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PSAT Math : Variables

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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 $\left ( 0.2x+ 5 \right )^{7}$ .

Possible Answers:

0.168

315,000

20.16

2,625

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 #6 : 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:

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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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:

0.5

2.5

0.7

1.3

-0.1

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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Example Question #41 : Algebra

Multiply the binomial.

(x-3)(x+4)

Possible Answers:

$x^{2}-x-12$

$x^{2}-12$

$x^{2}+x+12$

$x^{2}+x-12$

Correct answer:

$x^{2}+x-12$

Explanation:

By multiplying with the foil method, we multiply our first values giving x^2 , our outside values giving . our inside values which gives -3x , and out last values giving .

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Example Question #1 : How To Factor A Trinomial

Factor the following expression completely:

x^4-8x^3+12x^2

Possible Answers:

x^2(x-6)(x-2)

x^2(x-3)(x-4)

x^2(x+6)(x+2)

x^2(x^2-8x+12)

x^2(x+3)(x+4)

Correct answer:

x^2(x-6)(x-2)

Explanation:

We must begin by factoring out x^2 from each term.

x^2(x^2-8x+12)

Next, we must find two numbers that sum to and multiply to .

(-6)+(-2)=-8

-6*-2=12

Thus, our final answer is:

x^2(x-6)(x-2)

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Example Question #1 : How To Factor A Trinomial

Factor the following trinomial:

x^2-x-12

Possible Answers:

(x+3)(x-4)

(x-1)(x-12)

(x+4)(x-3)

(x-6)(x+2)

(x-3)(x-4)

Correct answer:

(x+3)(x-4)

Explanation:

x^2-x-12

To trinomial is in ax+bx+c form

In order to factor, find two numbers whose procuct is , in this case , and whose sum is , in this case

Factors of :

-1, 12; 1, -12; -3,4; 3,-4;-2,6; 2,-6

Which of these pairs has a sum of ?

and

Therefore the factored form of x^2-x-12 is:

(x+3)(x-4)

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Example Question #41 : Algebra

Factor the trinomial.

$x^{2}-5x-14$

Possible Answers:

(x+7)(x-2)

(x-7)(x+2)

(x-3)(x-2)

(x-8)(x+3)

Correct answer:

(x-7)(x+2)

Explanation:

Our factors will need to have a product of (-14) , and a sum of (-5) , so our factors must be (-7) and .

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Example Question #1 : Trinomials

(x^2+y^2+z^2)(x^2+y^2+z^2)

Possible Answers:

x^4+y^4+z^4+2x^2y^2+2y^2z^2+2x^2z^2

x^2y^2+y^2z^2+x^2z^2

x^4+x^2y^2+y^2z^2+z^4

2x^2y^2+z^4+y^4+2z^2y^2

x^4+y^4+z^4

Correct answer:

x^4+y^4+z^4+2x^2y^2+2y^2z^2+2x^2z^2

Explanation:

(x^2+y^2+z^2)(x^2+y^2+z^2)

Use the distributive property:

x^2(x^2+y^2+z^2)+y^2(x^2+y^2+z^2)+z^2(x^2+y^2+z^2)

x^4+x^2y^2+x^2z^2 + x^2y^2+y^4+y^2z^2+x^2z^2+y^2z^2+z^4

Combine like terms:

x^4+y^4+z^4+2x^2y^2+2y^2z^2+2x^2z^2

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Example Question #2 : How To Multiply Trinomials

Find the product:

(x^2+7x+4)(x^2-3x-2)

Possible Answers:

x^4+4x^3-19x^2-26x-8

x^4-10x^3+27x^2-14x-4

x^4-21x-8

2x^2+4x+2

2x^2-4x-2

Correct answer:

x^4+4x^3-19x^2-26x-8

Explanation:

Find the product:

(x^2+7x+4)(x^2-3x-2)

Step 1: Use the Distributive Property

x^2(x^2-3x-2)+7x(x^2-3x-2)+4(x^2-3x-2)

x^4-3x^3-2x^2+7x^3-21x^2-14x+4x^2-12x-8

Step 2: Combine like terms

x^4+4x^3-19x^2-26x-8

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Example Question #3 : How To Multiply Trinomials

Find the product:

(2x^2+3y+6)(7x^2+4y+1)

Possible Answers:

14x^2+8x^2y+12y^2+8

14x^2+29x^2y+44x^2+12y^2+27y+6

21x^2y+12y^2+3y

14x^4+8x^2y+2x^2

42x^2+24y+6

Correct answer:

14x^2+29x^2y+44x^2+12y^2+27y+6

Explanation:

Find the product:

(2x^2+3y+6)(7x^2+4y+1)

Use the distributive property:

2x^2(7x^2+4y+1)+3y(7x^2+4y+1)+6(7x^2+4y+1)

14x^4+8x^2y+2x^2+21x^2y+12y^2+3y+42x^2+24y+6

14x^4+29x^2y+44x^2+12y^2+27y+6

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PSAT Math : Variables

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

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

Example Question #6 : How To Find The Value Of The Coefficient

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

Example Question #41 : Algebra

Example Question #1 : How To Factor A Trinomial

Example Question #1 : How To Factor A Trinomial

Example Question #41 : Algebra

Example Question #1 : Trinomials

Example Question #2 : How To Multiply Trinomials

Example Question #3 : How To Multiply Trinomials

All PSAT Math Resources

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