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GED Math : GED Math

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4 Diagnostic Tests 263 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #5 : Simplifying, Distributing, And Factoring

Multiply:

$\left (100Y^{2}+ 110Y + 121 \right )(10Y-11)$

Possible Answers:

$1,000Y^{3}+ 1,100Y^{2}-1,210Y - 1,331$

$1,000Y^{3}- 1,331$

$1,000Y^{3}- 3,300Y^{2}+3,630Y - 1,331$

$1,000Y^{3}- 1,100Y^{2}+1,210Y - 1,331$

Correct answer:

$1,000Y^{3}- 1,331$

Explanation:

$\left (100Y^{2}+ 110Y + 121 \right )(10Y-11)$

$=\left [\left (10Y \right) ^{2}+ 10Y \cdot 11 + 11^{2} \right] (10Y-11)$

This product fits the difference of cubes pattern, where A = 10Y, B = 11 :

$(A^{2}+AB +B^{2})(A-B) = A^{3}-B^{3}$

$\left [\left (10Y \right) ^{2}+ 10Y \cdot 11 + 11^{2} \right] (10Y-11)$

$= \left ( 10Y\right )^{3} -11^{3} = 1,000Y^{3}-1,331$

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Example Question #4 : Algebra

Give the value of that makes the polynomial $121x^{2}- Nx + 49$ the square of a linear binomial.

Possible Answers:

N = 77

N = 154

N = 231

N = 198

Correct answer:

N = 154

Explanation:

A quadratic trinomial is a perfect square if and only if takes the form

$A^{2} \pm 2AB + B^{2}$ for some values of and .

$121x^{2}- Nx + 49= (11x)^{2} - Nx + 7^{2}$ , so

A = 11x and B = 7 .

For $121x^{2}- Nx + 49$ to be a perfect square, it must hold that

$Nx = 2AB = 2 \cdot 11x \cdot 7 = 154x$ ,

so N = 154 . This is the correct choice.

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Example Question #1 : Algebra

Which of the following is a factor of the polynomial $x^{4}-13x^{2}+ 36$ ?

Possible Answers:

x-9

x - 4

x-1

x-2

Correct answer:

x-2

Explanation:

Perhaps the easiest way to identify the factor is to take advantage of the factor theorem, which states that x-a is a factor of polynomial P(x) if and only if P(a)= 0 . We substitute 1, 2, 4, and 9 for in the polynomial to identify the factor.

x=1 :

$1^{4}-13 \cdot 1^{2}+ 36$

= 1 - 13 + 36

= 24

x=2 :

$2^{4}-13 \cdot 2^{2}+ 36$

$= 16- 13 \cdot 4 + 36$

= 16 -52 + 36

= 0

x =4 :

$4^{4}-13 \cdot 4^{2}+ 36$

$= 256- 13 \cdot 16 + 36$

= 256 -208 +36

= 84

x=9 :

$9^{4}-13 \cdot 9^{2}+ 36$

$= 6,561- 13 \cdot 81 + 36$

= 6,561- 1,053+ 36

= 5,544

Only x = 2 makes the polynomial equal to 0, so among the choices, only x-2 is a factor.

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Example Question #5 : Algebra

Which of the following is a prime factor of $x^{6} +1$ ?

Possible Answers:

$x^{2}-x+1$

x+1

$x^{2}+x+1$

$x^{2}+1$

Correct answer:

$x^{2}+1$

Explanation:

$x^{6} +1$ is the sum of two cubes:

$x^{6} +1 =\left ( x^{2} \right ) ^{3}+1^{3}$

As such, it can be factored using the pattern

$A^{3}+ B^{3}= (A+B)(A^{2}-AB+B^{2})$

where $A = x^{2}, B = 1$ ;

$x^{6} +1 =\left ( x^{2} \right ) ^{3}+1^{3}$

$= (x^{2}+1)((x^{2})^{2}-x^{2} \cdot 1 + 1^{2})$

$= (x^{2}+1)(x^{4}-x^{2} + 1)$

The first factor,as the sum of squares, is a prime.

We try to factor the second by noting that it is "quadratic-style" based on $x^{2}$ . and can be written as

$= (x^{2})^{2}-x^{2} + 1$ ;

we seek to factor it as $(x^{2}+\; \; \; )(x^{2}+\; \; \; )$

We want a pair of integers whose product is 1 and whose sum is . These integers do not exist, so $x^{4}-x^{2} + 1$ is a prime.

$(x^{2}+1)(x^{4}-x^{2} + 1)$ is the prime factorization and the correct response is $x^{2}+1$ .

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Example Question #1 : Algebra

Which of the following is a factor of the polynomial $x^{6} - 66x^{3} + 128$

Possible Answers:

x+2

x+4

x-4

x-2

Correct answer:

x-4

Explanation:

Perhaps the easiest way to identify the factor is to take advantage of the factor theorem, which states that x-a is a factor of polynomial P(x) if and only if P(a)= 0 . We substitute $\pm2$ and $\pm 4$ for in the polynomial to identify the factor.

x = 2 :

$x^{6} - 66x^{3} + 128$

$2^{6} - 66 \cdot 2^{3} + 128$

$= 64 - 66 \cdot 8 + 128$

= 64 - 528 + 128

= -336

x = -2 :

$x^{6} - 66x^{3} + 128$

$\left (-2 \right )^{6} - 66 \cdot \left (-2 \right )^{3} + 128$

$= 64 - 66 \cdot\left ( -8 \right )+ 128$

= 64 + 528 + 128

= 720

x = 4 :

$x^{6} - 66x^{3} + 128$

$4^{6} - 66 \cdot 4^{3} + 128$

$= 4,096- 66 \cdot 64 + 128$

= 4,096- 4,224 + 128

= 0

x = -4 :

$x^{6} - 66x^{3} + 128$

$\left (-4 \right )^{6} - 66 \cdot \left (-4 \right )^{3} + 128$

$= 4,096- 66 \cdot\left ( -64 \right )+ 128$

= 4,096+4,224 + 128

= 8,448

Only x = 4 makes the polynomial equal to 0, so of the four choices, only x - 4 is a factor of the polynomial.

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Example Question #11 : Algebra

Express 286 in base five.

Possible Answers:

$2101_{\textrm{five}}$

$2221_{\textrm{five}}$

$2121_{\textrm{five}}$

$2201_{\textrm{five}}$

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 - 3x - 3y

-5x^2 - 3x

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

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

-2x^2 - 3y^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 +4x-12

-x^2 +8x-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 #561 : Ged Math

Simplify:

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

Possible Answers:

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

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

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

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

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 #15 : Simplifying, Distributing, And Factoring

Factor completely:

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

Possible Answers:

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

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

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

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

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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All GED Math Resources

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GED Math : GED Math

Study concepts, example questions & explanations for GED Math

All GED Math Resources

Example Questions

Example Question #5 : Simplifying, Distributing, And Factoring

Example Question #4 : Algebra

Example Question #1 : Algebra

Example Question #5 : Algebra

Example Question #1 : Algebra

Example Question #11 : Algebra

Example Question #12 : Simplifying, Distributing, And Factoring

Example Question #13 : Simplifying, Distributing, And Factoring

Example Question #561 : Ged Math

Example Question #15 : Simplifying, Distributing, And Factoring

All GED Math Resources

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