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PSAT Math : How to find the value of the coefficient

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Example Questions

Example Question #2 : Binomials

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

$\left ( 2x + \frac{1}{3}\right ) \left ( x- \frac{1}{6}\right ) \left ( x+ \frac{1}{4}\right )$

Possible Answers:

$\frac{1}{2}$

$\frac{1}{6}$

$\frac{11}{12}$

$\frac{1}{4}$

$\frac{7}{6}$

Correct answer:

$\frac{1}{2}$

Explanation:

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:

$\left ( \underline{2x }+ \frac{1}{3}\right ) \left ( \underline{x}- \frac{1}{6}\right ) \left ( x\underline{+ \frac{1}{4}}\right )$

$2x \cdot x \cdot \frac{1}{4} = \frac{1}{2}x^{2}$

$\left ( \underline{2x }+ \frac{1}{3}\right ) \left ( x \underline{- \frac{1}{6}}\right ) \left ( \underline{x}+ \frac{1}{4}\right )$

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

$\left ( 2x\underline{ + \frac{1}{3}}\right ) \left ( \underline{x}- \frac{1}{6}\right ) \left ( \underline{x}+ \frac{1}{4}\right )$

$\frac{1}{3} \cdot x \cdot x= \frac{1}{3}x^{2}$

Add: $\frac{1}{2}x^{2} +\left ( - \frac{1}{3}x^{2} \right ) + \frac{1}{3}x^{2} = \frac{1}{2}x^{2}$

The correct response is $\frac{1}{2}$ .

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Example Question #1 : How To Find The Value Of The Coefficient

Give the coefficient of $x^{5}$ in the binomial expansion of $\left ( 2x+ 0.5\right )^{8}$ .

Possible Answers:

1,680

224

26,880

Correct answer:

224

Explanation:

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

A = 2, B =0.5, k=5, n = 8 :

$C(8,5) \cdot 2^{5} \cdot 0.5 ^{8-5}$

$=C(8,5) \cdot 2^{5} \cdot 0.5 ^{3}$

$= 56\cdot 32 \cdot 0.125$

=224

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Example Question #2 : How To Find The Value Of The Coefficient

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

Possible Answers:

210

140

$\frac{35}{6}$

5,040

Correct answer:

210

Explanation:

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

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

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

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

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

= 210

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Example Question #3 : How To Find The Value Of The Coefficient

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

Possible Answers:

20.16

2,625

315,000

0.168

Correct answer:

0.168

Explanation:

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

A = 0.2, B =5, k=5, n = 7

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

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

$= 21\cdot 0.00032 \cdot 25$

= 0.168

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Example Question #2 : How To Find The Value Of The Coefficient

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

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

Possible Answers:

158

Correct answer:

Explanation:

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

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

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

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

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

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

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

The correct response is -122.

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Example Question #1 : How To Find The Value Of The Coefficient

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

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

Possible Answers:

0.7

1.3

2.5

-0.1

0.5

Correct answer:

-0.1

Explanation:

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

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

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

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

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

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

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

The correct response is -0.1 .

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PSAT Math : How to find the value of the coefficient

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #2 : Binomials

Example Question #1 : How To Find The Value Of The Coefficient

Example Question #2 : How To Find The Value Of The Coefficient

Example Question #3 : How To Find The Value Of The Coefficient

Example Question #2 : How To Find The Value Of The Coefficient

Example Question #1 : How To Find The Value Of The Coefficient

All PSAT Math Resources

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