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Algebra II : Intermediate Single-Variable Algebra

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Example Questions

Example Question #641 : Intermediate Single Variable Algebra

$\frac{\frac{4}{x}}{\frac{6}{x^2}}$

Possible Answers:

$\frac{2}{3}$

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

$\frac{1}{3}$

$\frac{x}{3}$

$\frac{2x}{3}$

Correct answer:

$\frac{2x}{3}$

Explanation:

In this problem, we're dealing with dividing rational expressions. Therefore, we have to flip the second fraction and then multiply the two: $\frac{4}{x}\cdot \frac{x^2}{6}$ . Simplify and multiply straight across to get your answer: $\frac{2x}{3}$ .

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Example Question #641 : Intermediate Single Variable Algebra

$\frac{2a}{b^2}\cdot \frac{5b^4}{10a}$

Possible Answers:

b^2

$\frac{1}{b^2}$

b^4

Correct answer:

b^2

Explanation:

When multiplying fractions, you will multiply straight across.

But first, see if you can reduce diagonally.

The a's cross out, and you can take out a b^2 from the other diagonal.

The coefficients also reduce.

$\frac{2a}{b^2}\cdot \frac{5b^4}{10a}=\frac{2}{1}\cdot \frac{b^2}{2}=b^2$

Therefore, your answer is b^2 .

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Example Question #116 : Rational Expressions

Multiply: $\frac{3x}{x-3}\cdot \frac{x+3}{x-3}$

Possible Answers:

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

$\frac{ 3x^2+9}{x^2-9}$

$\frac{ 3x^2+9x}{x-3}$

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

$\frac{ 27x^3}{x^2-6x+9}$

Correct answer:

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

Explanation:

Multiply the numerator with the numerator and denominator with the denominator.

Use the distributive property to expand the numerator.

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

Use the FOIL method to simplify the denominator.

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

x^2-6x+9

Divide the numerator with the denominator. There are no common factors.

The answer is: $\frac{ 3x^2+9x}{x^2-6x+9}$

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Example Question #117 : Rational Expressions

Simplify: $\frac{x^2-2x}{2x}\cdot \frac{x^2}{x^2-x-2}$

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Factor all the binomial and trinomial in the expression.

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

Cancel all the common terms.

The expression becomes:

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

The answer is: $\frac{x^2}{2x+2}$

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Example Question #642 : Intermediate Single Variable Algebra

Divide: $\frac{x^2-1}{x}\div \frac{x^2}{x+1}$

Possible Answers:

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

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

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

x^2+x

x^2-x

Correct answer:

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

Explanation:

To divide this expression, take the reciprocal of the second term and change the division sign to a multiplication sign.

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

Expand the numerator by FOIL method.

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

x^3+x^2-x-1

Divide this by the denominator.

The answer is: $\frac{x^3+x^2-x-1}{x^3}$

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Example Question #643 : Intermediate Single Variable Algebra

Solve: $\frac{x+3}{x^2}\div\frac{x^4}{3}$

Possible Answers:

$\frac{3x+9}{x^6}$

$\frac{3x+9}{x^8}$

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

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

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

Correct answer:

$\frac{3x+9}{x^6}$

Explanation:

In order to divide the rational expressions, we will need to convert the division sign to a multiplication sign and take the reciprocal of the second term.

$\frac{x+3}{x^2}\div\frac{x^4}{3} = \frac{x+3}{x^2} \times \frac{3} {x^4}$

Multiply the numerator with the numerator and denominator with the denominator.

3(x+3)=3x+9

x^2(x^4) = x^6

The answer is: $\frac{3x+9}{x^6}$

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Example Question #644 : Intermediate Single Variable Algebra

Divide: $\frac{3x+3}{6x-7}\div \frac{6}{x}$

Possible Answers:

$\frac{x^2+x}{12x-14}$

$\frac{x^2-x}{12x-14}$

$\frac{18x-18}{6x^2-7}$

$\frac{18x+18}{6x^2-7}$

$\frac{x^2-1}{12x-14}$

Correct answer:

$\frac{x^2+x}{12x-14}$

Explanation:

To divide the rational expression, we will need to take the reciprocal of the second term and change the division sign to a multiplication sign.

$\frac{3x+3}{6x-7}\div \frac{6}{x} = \frac{3x+3}{6x-7}\times \frac{x}{6}$

Multiply the numerator with the numerator and the denominator with the denominator.

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

6(6x-7) =36x-42

Divide the numerator and denominator.

$\frac{ 3x^2+3x}{36x-42}$

Take out a three as a common factor.

$\frac{ 3x^2+3x}{36x-42}=\frac{3(x^2+x)}{3(12x-14)}=\frac{x^2+x}{12x-14}$

The answer is: $\frac{x^2+x}{12x-14}$

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Example Question #643 : Intermediate Single Variable Algebra

Divide: $\frac{x^3-1}{x+1} \div \frac{1}{x-1}$

Possible Answers:

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

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

$\frac{x^4-x^3-x+1}{x+1}$

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

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

Correct answer:

$\frac{x^4-x^3-x+1}{x+1}$

Explanation:

Be very careful not to apply the difference of cubes to simplify this expression!

Change the sign to multiplication and take the reciprocal of the second term.

$\frac{x^3-1}{x+1} \div \frac{1}{x-1}=\frac{x^3-1}{x+1} \cdot\frac{ x-1}{1}$

Multiply the top by FOIL method.

(x^3-1)(x-1) = (x^3)(x)+(x^3)(-1)+(-1)(x)+(-1)(-1)

x^4-x^3-x+1

Divide this by the denominator.

The answer is: $\frac{x^4-x^3-x+1}{x+1}$

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Example Question #645 : Intermediate Single Variable Algebra

Simplify: $\frac{x}{x-9}\div\frac{x-9}{x-18}$

Possible Answers:

$\frac{x-18}{x}$

$\frac{x+9}{x^2-18+81}$

$\frac{x}{x-18}$

$\frac{x+2}{x-9}$

$\frac{x^2-18}{x^2-18+81}$

Correct answer:

$\frac{x^2-18}{x^2-18+81}$

Explanation:

In order to simplify the rational expression, we will need to rewrite the expression as a multiplication sign and take the reciprocal of the second term.

$\frac{x}{x-9}\div\frac{x-9}{x-18}= \frac{x}{x-9} \times \frac{x-18}{x-9}$

Simplify the numerator.

x(x-18) = x^2-18x

Simplify the denominator by FOIL method.

(x-9)(x-9)= x^2-9x-9x+81 = x^2-18x+81

Divide the numerator with the denominator.

The answer is: $\frac{x^2-18}{x^2-18+81}$

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Example Question #646 : Intermediate Single Variable Algebra

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

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

Rewrite the left fraction using common factors.

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

Cancel out common terms.

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

Factorize the bottom term.

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

The answer is: $\frac{1}{x-1}$

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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 #641 : Intermediate Single Variable Algebra

Example Question #641 : Intermediate Single Variable Algebra

Example Question #116 : Rational Expressions

Example Question #117 : Rational Expressions

Example Question #642 : Intermediate Single Variable Algebra

Example Question #643 : Intermediate Single Variable Algebra

Example Question #644 : Intermediate Single Variable Algebra

Example Question #643 : Intermediate Single Variable Algebra

Example Question #645 : Intermediate Single Variable Algebra

Example Question #646 : Intermediate Single Variable Algebra

All Algebra II Resources

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