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Algebra II : Simplifying Expressions

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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 #11 : Simplifying Expressions

Simplify the expression.

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

Possible Answers:

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

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

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

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

None of the other answers are correct.

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 #12 : Simplifying Expressions

Simplify the following expression.

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

Possible Answers:

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

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

$5c^{2}-5c$

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 #1 : How To Subtract Polynomials

Simplify the following:

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

Possible Answers:

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

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

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

8x^3-2x+16

None of the other answers are correct.

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 #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 #12 : Simplifying Expressions

Simplify the expression.

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

Possible Answers:

$x^{80}$

$x^{21}$

$x^{11}$

$2x^{16}$

The expression cannot be simplified further.

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 #1 : How To Multiply Polynomials

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

Possible Answers:

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

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

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

None of the other answers are correct.

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

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 #11 : How To Divide Polynomials

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 #12 : Simplifying Expressions

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

Possible Answers:

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

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

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

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

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

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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Example Question #13 : Simplifying Expressions

Simplify the expression:

$\frac{x^{2}y^{15}z^{-2}}{x^{5}y^{4}}$

Possible Answers:

$\frac{x^{2}y^{2}}{z^{2}}$

The fraction cannot be simplified further.

$\frac{y^{12}}{x^{2}z^{2}}$

$\frac{y^{9}}{z^{2}}$

$\frac{y^{11}}{x^{3}z^{2}}$

Correct answer:

$\frac{y^{11}}{x^{3}z^{2}}$

Explanation:

When dividing polynomials, subtract the exponent of the variable in the numberator by the exponent of the same variable in the denominator.

If the power is negative, move the variable to the denominator instead.

First move the negative power in the numerator to the denominator:

$\frac{x^{2}y^{15}}{x^{5}y^{4}z^{2}}$

Then subtract the powers of the like variables:

$\frac{y^{11}}{x^{3}z^{2}}$

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Algebra II : Simplifying Expressions

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

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

Example Question #11 : Simplifying Expressions

Example Question #12 : Simplifying Expressions

Example Question #1 : How To Subtract Polynomials

Example Question #11 : Polynomials

Example Question #12 : Simplifying Expressions

Example Question #1 : How To Multiply Polynomials

Example Question #11 : How To Divide Polynomials

Example Question #12 : Simplifying Expressions

Example Question #13 : Simplifying Expressions

All Algebra II Resources

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