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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #22 : Least Common Denominator

Determine the least common denominator: $\frac{1}{x^2-x}+\frac{1}{2x-2}$

Possible Answers:

x^2-2x

2x^2-2x

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

x^2-x-2

4x^2-x

Correct answer:

2x^2-2x

Explanation:

In order to determine the least common denominator, we will need to break up both denominator with their simplest forms.

x^2-x = x(x-1)

2x-2 =2(x-1)

Notice that both of the values share an (x-1) term. The first expression lacks a two coefficient, and the second expression lacks an term.

This means that the LCD will be: 2x(x-1)

Simplify this expression by distribution.

The answer is: 2x^2-2x

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Example Question #1 : Simplifying Rational Expressions

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

Possible Answers:

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

$\frac{2x^{2}-x+11}{x^{2}+2x-3}$

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

$\frac{4x^{2}-3x+5}{x^{2}+3x-2}$

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

Correct answer:

$\frac{2x^{2}-x+11}{x^{2}+2x-3}$

Explanation:

This is a more complicated form of $\frac{1}{2}+\frac{1}{3}= \frac{3}{6}+\frac{2}{6}=\frac{5}{6}$

Find the least common denominator (LCD) and convert each fraction to the LCD, then add the numerators. Simplify as needed.

$\frac{x+2}{x-1}+\frac{x-5}{x+3}=\frac{x+2}{x-1}\cdot \frac{x+3}{x+3}+\frac{x-5}{x+3}\cdot \frac{x-1}{x-1}$

which is equivalent to $\frac{(x+2)\cdot (x+3)+(x-5)\cdot (x-1)}{(x+3)\cdot (x-1)}$

Simplify to get $\frac{2x^{2}-x+11}{x^{2}+2x-3}$

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Example Question #2291 : Algebra 1

Simplify:

$\frac{4x}{x+3}+\frac{2x}{x+3}$

Possible Answers:

$\frac{8x}{x+3}$

$\frac{6}{x}$

$\frac{6x}{x+3}$

Correct answer:

$\frac{6x}{x+3}$

Explanation:

$\frac{4x}{x+3}+\frac{2x}{x+3}$

Because the two rational expressions have the same denominator, we can simply add straight across the top. The denominator stays the same.

Therefore the answer is $\frac{6x}{x+3}$ .

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Example Question #1 : Solving Rational Expressions

Simplify

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

Possible Answers:

$\frac{15x-1}{3x^3}$

The expression cannot be simplified.

$\frac{4}{x^2-3x^3}$

$\frac{4}{3x^3}$

$\frac{15x-1}{1}$

Correct answer:

$\frac{15x-1}{3x^3}$

Explanation:

a. Find a common denominator by identifying the Least Common Multiple of both denominators. The LCM of 3 and 1 is 3. The LCM of x^2 and x^3 is x^3 . Therefore, the common denominator is 3x^3 .

b. Write an equivialent fraction to $\frac{5}{x^2}$ using 3x^3 as the denominator. Multiply both the numerator and the denominator by to get $\frac{15x}{3x^3}$ . Notice that the second fraction in the original expression already has 3x^3 as a denominator, so it does not need to be converted.

The expression should now look like: $\frac{15x}{3x^3}-\frac{1}{3x^3}$ .

c. Subtract the numerators, putting the difference over the common denominator.

$\frac{15x-1}{3x^3}$

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Example Question #2 : Solving Rational Expressions

Combine the following expression into one fraction:

$\small \frac{x^2+y}{25x}+\frac{z+3}{y}$

Possible Answers:

$\small \frac{25x^3+25xy+yz+3y}{25xy}$

$\small \frac{x^2y+y^2+25xz+75x}{25xy}$

The two fractions cannot be combined as they have different denominators.

$\small \frac{x^2z+yz+3x^2+3y}{25xy}$

$\small \frac{x^2+y+z+3}{25xy}$

Correct answer:

$\small \frac{x^2y+y^2+25xz+75x}{25xy}$

Explanation:

To combine fractions of different denominators, we must first find a common denominator between the two. We can do this by multiplying the first fraction by $\small \frac{y}{y}$ and the second fraction by $\small \frac{25x}{25x}$ . We therefore obtain:

$\small \frac{x^2+y}{25x}\times \frac{y}{y}+\frac{z+3}{y}\times\frac{25x}{25x}$

$\small \small \frac{x^2y+y^2}{25xy}+\frac{25xz+75x}{25xy}$

Since these fractions have the same denominators, we can now combine them, and our final answer is therefore:

$\small \frac{x^2y+y^2+25xz+75x}{25xy}$

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Example Question #3 : Solving Rational Expressions

What is $\frac{x+1}{3} + \frac{2x-5}{2}$ ?

Possible Answers:

$\frac{8x-13}{6}$

$\frac{3x-4}{5}$

$\frac{6x-2}{3}$

$\frac{8x+17}{6}$

$\frac{8x+13}{3}$

Correct answer:

$\frac{8x-13}{6}$

Explanation:

We start by adjusting both terms to the same denominator which is 2 x 3 = 6

Then we adjust the numerators by multiplying x+1 by 2 and 2x-5 by 3

$\frac{(x+1)\cdot 2}{3 \cdot 2}+\frac{(2x-5) \cdot 3}{2 \cdot 3}$

The results are:

$\frac{2x+2}{6}+\frac{6x-15}{6}$

So the final answer is,

$\frac{8x-13}{6}$

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Example Question #1 : Solving Rational Expressions

What is $\frac{3x-4}{x+2}-\frac{5x+2}{2x+4}$ ?

Possible Answers:

$\frac{x-10}{2x+4}$

$\frac{8x+6}{x+2}$

$\frac{-2x-6}{3x+6}$

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

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

Correct answer:

$\frac{x-10}{2x+4}$

Explanation:

Start by putting both equations at the same denominator.

2x+4 = (x+2) x 2 so we only need to adjust the first term:

$\frac{(3x-4)\cdot 2}{(x+2)\cdot 2}-\frac{5x+2}{2x+4}$

$=\frac{6x-8}{2x+4}-\frac{5x+2}{2x+4}$

Then we subtract the numerators, remembering to distribute the negative sign to all terms of the second fraction's numerator:

$=\frac{6x -8 -5x-2}{2x+4}=\frac{x-10}{2x+4}$

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Example Question #1 : Adding And Subtracting Rational Expressions

$\frac{15x+A}{x^{2}+8x+15}=\frac{7}{x+5}+\frac{8}{x+3}$

Determine the value of .

Possible Answers:

Correct answer:

Explanation:

(x+5)(x+3) is the common denominator for this problem making the numerators 7(x+3) and 8(x+5).

7(x+3)+8(x+5)= 7x+21+8x+40= 15x+61

A=61

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Example Question #6 : How To Add Polynomials

Add:

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

Possible Answers:

$\frac{1}{x-3}$

$\frac{x-3}{x+1}$

$\frac{1}{x+1}$

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

Correct answer:

$\frac{1}{x-3}$

Explanation:

First factor the denominators which gives us the following:

$\frac{x}{\left ( x+1 \right )\left ( x-3 \right )} + \frac{1}{\left ( x+1 \right )\left ( x-3 \right )}$

The two rational fractions have a common denominator hence they are like "like fractions". Hence we get:

$\frac{x+1}{\left ( x+1 \right )\left ( x-3 \right )}$

Simplifying gives us

$\frac{1}{x-3}$

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Example Question #571 : Intermediate Single Variable Algebra

Subtract:

$\frac{2x}{x-1} - \frac{3}{x^{2}-1}$

Possible Answers:

$\frac{2x^{2}-3}{\left ( x+1 \right )\left ( x-1 \right )}$

$\frac{2x-3}{x-1}$

$\frac{2}{x-1}$

$\frac{2x^{2}+2x-3}{\left ( x+1 \right )\left ( x-1 \right )}$

$\frac{2x-3}{x+1}$

Correct answer:

$\frac{2x^{2}+2x-3}{\left ( x+1 \right )\left ( x-1 \right )}$

Explanation:

First let us find a common denominator as follows:

$\frac{2x\left ( x+1 \right )}{\left ( x+1 \right )\left ( x-1 \right )} - \frac{3}{\left ( x+1 \right )\left ( x-1 \right )}$

Now we can subtract the numerators which gives us : $2x^{2}+2x-3$

So the final answer is $\frac{2x^{2}+2x-3}{\left ( x+1 \right )\left ( x-1 \right )}$

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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #22 : Least Common Denominator

Example Question #1 : Simplifying Rational Expressions

Example Question #2291 : Algebra 1

Example Question #1 : Solving Rational Expressions

Example Question #2 : Solving Rational Expressions

Example Question #3 : Solving Rational Expressions

Example Question #1 : Solving Rational Expressions

Example Question #1 : Adding And Subtracting Rational Expressions

Example Question #6 : How To Add Polynomials

Example Question #571 : Intermediate Single Variable Algebra

All Algebra II Resources

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