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SAT Math : Polynomials

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Example Questions

Example Question #1 : How To Divide Polynomials

Simplify: $\frac{6x^7y^3z^9}{3x^6y^3z}$

Possible Answers:

xyz^8

2xz^8

6xyz^8

xz^8

xyz

Correct answer:

2xz^8

Explanation:

Cancel by subtracting the exponents of like terms:

$\frac{6x^7y^3z^9}{3x^6y^3z} = 2x^{7-6}y^{3-3}z^{9-1}=2xy^0z^8=2xz^8$

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Example Question #12 : Polynomials

Divide $x^{3} + 125y ^{3}$ by x+ 5y .

Possible Answers:

$x ^{2}+ 5xy + 25y^{2}$

$x ^{2}+10xy + 25y^{2}$

$x ^{2} + 25y^{2}$

$x ^{2}-10xy + 25y^{2}$

$x ^{2}-5xy + 25y^{2}$

Correct answer:

$x ^{2}+ 5xy + 25y^{2}$

Explanation:

It is not necessary to work a long division if you recognize $x^{3} + 125y ^{3}$ as the sum of two perfect cube expressions:

$x^{3} + 125y ^{3} = x^{3} + 5^{3} y ^{3} = x^{3} + (5y )^{3}$

A sum of cubes can be factored according to the pattern

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

so, setting A = x, B = 5y ,

$x^{3} + (5y )^{3} =(x+5y) [x ^{2}+ x \cdot 5y + (5y)^{2}]$

$=(x+5y)(x ^{2}+ 5xy + 25y^{2})$

Therefore,

$\frac{x^{3} + 125y ^{3}}{x+ 5y} = \frac{(x+5y)(x ^{2}+ 5xy + 25y^{2})}{x+ 5y} = x ^{2}+ 5xy + 25y^{2}$

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Example Question #374 : Algebra

By what expression can 3x+7 be multiplied to yield the product $6x^{3} +20x^{2} +5x-21$ ?

Possible Answers:

$2x^{2 }+ x - 3$

$2x^{2 } - 3$

$2x^{2 }-x - 3$

$2x^{2 }+ 2x - 3$

Correct answer:

$2x^{2 }+ 2x - 3$

Explanation:

Divide $6x^{3} +20x^{2} +5x-21$ by 3x+7 by setting up a long division.

Divide the lead term of the dividend, $6x^{3}$ , by that of the divisor, ; the result is $\frac{6x^{3}}{3x} = 2 x^{2}$

Enter that as the first term of the quotient. Multiply this by the divisor:

$2 x^{2} (3x+7) = 2 x^{2} (3x)+2 x^{2} ( 7) = 2x^{3} + 14x^{2}$

Subtract this from the dividend. This is shown in the figure below.

Division poly

Repeat the process with the new difference:

$\frac{6x^{2}}{3x} = 2 x$

$2 x (3x+7) = 2 x (3x)+2 x ( 7) = 2x^{2} + 14x$

Division poly

Repeating:

$\frac{-9x}{3x} = -3$

-3 (3x+7) = -3 (3x)+ (-3) ( 7) = -9x+ 21

Division poly

The quotient - and the correct response - is $2x^{2 }+ 2x - 3$ .

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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:

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

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

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

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

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

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

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

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

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

x^8+7x^4-10x^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 #12 : 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:

The correct answer is not among the other choices.

237

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 #14 : 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 #2 : How To Multiply Polynomials

and represent positive quantities.

$x^{2} = 20$

$y^{2} = 5$

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

Possible Answers:

$5 \sqrt{5}$

$45 \sqrt{15 }$

$35 \sqrt{15 }$

$35 \sqrt{5}$

$45\sqrt{5}$

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 #13 : Polynomials

Simplify the following expression:

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

Possible Answers:

$-3q^{2}+12$

-q^2+4q+2

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

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

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:

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

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

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

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

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

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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SAT Math : Polynomials

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #1 : How To Divide Polynomials

Example Question #12 : Polynomials

Example Question #374 : Algebra

Example Question #1 : How To Multiply Polynomials

Example Question #11 : Polynomials

Example Question #12 : Polynomials

Example Question #14 : Polynomials

Example Question #2 : How To Multiply Polynomials

Example Question #13 : Polynomials

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

All SAT Math Resources

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