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Algebra II : Basic Single-Variable Algebra

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Example Questions

Example Question #3 : Simplifying Expressions

Simplify:

-5(x+15)-3x

Possible Answers:

-8x+15

2x-75

None of the other answers are correct.

-8x-75

2x+15

Correct answer:

-8x-75

Explanation:

First, distribute –5 through the parentheses by multiplying both terms by –5.

-5(x+15)-3x=-5x-75-3x

Then, combine the like-termed variables (–5x and –3x).

-5x-75-3x=-8x-75

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Example Question #1 : How To Multiply Binomials With The Distributive Property

Expand:

$\frac{1}{3}(2x^2-6)(4x-\frac{2}{3})$

Possible Answers:

$2\frac{2}{3}x^3-\frac{4}{9}x^2-8x-1\frac{1}{3}$

$2\frac{2}{3}x^3+\frac{4}{9}x^2-8x+1\frac{1}{3}$

$2\frac{2}{3}x^3-\frac{4}{9}x^2-8x+1\frac{1}{3}$

$2\frac{2}{3}x^3-\frac{4}{9}x^2+8x+1\frac{1}{3}$

$-2\frac{2}{3}x^3-\frac{4}{9}x^2-8x+1\frac{1}{3}$

Correct answer:

$2\frac{2}{3}x^3-\frac{4}{9}x^2-8x+1\frac{1}{3}$

Explanation:

First, FOIL:

$\frac{1}{3}(8x^3-\frac{4}{3}x^2-24x+\frac{12}{3})$

Simplify:

$\frac{1}{3}(8x^3-\frac{4}{3}x^2-24x+4)$

Distribute the $\frac{1}{3}$ through the parentheses:

$\frac{8}{3}x^3-\frac{4}{9}x^2-\frac{24}{3}x+\frac{4}{3}$

Rewrite to make the expression look like one of the answer choices:

$2\frac{2}{3}x^3-\frac{4}{9}x^2-8x+1\frac{1}{3}$

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Example Question #101 : Basic Single Variable Algebra

Simplify the expression.

$15x^{3}y^{2}+8x^{2}+3x^{3}y^{2}-4x^{2}$

Possible Answers:

$18x^{3}y^{2}+4x^{2}$

None of the other answers are correct.

$22x^{6}y^{4}$

$18x^{3}y^{2}-4x^{2}$

$22x^3y^{2}x^{2}$

Correct answer:

$18x^{3}y^{2}+4x^{2}$

Explanation:

When simplifying polynomials, only combine the variables with like terms.

$15x^{3}y^{2}$ can be added to $3x^{3}y^{2}$ , giving $18x^{3}y^{2}$ .

$4x^{2}$ can be subtracted from $8x^{2}$ to give $4x^{2}$ .

Combine both of the terms into one expression to find the answer: $18x^{3}y^{2}+4x^{2}$

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Example Question #102 : Basic Single Variable Algebra

Simplify the following expression.

$(5c^{2}+5)+(-5c-5)$

Possible Answers:

$5c^{2}-5c+10$

$5c^{2}-5c$

$5c^{2}-5c-25$

Correct answer:

$5c^{2}-5c$

Explanation:

$(5c^{2}+5)+(-5c-5)$

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

Add like terms to solve.

$5c^{2}$ and -5c have no like terms and cannot be combined with anything.

5 and -5 can be combined however:

5-5=0

This leaves us with $5c^{2}-5c$ .

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Example Question #103 : Basic Single Variable Algebra

Simplify the following:

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

Possible Answers:

None of the other answers are correct.

8x^3-2x+16

2x^3-2x^2+12x

4x^3-2x(x^2+x-6)

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

Correct answer:

2x^3-2x^2+12x

Explanation:

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

First, FOIL the two binomials:

4x^3-2x(x^2+x-6)

Then distribute the through the terms in parentheses:

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

Combine like terms:

2x^3-2x^2+12x

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Example Question #1 : How To Subtract Polynomials

Simplify the following expression:

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

Possible Answers:

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

$-3q^{2}+12$

-q^2+4q+2

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

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 #104 : Basic Single Variable Algebra

Simplify the expression.

$x^{5}*x^{16}$

Possible Answers:

The expression cannot be simplified further.

$x^{11}$

$x^{21}$

$2x^{16}$

$x^{80}$

Correct answer:

$x^{21}$

Explanation:

When multiplying exponential components, you must add the powers of each term together.

5+16=21

$x^{5}*x^{16}=x^{21}$

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Example Question #105 : Basic Single Variable Algebra

$x^{5}y^{17}*x^{2}y^{2}=?$

Possible Answers:

$2x^{2}x^{7}y^{19}$

$x^{2}y^{8}$

$x^{10}y^{34}$

None of the other answers are correct.

$x^{7}y^{19}$

Correct answer:

$x^{7}y^{19}$

Explanation:

When multiplying polynomials, add the powers of each like-termed variable together.

For x: 5 + 2 = 7

For y: 17 + 2 = 19

Therefore the answer is $x^{7}y^{19}$ .

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Example Question #106 : Basic Single Variable Algebra

Simplify the following:

$\frac{x^2-10x+25}{x^2-25}$

Possible Answers:

$\frac{(x-5)(x-5)}{(x-5)(x+5)}$

$\frac{x-5}{x+5}$

This fraction cannot be simplified.

Correct answer:

$\frac{x-5}{x+5}$

Explanation:

$\frac{x^2-10x+25}{x^2-25}$

First we will factor the numerator:

x^2-10x+25=(x-5)(x-5)

Then factor the denominator:

x^2-25=(x+5)(x-5)

We can re-write the original fraction with these factors and then cancel an (x-5) term from both parts:

$\frac{(x-5)(x-5)}{(x-5)(x+5)}=\frac{x-5}{x+5}$

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Example Question #1 : Operations With Polynomials

Divide 3x^3+5x^2-x+8 by x^2-1 .

Possible Answers:

$2x+13+\frac{3x+5}{x^2-1}$

$3x+5-\frac{2x+13}{x^2-1}$

$3x+5+\frac{x^2+1}{2x-13}$

$3x+5+\frac{2x+13}{x^2-1}$

$x^2-1-\frac{3x+5}{2x+13}$

Correct answer:

$3x+5+\frac{2x+13}{x^2-1}$

Explanation:

First, set up the division as the following:

$\begin{array}{*2r @{\hskip\arraycolsep}c@{\hskip\arraycolsep} *4r} & && & & & \\ \cline{3-7} x^2 & -1 &\big)& 3x^3 & 5x^2 & -x & 8 \end{array}$

Look at the leading term x^2 in the divisor and 3x^3 in the dividend. Divide 3x^3 by x^2 gives ; therefore, put on the top:

$\begin{array}{*2r @{\hskip\arraycolsep}c@{\hskip\arraycolsep} *4r} & && 3x & & & \\ \cline{3-7} x^2 & -1 &\big)& 3x^3 & 5x^2 & -x & 8 \end{array}$

Then take that and multiply it by the divisor, x^2-1 , to get 3x^3-3x . Place that 3x^3-3x under the division sign:

$\begin{array}{*2r @{\hskip\arraycolsep}c@{\hskip\arraycolsep} *4r} & && 3x & & & \\ \cline{3-7} x^2 & -1 &\big)& 3x^3 & 5x^2 & -x & 8 \\ & && 3x^3 & -3x \end{array}$

Subtract the dividend by that same 3x^3-3x and place the result at the bottom. The new result is 5x^2+2x+8 , which is the new dividend.

$\begin{array}{*2r @{\hskip\arraycolsep}c@{\hskip\arraycolsep} *4r} & && 3x & & & \\ \cline{3-7} x^2 & -1 &\big)& 3x^3 & 5x^2 & -x & 8 \\ & && 3x^3 & -3x \\ \cline{4-7} & && & 5x^2 & 2x & 8 \end{array}$

Now, 5x^2 is the new leading term of the dividend. Dividing 5x^2 by x^2 gives 5. Therefore, put 5 on top:

$\begin{array}{*2r @{\hskip\arraycolsep}c@{\hskip\arraycolsep} *4r} & && 3x & 5 & & \\ \cline{3-7} x^2 & -1 &\big)& 3x^3 & 5x^2 & -x & 8 \\ & && 3x^3 & -3x \\ \cline{4-7} & && & 5x^2 & 2x & 8 \end{array}$

Multiply that 5 by the divisor and place the result, 5x^2-5 , at the bottom:

Perform the usual subtraction:

Therefore the answer is 3x+5 with a remainder of 2x+13 , or $3x+5+\frac{2x+13}{x^2-1}$ .

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Algebra II : Basic Single-Variable Algebra

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #3 : Simplifying Expressions

Example Question #1 : How To Multiply Binomials With The Distributive Property

Example Question #101 : Basic Single Variable Algebra

Example Question #102 : Basic Single Variable Algebra

Example Question #103 : Basic Single Variable Algebra

Example Question #1 : How To Subtract Polynomials

Example Question #104 : Basic Single Variable Algebra

Example Question #105 : Basic Single Variable Algebra

Example Question #106 : Basic Single Variable Algebra

Example Question #1 : Operations With Polynomials

All Algebra II Resources

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