Binomials - ACT Math

Card 0 of 279

Question

Solve for x when 6x – 4 = 2x + 5

Answer

Solve by simplifying:

6x – 4 = 2x + 5

6x = 2x + 9

4x = 9

x = 9/4

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Question

What is the value of the following equation if and ?

$\small 2t^2+4vt+2v^2$

Answer

Substitute the numbers 3 and –4 for t and v, respectively.

$\small 2(3)^2+4(-4)(3)+2(-4)^2$

$\small 18+(-48)+32=2$

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Question

Simplify the following binomial:

$4x^2 + 16x - 4 - 2x^{2}$

Answer

The equation that is presented is:

$4x^2 + 16x - 4 - 2x^{2}$

To get the correct answer, you first need to combine all of the like terms. So, you can subtract the from the , leaving you with:

From there, you can reduce the numbers by their greatest common denominator, in this case, :

Then you have arrived at your final answer.

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Question

Simplify the following binomial:

$17x^{2} - 15x + 3x^{2 }+10x$

Answer

The equation presented in the problem is:

$17x^{2} - 15x + 3x^{2 }+10x$

First you have to combine the like terms, i.e. combining all instances of and :

Then, you can factor out the common to get your answer

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Question

Simplify the following binomial expression:

Answer

First, combine all of the like terms that you are able:

Then, reduce by the greatest common denominator (in this case, ):

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Question

Solve for x when 6x – 4 = 2x + 5

Answer

Solve by simplifying:

6x – 4 = 2x + 5

6x = 2x + 9

4x = 9

x = 9/4

Compare your answer with the correct one above

Question

What is the value of the following equation if and ?

$\small 2t^2+4vt+2v^2$

Answer

Substitute the numbers 3 and –4 for t and v, respectively.

$\small 2(3)^2+4(-4)(3)+2(-4)^2$

$\small 18+(-48)+32=2$

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Question

Simplify the following binomial:

$4x^2 + 16x - 4 - 2x^{2}$

Answer

The equation that is presented is:

$4x^2 + 16x - 4 - 2x^{2}$

To get the correct answer, you first need to combine all of the like terms. So, you can subtract the from the , leaving you with:

From there, you can reduce the numbers by their greatest common denominator, in this case, :

Then you have arrived at your final answer.

Compare your answer with the correct one above

Question

Simplify the following binomial:

$17x^{2} - 15x + 3x^{2 }+10x$

Answer

The equation presented in the problem is:

$17x^{2} - 15x + 3x^{2 }+10x$

First you have to combine the like terms, i.e. combining all instances of and :

Then, you can factor out the common to get your answer

Compare your answer with the correct one above

Question

Simplify the following binomial expression:

Answer

First, combine all of the like terms that you are able:

Then, reduce by the greatest common denominator (in this case, ):

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Question

Solve for x when 6x – 4 = 2x + 5

Answer

Solve by simplifying:

6x – 4 = 2x + 5

6x = 2x + 9

4x = 9

x = 9/4

Compare your answer with the correct one above

Question

What is the value of the following equation if and ?

$\small 2t^2+4vt+2v^2$

Answer

Substitute the numbers 3 and –4 for t and v, respectively.

$\small 2(3)^2+4(-4)(3)+2(-4)^2$

$\small 18+(-48)+32=2$

Compare your answer with the correct one above

Question

Simplify the following binomial:

$4x^2 + 16x - 4 - 2x^{2}$

Answer

The equation that is presented is:

$4x^2 + 16x - 4 - 2x^{2}$

To get the correct answer, you first need to combine all of the like terms. So, you can subtract the from the , leaving you with:

From there, you can reduce the numbers by their greatest common denominator, in this case, :

Then you have arrived at your final answer.

Compare your answer with the correct one above

Question

Simplify the following binomial:

$17x^{2} - 15x + 3x^{2 }+10x$

Answer

The equation presented in the problem is:

$17x^{2} - 15x + 3x^{2 }+10x$

First you have to combine the like terms, i.e. combining all instances of and :

Then, you can factor out the common to get your answer

Compare your answer with the correct one above

Question

Simplify the following binomial expression:

Answer

First, combine all of the like terms that you are able:

Then, reduce by the greatest common denominator (in this case, ):

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Question

What is the value of the coefficient in front of the term that includes $x^{2}y^{7}$ in the expansion of $\left ( 2x-y \right )^{9}$ ?

Answer

Using the binomial theorem, the term containing the _x_2 _y_7 will be equal to

(2_x_)2(–y)7

=36(–4_x_2 _y_7)= -144_x_2_y_7

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Question

Give the coefficient of $x^{5}$ in the binomial expansion of $\left ( 0.2x+ 5 \right )^{7}$ .

Answer

If the expression $\left ( A x + B\right )^{n}$ is expanded, then by the binomial theorem, the $x^{k}$ term is

$C(n, k) \cdot\left ( Ax \right )^{k} \cdot B ^{n - k}$

$= C(n, k) \cdot A ^{k} \cdot B ^{n - k} \cdot x^{k}$

or, equivalently, the coefficient of $x^{k}$ is

$C(n, k) \cdot A ^{k} \cdot B ^{n - k}$

Therefore, the $x^{5}$ coefficient can be determined by setting

$C(7,5) \cdot 0.2^{5} \cdot 5 ^{7 - 5}$

$= C(7,5) \cdot 0.2 ^{5} \cdot 5 ^{2}$

$= 21\cdot 0.00032 \cdot 25$

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Question

Give the coefficient of $x^{4}$ in the binomial expansion of $\left (6x+ \frac{1}{6}\right )^{7}$ .

Answer

If the expression $\left ( A x + B\right )^{n}$ is expanded, then by the binomial theorem, the $x^{k}$ term is

$C(n, k) \cdot\left ( Ax \right )^{k} \cdot B ^{n - k}$

$= C(n, k) \cdot A ^{k} \cdot B ^{n - k} \cdot x^{k}$

or, equivalently, the coefficient of $x^{k}$ is

$C(n, k) \cdot A ^{k} \cdot B ^{n - k}$

Therefore, the $x^{4}$ coefficient can be determined by setting

$A = 6, B = \frac{1}{6}, n = 7, k= 4$ :

$C(7,4) \cdot 6 ^{4} \cdot\left ( \frac{1}{6} \right ) ^{7-4}$

$=C(7,4) \cdot 6 ^{4} \cdot\left ( \frac{1}{6} \right ) ^{3}$

$=35 \cdot 1,296 \cdot \frac{1}{216}$

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Question

Give the coefficient of $x^{2}$ in the product

$\left ( x+ 0.4\right ) (x - 0.2) (3x-0.7)$ .

Answer

While this problem can be answered by multiplying the three binomials, it is not necessary. There are three ways to multiply one term from each binomial such that two terms and one constant are multiplied; find the three products and add them, as follows:

$\left (\underline{ x}+ 0.4\right ) (\underline{x} - 0.2) (3x\underline{-0.7})$

$x \cdot x \cdot (-0.7) = -0.7x^{2}$

$\left (\underline{ x}+ 0.4\right ) (x \underline{-0.2})(\underline{3x}-0.7)$

$x \cdot (-0.2) \cdot 3x= -0.6x^{2}$

$\left (x+ \underline{0.4}\right ) (\underline{x} - 0.2)(\underline{3x}-0.7)$

$0.4 \cdot x \cdot 3x = 1.2 x^{2}$

Add: $-0.7x^{2}+ (-0.6x^{2})+ 1.2x^{2} = -0.1x^{2}$ .

The correct response is .

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Question

Give the coefficient of $x^{2}$ in the product

$\left (3x- 7 \right ) \left ( 4x+3\right )\left ( 2x-7\right )$ .

Answer

While this problem can be answered by multiplying the three binomials, it is not necessary. There are three ways to multiply one term from each binomial such that two terms and one constant are multiplied; find the three products and add them, as follows:

$\left (\underline{3x}- 7 \right ) \left ( \underline{4x}+3\right )\left ( 2x\underline{-7}\right )$

$3x \cdot 4x \cdot (-7) = -84x^{2}$

$\left (\underline{3x}- 7 \right ) \left ( 4x\underline{+3}\right )\left ( \underline{2x}-7\right )$

$3x \cdot 3 \cdot 2x = 18x^{2}$

$\left (3x\underline{- 7} \right ) \left ( \underline{4x}+3\right )\left ( \underline{2x}-7\right )$

$-7 \cdot 4x \cdot 2x = -56x^{2}$

Add: $-84x^{2} + 18x^{2} - 56x^{2} = -122x^{2}$

The correct response is -122.

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