+ 71

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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 #11 : Simplifying, Distributing, And Factoring

Example Question #12 : Simplifying, Distributing, And Factoring

Example Question #13 : Simplifying, Distributing, And Factoring

Example Question #14 : Simplifying, Distributing, And Factoring

Example Question #11 : Simplifying, Distributing, And Factoring

Example Question #16 : Simplifying, Distributing, And Factoring

Example Question #17 : Simplifying, Distributing, And Factoring

Example Question #11 : Algebra

Example Question #11 : Simplifying, Distributing, And Factoring

Example Question #12 : Algebra

All GED Math Resources

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