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Algebra II : Intermediate Single-Variable Algebra

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 $\frac{x-2}{2x-3}$ 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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Example Question #31 : Understanding Rational Expressions

What is the least common denominator for the following fractions?

$\small \frac{1}{4},\frac{2}{3},\frac{1}{6},\frac{3}{8},\frac{5}{12}$

Possible Answers:

Correct answer:

Explanation:

Remember that the least common denominator is the smallest number such that all fractions' denominators divide it evenly.

Always start with the largest number. In this case, it is 12. 3, 4, 6, and 12 all divide evenly into 12, but 8 does not. Think of the next multiple of the largest number. In this case, 24. Here we see that 3, 4, 6, 8, and 12 all divide 24 evenly. Thus, our least common denominator is 24.

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Example Question #32 : Rational Expressions

Simplify:

$\frac{4}{x}+\frac{3}{x+5}+\frac{1}{x^2+5x}$

Possible Answers:

$\frac{7x+21}{x^2+5x}$

$\frac{28}{x+5}$

$\frac{7x+6}{x^2+5x}$

$\frac{8}{x^2+5x}$

Correct answer:

$\frac{7x+21}{x^2+5x}$

Explanation:

To simplify the expression, we must find the common denominator, which in this case is

x(x+5)

Note that when we distribute the x, we get the denominator of the third term, x^2+5x .

Now, we multiply each term by the LCD divided by itself:

$\frac{x(x+5)}{x(x+5)}\cdot \frac{4}{x}+\frac{x(x+5)}{x(x+5)}\cdot \frac{3}{x+5}+\frac{x(x+5)}{x(x+5)}\cdot \frac{1}{x^2+5x}$

Which, after some canceling, becomes

$\frac{4(x+5)}{x(x+5)}+\frac{3(x)}{x(x+5)}+\frac{1}{x^2+5x}$

Now that all of the terms have the same denominator, we can add their numerators together:

$\frac{4(x+5)+3x+1}{x^2+5x}=\frac{4x+20+3x+1}{x^2+5x}=\frac{7x+21}{x^2+5x}$

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Example Question #16 : Least Common Denominator

What is the least common denominator of the following fractions?

$\frac{2}{3x}+\frac{3}{2+x}-\frac{1}{6}$

Possible Answers:

18x^2+36x

10x^2+2

3x^2+6x

3x^2+3x+8

4x+8

Correct answer:

18x^2+36x

Explanation:

In order to determine the least common denominator, multiply all the uncommon denominators together.

3x(2+x)(6)

Use the distributive property to simplify these terms.

3x(2+x)(6) = (6x+3x^2)(6) = 36x+18x^2

The least common denominator is: 18x^2+36x

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Example Question #17 : Least Common Denominator

What is the least common denominator needed to continue with the following problem?

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

Possible Answers:

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

x^2+5x-6

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

x^2-x-6

Correct answer:

x^2-x-6

Explanation:

In order to continue to simplify this problem, we will need to multiply both denominators together in order to simplify the numerators.

Multiply both of the denominators by using the FOIL method.

(x+2)(x-3) = (x)(x)+(x)(-3)+(2)(x)+(2)(-3)

Simplify this expression.

The least common denominator required is: x^2-x-6

Be careful not to continue and solve the problem!

The answer is: x^2-x-6

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Example Question #18 : Least Common Denominator

Determine the least common denominator for: $\frac{1}{3x}-\frac{2}{3-x}+\frac{1}{3+x}$

Possible Answers:

x^3+3x-9

x^3+27x-3

-3x^3+27x+9

x^2+3

-3x^3+27x

Correct answer:

-3x^3+27x

Explanation:

Determine the least common denominator by multiplying the denominators together.

3x(3-x)(3+x)

Multiply the first two terms together.

9x-3x^2

Multiply this quantity with the third term.

(9x-3x^2)(3+x)

Use the FOIL method to expand this expression.

= (9x)(3)+(9x)(x)+(-3x^2)(3)+(-3x^2)(x)

Simplify and combine like terms.

27x+9x^2-9x^2-3x^3

The answer is: -3x^3+27x

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Example Question #19 : Least Common Denominator

Determine the least common denominator: $\frac{1}{9-x}+\frac{2}{9-2x}$

Possible Answers:

-4x+27

4x-27

2x^2-27x-81

2x^2-27x+81

2x^2-81

Correct answer:

2x^2-27x+81

Explanation:

In order to determine the least common denominator, we will need to use the FOIL method to expand the denominators.

(9-x)(9-2x)

Multiply the terms.

(9)(9)+(9)(-2x)+(-x)(9)+(-x)(-2x)

81-18x-9x+2x^2

Combine like-terms.

The answer is: 2x^2-27x+81

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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 #1 : Least Common Denominator

Example Question #11 : Least Common Denominator

Example Question #12 : Least Common Denominator

Example Question #13 : Least Common Denominator

Example Question #31 : Understanding Rational Expressions

Example Question #32 : Rational Expressions

Example Question #16 : Least Common Denominator

Example Question #17 : Least Common Denominator

Example Question #18 : Least Common Denominator

Example Question #19 : Least Common Denominator

All Algebra II Resources

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