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High School Math : Intermediate Single-Variable Algebra

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Example Questions

Example Question #1 : Polynomials

If $f(x)=x^{3}-1$ and $g(x)=x^{2}+1$ , what is g(f(2)) ?

Possible Answers:

124

Correct answer:

Explanation:

g(f(2)) is a composite function where f(2) is plugged into g(x) :

$f(2)=(2)^{3}-1=8-1=7$

$g(7)=(7)^{2}+1=49+1=50$

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Example Question #1 : Simplifying Polynomials

Simplify the following expression:

$3x^9y^4 \cdot (2y)^3$ .

Possible Answers:

$24x^9y^{12}$

11x^9y^7

18x^9y^6

24x^9y^7

Correct answer:

24x^9y^7

Explanation:

First, multiply out the second expression so that you get 8y^3 .

Then, multiply your like terms, taking care to remember that when multiplying terms that have the same base, you add the exponents. Thus, you get $24x^9y^{12}$ .

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Example Question #2 : Simplifying Polynomials

Simplify:

$\frac{13x^2y^5z^9}{169xy^5z^4}$

Possible Answers:

xy^5

$\frac{xz^5}{169}$

$\frac{xz^5}{13}$

xz^5

$\frac{x}{13}$

Correct answer:

$\frac{xz^5}{13}$

Explanation:

Focus on each pair of like terms. The y's completely cancel out, there is one left on top, and five z's left on the bottom.

$\frac{13}{169}$ reduces to $\frac{1}{13}$ .

Put that all together to get $\frac{xz^5}{13}$ .

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Example Question #1 : Simplifying Exponents

Simplify.

$x^{-6}x^3x^9$

Possible Answers:

x^6

$x^{-12}$

$x^{-6}$

$x^{12}$

Correct answer:

x^6

Explanation:

Put the negative exponent on the bottom so that you have $\frac{x^{12}}{x^{6}}$ which simplifies further to $x^{6}$ .

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Example Question #1 : Simplifying Polynomials

Expand this expression:

(x-4) (2x+4)

Possible Answers:

2x^2 - 4x - 16

2x^2 - 4x + 16

2x - 4x - 16

2x^2 + 4x + 16

2x^2 + 4x - 16

Correct answer:

2x^2 - 4x - 16

Explanation:

Use the FOIL method (First, Outer, Inner, Last):

x * 2x = 2x^2

x * 4 = 4x

-4 * 2x = -8x

-4 * 4 = -16

Put all of these terms together:

2x^2 + 4x - 8x - 16

Combine like terms:

2x^2 - 4x - 16

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Example Question #1 : Simplifying Polynomials

Simplify the following polynomial:

$ab^{-2}(a^2b+a^{-1}b^3-a^3b^{-1})$

Possible Answers:

$\frac{a^3}{b}+a-\frac{a^4}{b^3}$

$\frac{a^2}{b}+b-\frac{a^4}{b^3}$

$\frac{a^3}{b}+b-\frac{a^3}{b^4}$

$\frac{a^3}{b}+b-\frac{a^4}{b^3}$

$\frac{a^3}{b}+b^2-\frac{a^4}{b^3}$

Correct answer:

$\frac{a^3}{b}+b-\frac{a^4}{b^3}$

Explanation:

To simplify the polynomial, begin by multiplying the first binomial by every term within the parentheses:

$ab^{-2}(a^2b+a^{-1}b^3-a^3b^{-1})$

$aa^2b^{-2}b+aa^{-1}b^{-2}b^3-aa^3b^{-2}b^{-1}$

Now, combine like terms:

$a^3b^{-1}+b^1-a^4b^{-3}$

Convert the polynomial into fraction form:

$\frac{a^3}{b}+b-\frac{a^4}{b^3}$

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Example Question #1 : Simplifying Polynomials

Simplify the following polynomial:

$\frac{-18a^{-2}b^3c^{-4}}{12ab^{-2}c^{-3}}$

Possible Answers:

$\frac{-3b^5}{2a^3c^2}$

$\frac{-3b^4}{2a^3c}$

$\frac{-3b^5}{2a^3c}$

$\frac{-3b^5}{2ac}$

$\frac{-5b^5}{2a^3c}$

Correct answer:

$\frac{-3b^5}{2a^3c}$

Explanation:

To simplify the polynomial, begin by rearranging the terms to have positive exponents:

$\frac{-18a^{-2}b^3c^{-4}}{12ab^{-2}c^{-3}}$

$\frac{-18b^3b^2c^{3}}{12aa^2c^{4}}$

Now, combine like terms:

$\frac{-18b^5}{12a^3c^{1}}$

Simplify the integers:

$\frac{-3b^5}{2a^3c}$

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Example Question #1 : Simplifying Polynomials

Simplify the following polynomial:

$(5n^{3x}-2m^{x+2})^2$

Possible Answers:

$5n^{6x}-20n^{3x}m^{x+2}+4m^{2x+4}$

$25n^{6x}-20n^{3x}m^{x+2}+4m^{2x+4}$

$10n^{6x}-20n^{3x}m^{x+2}+2m^{2x+4}$

$25n^{6x}-20n^{3x}m^{x+2}+2m^{2x+4}$

$25n^{6x}-10n^{3x}m^{x+2}+4m^{2x+4}$

Correct answer:

$25n^{6x}-20n^{3x}m^{x+2}+4m^{2x+4}$

Explanation:

Squaring the polynomial is equivalent to:

$(5n^{3x}-2m^{x+2})^2$

$(5n^{3x}-2m^{x+2})(5n^{3x}-2m^{x+2})$

Use the FOIL method to multiply the terms:

F - First

O - Outer

I - Inner

L - Last

$25n^{6x}-10n^{3x}m^{x+2}-10n^{3x}m^{x+2}+4m^{2x+4}$

Combine like terms:

$25n^{6x}-20n^{3x}m^{x+2}+4m^{2x+4}$

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Example Question #1 : Simplifying Polynomials

Simplify the following polynomial:

$(\frac{-a^{-2}b^3c^{-1}}{3a^{-4}b^{-1}c^3})^{-2}$

Possible Answers:

$\frac{9c^8}{a^{4}b^8}$

$\frac{9c^8}{a^{4}b^4}$

$\frac{c^8}{9a^{2}b^8}$

$\frac{c^8}{9a^{4}b^8}$

$\frac{9c^8}{a^{2}b^8}$

Correct answer:

$\frac{9c^8}{a^{4}b^8}$

Explanation:

To simplify the polynomial, begin by rearranging the terms to have positive exponents:

$(\frac{-a^{-2}b^3c^{-1}}{3a^{-4}b^{-1}c^3})^{-2}$

$(\frac{-a^{4}b^3b^1}{3a^{2}c^3c^1})^{-2}$

Rearrange the terms once again so that the outer exponent is positive. Also, combine like terms:

$(\frac{-3c^4}{a^{2}b^4})^{2}$

Now, square the polynomial:

$\frac{9c^8}{a^{4}b^8}$

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Example Question #1 : Polynomials

Simplify the following polynomial:

(5a)(6a^2b)(3ab^3)+(2a)^2(3b^3)(2a^2b)

Possible Answers:

114a^4b^2

114a^2b^4

114a^3b^4

114a^4b^4

114a^4b^3

Correct answer:

114a^4b^4

Explanation:

To simplify the polynomial, begin by combining the terms within the parentheses and multiplying the integers:

(5a)(6a^2b)(3ab^3)+(2a)^2(3b^3)(2a^2b)

90a^4b^4+24a^4b^4

Now, add together like terms:

90a^4b^4+24a^4b^4=114a^4b^4

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High School Math : Intermediate Single-Variable Algebra

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Polynomials

Example Question #1 : Simplifying Polynomials

Example Question #2 : Simplifying Polynomials

Example Question #1 : Simplifying Exponents

Example Question #1 : Simplifying Polynomials

Example Question #1 : Simplifying Polynomials

Example Question #1 : Simplifying Polynomials

Example Question #1 : Simplifying Polynomials

Example Question #1 : Simplifying Polynomials

Example Question #1 : Polynomials

All High School Math Resources

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