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Algebra II : Rational Expressions

Study concepts, example questions & explanations for Algebra II

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Example Questions

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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Example Question #1 : Least Common Denominator

Find the least common denominator of $\frac{x-2}{2x-3}$ and $\frac{6-x}{x+8}$

Possible Answers:

2x^2+13x-24

x^2+10x+9

x^2-3x-18

2x-24

Correct answer:

2x^2+13x-24

Explanation:

To find the least common denominator, multply the two denominators together.

(2x-3)(x+8)=2x^2+13x-24

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Example Question #11 : Least Common Denominator

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

Possible Answers:

6x^2+9

5x+10

6x^2+18

6x^2+21x+9

Correct answer:

6x^2+21x+9

Explanation:

To find the least common denominator, multply the two denominators together.

(2x+1)(3x+9)=6x^2+21x+9

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Example Question #12 : Least Common Denominator

Find the least common denominator between $\frac{x^2-2}{x+4}$ and $\frac{3x^2-2}{x-9}$

Possible Answers:

x^2-13x-36

x^2-5x-36

x^2+5x-36

x^2-36

Correct answer:

x^2-5x-36

Explanation:

To find the least common denominator, multply the two denominators together.

(x-9)(x+4)=x^2-5x-36

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Example Question #13 : Least Common Denominator

Find the least common denominator for $\frac{x-2}{x^2-4}$ and $\frac{x-3}{x^2-9}$ .

Possible Answers:

x-4

x^2+5x+6

x^2+6

x^2-5x-6

Correct answer:

x^2+5x+6

Explanation:

Start by simplifying both fractions.

$\frac{x-2}{x^2-4}=\frac{x-2}{(x-2)(x+2)}=\frac{1}{x+2}$

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

Now, to find the least common denominator for the two simplified fractions, multiply the denominators together.

(x+2)(x+3)=x^2+5x+6

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Algebra II : Rational Expressions

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

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

Example Question #1 : Least Common Denominator

Example Question #11 : Least Common Denominator

Example Question #12 : Least Common Denominator

Example Question #13 : Least Common Denominator

All Algebra II Resources

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