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Algebra II : Intermediate Single-Variable Algebra

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Example Questions

Example Question #13 : Rational Expressions

Simply the expression:

$\frac{3}{x+5}+ \frac{7}{x^{2}+3x -10}$

Possible Answers:

$\frac{5x-3}{x^{2}+2x-10}$

$\frac{7x+5}{x^{2}+4x-5}$

$\frac{3x+1}{x^{2}+3x-10}$

$\frac{7x-3}{x^{2}+3x-10}$

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

Correct answer:

$\frac{3x+1}{x^{2}+3x-10}$

Explanation:

In order to simplify the expression $\frac{3}{x+5}+ \frac{7}{x^{2}+3x -10}$ , first note that the denominators in both terms share a factor:

$\frac{3}{x+5}+ \frac{7}{(x+5)(x-2)}$

Find the Least Common Denominator (LCD) of both terms, and then simplify the expression:

$\frac{3(x-2)}{(x+5)(x-2)}+ \frac{7}{(x+5)(x-2)}$

$=\frac{3x-6}{(x+5)(x-2)}+ \frac{7}{(x+5)(x-2)}$

$=\frac{3x-6+7}{(x+5)(x-2)}$

Which equals:

$\frac{3x+1}{x^{2}+3x-10}$

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Example Question #1 : Least Common Denominator

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

Which of the following equations is equivalent to f(x) ?

Possible Answers:

$f(x)=\frac{5x^{2}+3x+8}{x^{2}+2x-3}$

$f(x)=\frac{4}{5x-4}$

$f(x)=\frac{5x-4}{4}$

$f(x)=\frac{x^{2}+2x-3}{5x^{2}+3x+8}$

Correct answer:

$f(x)=\frac{5x^{2}+3x+8}{x^{2}+2x-3}$

Explanation:

By looking at the answer choices, we can assume that the problem wants us to simplify f(x) . To do that, we need to combine the two terms within f(x) into one fraction.

First, let's remember how to add or subtract fractions:

Make sure the fractions have the same denominator.
Add or subtract the numerators, leaving the denominator alone.

The process looks like this:

$\frac{a}{b}+\frac{c}{d}=\frac{a}{b}\left( \frac{d}{d} \right) + \frac{c}{d}\left( \frac{b}{b} \right) = \frac{ad}{bd}+\frac{cb}{bd}=\frac{ad+cb}{bd}$

This is exactly what we're going to have to do to f(x) .

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

First, we find a common denominator between the two terms. No matter what ends up being equal to, a common denominator can always be found by multiplying the two terms together. In other words, we can use (x+3)(x-1) as our common denominator.

$f(x)=\frac{2x-5}{x+3}\left ( \frac{x-1}{x-1} \right )-\frac{-3x-1}{x-1} \left ( \frac{x+3}{x+3} \right )$

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

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

Now, all that's left is getting rid of these parentheses.

$f(x)=\frac{(2x^{2}-2x-5x+5)-(-3x^{2}-9x-x-3)}{x^{2}-x+3x-3}$

$f(x)=\frac{2x^{2}-7x+5+3x^{2}+10x+3}{x^{2}+2x-3}$

$f(x)=\frac{5x^{2}+3x+8}{x^{2}+2x-3}$

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Example Question #1 : Least Common Denominator

Simplify the expression:

$\frac{5}{x+ 6} + \frac{2}{x^{2}+ 4x-12}$

Possible Answers:

$\frac{5x }{x^{2}+ 4x-12}$

$\frac{5x-8}{ x^{2}+ 4x-12}$

$\frac{5x-12}{x^{2}+ 4x-12}$

$\frac{5x-2}{x-2}$

$\frac{5x+2}{x-2}$

Correct answer:

$\frac{5x-8}{ x^{2}+ 4x-12}$

Explanation:

Factor the second denominator, then simplify:

$\frac{5}{x+ 6} + \frac{2}{x^{2}+ 4x-12}$

$=\frac{5}{x+ 6} + \frac{2}{(x-2)(x+6)}$

$=\frac{5 (x-2)}{ (x-2)(x+6)} + \frac{2}{(x-2)(x+6)}$

$=\frac{5x-10}{ (x-2)(x+6)} + \frac{2}{(x-2)(x+6)}$

$=\frac{5x-10 + 2}{ (x-2)(x+6)}$

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

$=\frac{5x-8}{x^{2}+ 4x-12}$

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Example Question #1 : Least Common Denominator

$\frac{(x^{2}-1x-6)}{x+3}+\frac{(x^{2}-6x+9)}{x+2}$

What is the least common denominator of the above expression?

Possible Answers:

$(x+3)^{2}+(x-2)$

(x+3)+(x+2)

(x+3)(x-2)

None of these answer choices

(x+3)(x+2)

Correct answer:

(x+3)(x+2)

Explanation:

The least common denominator is the least common multiple of the denominators of a set of fractions.

Simply multiply the two denominators together to find the LCD: (x+3)(x+2)

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Example Question #1 : Least Common Denominator

Find the least common denominator of the following fractions:

$\frac{6}{7} , \frac{4}{3} , \frac{5}{9}$

Possible Answers:

189

Correct answer:

Explanation:

The denominators are 7, 3, and 9. We have to find the common multiple of 7, 3, and 9.

Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63

Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90

The least common multiple of the 3 denominators is 63.

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Example Question #1 : Least Common Denominator

What is the least common denominator of the following fractions?

$\frac{7}{19} , \frac{3}{5}$

Possible Answers:

145

190

Correct answer:

Explanation:

Solution 1

The least common denominator is the least common multiple of the denominators.

We list the multiples of each denominator and we find the lowest common multiple.

Multiples of 19: 19, 38, 57, 76, 95, 114, 133, 152, 171, 190

Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100

The lowest common multiple in both lists is 95.

Solution 2

19 and 5 are prime numbers. They have no positive divisors other than 1 and themselves.

The least common denominator of two prime numbers is their product.

$LCD= 19\times5= 95$

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Example Question #1 : Least Common Denominator

Find the least common denominator of $\frac{x-2}{x-1}$ and $\frac{3}{x+9}$ .

Possible Answers:

x^2-9

x^2+5x+7

x^2

x^2+8x-9

Correct answer:

x^2+8x-9

Explanation:

To find the least common denominator for these two fractions, multiply the denominators together.

(x-1)(x+9)=x^2+8x-9

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Example Question #1 : Least Common Denominator

Find the least common denominator for $\frac{x^2-2x-1}{x+8}$ and $\frac{x^3-2x-2}{x-9}$

Possible Answers:

x^2-10x-72

x^2-8x+7

x^2-2x-2

x^2-x-72

Correct answer:

x^2-x-72

Explanation:

To find the least common denominator for these two fractions, multiply the denominators together.

(x+8)(x-9)=x^2-x-72

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Example Question #8 : Least Common Denominator

Find the least common denominator between $\frac{x^2-2x-3}{x^2-5x+6}$ and $\frac{x^2+x-12}{x^2+2x-8}$ .

Possible Answers:

x^2-9x+14

x^4-3x^3-2x^2+52x-48

x+1

x-2

Correct answer:

x-2

Explanation:

Start by factoring the numerator and denominator for each fraction.

$\frac{x^2-2x-3}{x^2-5x+6}=\frac{(x-3)(x+1)}{(x-3)(x-2)}=\frac{x+1}{x-2}$

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

So when the two simplified fractions are compared, they actually have the same denominator, which will be the least common denominator.

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Example Question #9 : Least Common Denominator

Find the least common denominator of $\frac{x^2-8x+12}{x^2-2x+24}$ and $\frac{x^2+7x+12}{x^2+10x+24}$

Possible Answers:

x^2+10x+24

x^2+3x-24

x^2+2x-24

x^2-2x-24

Correct answer:

x^2+2x-24

Explanation:

Start by simplifying both fractions.

$\frac{x^2-8x+12}{x^2-2x+24}=\frac{(x-2)(x-6)}{(x-6)(x-4)}=\frac{x-2}{x-4}$

$\frac{x^2+7x+12}{x^2+10x+24}=\frac{(x+3)(x+4)}{(x+6)(x+4)}=\frac{x+3}{x+6}$

Now, to find the least common denominator, multiply the denominators together.

(x-4)(x+6)=x^2+2x-24

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Algebra II : Intermediate Single-Variable Algebra

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #13 : Rational Expressions

Example Question #1 : Least Common Denominator

Example Question #1 : Least Common Denominator

Example Question #1 : Least Common Denominator

Example Question #1 : Least Common Denominator

Example Question #1 : Least Common Denominator

Example Question #1 : Least Common Denominator

Example Question #1 : Least Common Denominator

Example Question #8 : Least Common Denominator

Example Question #9 : Least Common Denominator

All Algebra II Resources

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