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Algebra II : Expressions

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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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Example Question #1521 : High School Math

Simplify:

$x^{3}\times{x^{4}}+x$

Possible Answers:

$x^{7}+x$

$x^{13}$

$x^{8}$

$x^{12} + x$

$3x^{8}$

Correct answer:

$x^{7}+x$

Explanation:

$x^{m}\times{x^{n}} = x^{m+n}$ . However, $x^{7}+x$ cannot be simplified any further because the terms have different exponents.

(Like terms are terms that have the same variables with the same exponents. Only like terms can be combined together.)

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Example Question #21 : Simplifying Expressions

Simplify:

$\left (\frac{6x^{2}}{t} \right )^{-3}$

Possible Answers:

$\frac{18 t^{3} } {x^{6} }$

$\frac{216 t^{3} } { x^{6} }$

$\frac{t^{3} } { 216 x^{5} }$

$\frac{t^{3} } {18 x^{6} }$

$\frac{t^{3} } { 216 x^{6} }$

Correct answer:

$\frac{t^{3} } { 216 x^{6} }$

Explanation:

Apply the laws of exponents as follows:

$\left (\frac{6x^{2}}{t} \right )^{-3}$

$= \left (\frac{t} {6x^{2}}\right )^{3}$

$= \frac{t^{3} } { \left ( 6x^{2} \right ) ^{3} }$

$= \frac{t^{3} } { 6 ^{3} \left ( x^{2} \right ) ^{3}}$

$= \frac{t^{3} } { 216 \left ( x^{2 \cdot 3} \right ) }$

$= \frac{t^{3} } { 216 x^{6} }$

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Example Question #22 : Simplifying Expressions

Simplify the expression:

$\frac{17}{x^{2}-x-6} + \frac{x+5}{x-3}$

Possible Answers:

$\frac{x+22}{x^{2}-x-6}$

$\frac{x^{2}+7x+27}{x^{2}-x-6}$

$\frac{x^{2}+7x+27}{x+2}$

$\frac{x^{2}+7x+10}{x^{2}-x-6}$

$\frac{x^{2}+7x+27}{x-3}$

Correct answer:

$\frac{x^{2}+7x+27}{x^{2}-x-6}$

Explanation:

$\frac{17}{x^{2}-x-6} + \frac{x+5}{x-3}$

1. Factor $x^{2}-x-6$

(x+2)(x-3)

***notice that the two fractions now share a factor in the denominator***

$\frac{17}{(x+2)(x-3)} + \frac{x+5}{x-3}$

2. Create a common denominator between the two terms

$\frac{17}{(x+2)(x-3)} + \frac{(x+5)(x+2)}{(x-3)(x+2)}$

3. Simplify

$=\frac{17}{x^{2}-x-6} + \frac{x^{2}+7x+10}{x^{2}-x-6}$

$=\frac{17 +x^{2}+7x+10}{x^{2}-x-6}$

$=\frac{x^{2}+7x+27}{x^{2}-x-6}$

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Example Question #23 : Simplifying Expressions

Add and simplify the following rational expression:

$\frac{\sqrt{5}}{\sqrt{7}}+\frac{\sqrt{10}}{\sqrt{3}}$

Possible Answers:

$\frac{\sqrt{85}}{\sqrt{21}}$

$\sqrt{\frac{3}{2}}$

$17\sqrt{\frac{5}{21}}$

$\small \frac{\sqrt{35}}{7}+\frac{\sqrt{30}}{3}$

No real solution

Correct answer:

$\small \frac{\sqrt{35}}{7}+\frac{\sqrt{30}}{3}$

Explanation:

To add any fractions together, they must first have a common denominator. We can obtain a common denominator of $\sqrt{21}$ if we multiply the first fraction by $\frac{\sqrt{3}}{\sqrt{3}}$ and the second one by $\frac{\sqrt{7}}{\sqrt{7}}$ . We therefore obtain:

$\frac{\sqrt{5}}{\sqrt{7}}\times\frac{\sqrt{3}}{\sqrt{3}} + \frac{\sqrt{10}}{\sqrt{3}}\times\frac{\sqrt{7}}{\sqrt{7}}$

$\small \frac{\sqrt{15}}{\sqrt{21}}+\frac{\sqrt{70}}{\sqrt{21}}=\frac{\sqrt{15}+\sqrt{70}}{\sqrt{21}}$

From there, we need to take out the radical in the denominator by multiplying by $\small \frac{\sqrt{21}}{\sqrt{21}}$ , as follows:

$\small \small \frac{\sqrt{15}+\sqrt{70}}{\sqrt{21}}\times\frac{\sqrt{21}}{\sqrt{21}}=\frac{\sqrt{315}+\sqrt{1470}}{21}$

From here, we can simplify the radicals above by finding their prime factors:

$\small \sqrt{315}=\sqrt{3*3*5*7}=3\sqrt{35}$

and

$\small \sqrt{1470}=\sqrt{2*5*3*7*7}=7\sqrt{30}$ .

We are therefore left with $\small \frac{3\sqrt{35}+7\sqrt{30}}{21}$ , which can be separated and reduced to our final answer,

$\small \frac{\sqrt{35}}{7}+\frac{\sqrt{30}}{3}$

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Algebra II : Expressions

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

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

Example Question #1521 : High School Math

Example Question #21 : Simplifying Expressions

Example Question #22 : Simplifying Expressions

Example Question #23 : Simplifying Expressions

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