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SAT Math : Variables

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Example Question #1 : How To Multiply Polynomials

$F(x) = x^{3} + x^{2} - x + 2$

and

$G(x) = x^{2} + 5$

What is FG(x) ?

Possible Answers:

$(FG)(x) = x^{5} + x^{4} - x - 2$

$(FG)(x) = x^{3} + 2x^{2} - x + 7$

$(FG)(x) = x^{5} + x^{4} +4x^{3} + 7x^{2} - 5x +10$

$(FG)(x) = x^{5} + x^{4} - x^{3} + 2x^{2} - 5x -10$

$(FG)(x) = x^{3} - x - 3$

Correct answer:

$(FG)(x) = x^{5} + x^{4} +4x^{3} + 7x^{2} - 5x +10$

Explanation:

$(FG)(x) = F(x)G(x)$ so we multiply the two function to get the answer. We use $x^{m}x^{n} = x^{m+n}$

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Example Question #11 : Polynomials

Find the product:

( x^4-5x+7)(x^4+2x+1)

Possible Answers:

x^8+7x^4-10x^2

x^4+8x^2+10x-9

x^8-3x^5+8x^4-10x^2+9x+7

x^7+9x^4-10x^2+8

x^6+5x^5+3x^4+x^2

Correct answer:

x^8-3x^5+8x^4-10x^2+9x+7

Explanation:

Find the product:

( x^4-5x+7)(x^4+2x+1)

Step 1: Use the distributive property.

x^4(x^4+2x+1)-5x(x^4+2x+1)+7(x^4+2x+1)

x^8+2x^5+x^4-5x^5-10x^2-5x+7x^4+14x+7

Step 2: Combine like terms.

x^8+2x^5-5x^5+x^4+7x^4-10x^2-5x+14x+7

x^8-3x^5+8x^4-10x^2+9x+7

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Example Question #3 : How To Multiply Polynomials

represents a positive quantity; represents a negative quantity.

$x^{4} = 116$

$y^{2} = 11$

Evaluate $(x- y)(x+y)(x^{2} + y^{2})$

Possible Answers:

237

The correct answer is not among the other choices.

Correct answer:

Explanation:

The first two binomials are the difference and the sum of the same two expressions, which, when multiplied, yield the difference of their squares:

$(x- y)(x+y)(x^{2} + y^{2})$

$=(x^{2}- y^{2}) (x^{2} + y^{2})$

Again, a sum is multiplied by a difference to yield a difference of squares, which by the Power of a Power Property, is equal to:

$(x^{2}) ^{2} - (y^{2}) ^{2}$

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

$= x^{4} - y^{4}$

$y^{2} = 11$ , so by the Power of a Power Property,

$(y^{2} )^{2}= 11^{2}$

$y^{2\cdot 2} = 121$

$y^{4} = 121$

Also, $x^{4} = 116$ , so we can now substitute accordingly:

$(x- y)(x+y)(x^{2} + y^{2})$

$= x^{4} - y^{4}$

= 116 - 121

= -5

Note that the signs of and are actually irrelevant to the problem.

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Example Question #3 : How To Multiply Polynomials

represents a positive quantity; represents a negative quantity.

$x^{6} = 625$

$y ^{6} = 64$

Evaluate $(x-y) (x^{2}+xy+y^{2})$ .

Possible Answers:

Correct answer:

Explanation:

$(x-y) (x^{2}+xy+y^{2})$ can be recognized as the pattern conforming to that of the difference of two perfect cubes:

$(x-y) (x^{2}+xy+y^{2}) = x^{3} - y^{3}$

Additionally, by way of the Power of a Power Property,

$(x^{3})^{ 2 } = x^{3 \cdot 2} = x^{6 }$ , making $x^{3}$ a square root of $x^{6}$ , or 625; since is positive, so is $x^{3}$ , so

$x^{3} = \sqrt{625} = 25$ .

Similarly, $y ^{3}$ is a square root of $y ^{6}$ , or 64; since is negative, so is $y^{3}$ (as an odd power of a negative number is negative), so

$y^{3} = - \sqrt{64} = -8$ .

Therefore, substituting:

$(x-y) (x^{2}+xy+y^{2}) = x^{3} - y^{3} = 25 - (-8) = 25+8= 33$ .

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Example Question #5 : How To Multiply Polynomials

and represent positive quantities.

$x^{2} = 20$

$y^{2} = 5$

Evaluate $(x-y) (x^{2}+xy+y^{2})$ .

Possible Answers:

$35 \sqrt{5}$

$5 \sqrt{5}$

$45 \sqrt{15 }$

$45\sqrt{5}$

$35 \sqrt{15 }$

Correct answer:

$35 \sqrt{5}$

Explanation:

$(x-y) (x^{2}+xy+y^{2})$ can be recognized as the pattern conforming to that of the difference of two perfect cubes:

$(x-y) (x^{2}+xy+y^{2}) = x^{3} - y^{3}$

Additionally,

$x^{2} = 20$ and is positive, so

$x = \sqrt{20}$

Using the product of radicals property, we see that

$x = \sqrt{4 \cdot 5} = \sqrt{4 } \cdot \sqrt{ 5} = 2 \sqrt{ 5}$

and

$x ^{3 } = x^{2} \cdot x = 20 \cdot 2 \sqrt{5} = 40 \sqrt{5}$

$y^{2} = 5$ and is positive, so

$y = \sqrt{5}$ ,

and

$y^{3}= y^{2} \cdot y = 5 \sqrt{5}$

Substituting for $x ^{3 }$ and $y^{3}$ , then collecting the like radicals,

$(x-y) (x^{2}+xy+y^{2})$

$= x^{3} - y^{3}$

$= 40 \sqrt{5}-5 \sqrt{5}$

$=( 40 -5 )\sqrt{5}$

$=35 \sqrt{5}$ .

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Example Question #11 : Polynomials

Simplify the following expression:

$(2q^{2}+4q+7)-(3q^{2}-5)$

Possible Answers:

-q^2+4q+2

$-q^{2}+4q+12$

$-3q^{2}+12$

$-5q^{2}+4q+2$

Correct answer:

$-q^{2}+4q+12$

Explanation:

This is not a FOIL problem, as we are adding rather than multiplying the terms in parentheses.

Add like terms together:

$2q^{2}-3q^{2}=-1q^{2}=-q^2$

has no like terms.

7-(-5)=12

Combine these terms into one expression to find the answer:

$-q^{2}+4q+12$

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Example Question #1 : How To Find The Solution To A Binomial Problem

Define an operation $\heartsuit$ on the set of real numbers as follows:

For all real m, n ,

$m \heartsuit n = \left ( m-4n\right )^{3}$

How else could this operation be defined?

Possible Answers:

$m \heartsuit n = m ^{3} -64n ^{3}$

$m \heartsuit n = m ^{3} +12 m ^{2} n - 48mn ^{2}-64n ^{3}$

$m \heartsuit n = m ^{3} -4 m ^{2} n + 16mn ^{2}- 64n ^{3}$

$m \heartsuit n = m ^{3} -12 m ^{2} n + 48mn ^{2}- 64n ^{3}$

$m \heartsuit n = m ^{3} +4 m ^{2} n - 16mn ^{2}- 64n ^{3}$

Correct answer:

$m \heartsuit n = m ^{3} -12 m ^{2} n + 48mn ^{2}- 64n ^{3}$

Explanation:

$\left ( m-4n\right )^{3}$ , as the cube of a binomial, can be rewritten using the following pattern:

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot m ^{2} \cdot 4n + 3 \cdot m \cdot (4n) ^{2}- (4n) ^{3}$

Applying the rules of exponents to simplify this:

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot 4 \cdot m ^{2} \cdot n + 3 \cdot m \cdot 4 ^{2}\cdot n ^{2}- 4^{3} \cdot n ^{3}$

$\left ( m-4n\right )^{3} = m ^{3} - 3 \cdot 4 \cdot m ^{2} \cdot n + 3 \cdot 16 \cdot m\cdot n ^{2}- 4^{3} \cdot n ^{3}$

$\left ( m-4n\right )^{3} = m ^{3} -12 m ^{2} n + 48mn ^{2}- 64n ^{3}$

Therefore, the correct choice is that, alternatively stated,

$m \heartsuit n = m ^{3} -12 m ^{2} n + 48mn ^{2}- 64n ^{3}$ .

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Example Question #31 : Variables

Solve for .

$\frac{x^{2}+2x-8}{x^{2}+5x-14}=\frac{3}{4}$

Possible Answers:

Correct answer:

Explanation:

$\frac{x^{2}+2x-8}{x^{2}+5x-14}=\frac{3}{4}$

Factor the expression

numerator: find two numbers that add to 2 and multiply to -8 [use 4,-2]

denominator: find two numbers that add to 5 and multiply to -14 [use 7,-2]

new expression:

$\frac{(x+4)(x-2)}{(x+7)(x-2)}=\frac{3}{4}$

Cancel the x-2 and cross multiply.

4(x+4)=3(x+7)

4x+16=3x+21

x=5

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Example Question #1 : How To Simplify Binomials

If 〖(x+y)〗²= 144 and 〖(x-y)〗²= 64, what is the value of xy?

Possible Answers:

Correct answer:

Explanation:

We first expand each binomial to get x² + 2xy + y² = 144 and x² - 2xy + y² = 64. We then subtract the second equation from the first to find 4xy = 80. Finally, we divide each side by 4 to find xy = 20.

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Example Question #1 : How To Factor A Trinomial

Solve each problem and decide which is the best of the choices given.

What are the zeros of the following trinomial?

3x^3-30x^2+63x

Possible Answers:

x=7,3

x=-7,-3

x=3,7,-3

x=0,7,3

x=0,-7,-3

Correct answer:

x=0,7,3

Explanation:

First factor out a . Then the factors of the remaining polynomial,

3x(x^2-10x+21)=0 , are (x-7) and (x-3) .

Set everything equal to zero and you get x=3 , x=7 , and x=0 because you cant forget to set equal to zero.

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SAT Math : Variables

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #1 : How To Multiply Polynomials

Example Question #11 : Polynomials

Example Question #3 : How To Multiply Polynomials

Example Question #3 : How To Multiply Polynomials

Example Question #5 : How To Multiply Polynomials

Example Question #11 : Polynomials

Example Question #1 : How To Find The Solution To A Binomial Problem

Example Question #31 : Variables

Example Question #1 : How To Simplify Binomials

Example Question #1 : How To Factor A Trinomial

All SAT Math Resources

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