Algebra Examples

$\frac{1 - \frac{4}{x} + \frac{3}{x^{2}}}{1 + \frac{1}{x} - \frac{12}{x^{2}}}$

Step 1

Multiply the numerator and denominator of the fraction by $x^{2}$ .

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Step 1.1

Multiply $\frac{1 - \frac{4}{x} + \frac{3}{x^{2}}}{1 + \frac{1}{x} - \frac{12}{x^{2}}}$ by $\frac{x^{2}}{x^{2}}$ .

$\frac{x^{2}}{x^{2}} \cdot \frac{1 - \frac{4}{x} + \frac{3}{x^{2}}}{1 + \frac{1}{x} - \frac{12}{x^{2}}}$

Step 1.2

Combine.

$\frac{x^{2} (1 - \frac{4}{x} + \frac{3}{x^{2}})}{x^{2} (1 + \frac{1}{x} - \frac{12}{x^{2}})}$

Step 2

Apply the distributive property.

$\frac{x^{2} \cdot 1 + x^{2} (- \frac{4}{x}) + x^{2} \frac{3}{x^{2}}}{x^{2} \cdot 1 + x^{2} \frac{1}{x} + x^{2} (- \frac{12}{x^{2}})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

Move the leading negative in $- \frac{4}{x}$ into the numerator.

$\frac{x^{2} \cdot 1 + x^{2} \frac{- 4}{x} + x^{2} \frac{3}{x^{2}}}{x^{2} \cdot 1 + x^{2} \frac{1}{x} + x^{2} (- \frac{12}{x^{2}})}$

Step 3.1.2

Factor $x$ out of $x^{2}$ .

$\frac{x^{2} \cdot 1 + x \cdot x \frac{- 4}{x} + x^{2} \frac{3}{x^{2}}}{x^{2} \cdot 1 + x^{2} \frac{1}{x} + x^{2} (- \frac{12}{x^{2}})}$

Step 3.1.3

Cancel the common factor.

$\frac{x^{2} \cdot 1 + x \cdot x \frac{- 4}{x} + x^{2} \frac{3}{x^{2}}}{x^{2} \cdot 1 + x^{2} \frac{1}{x} + x^{2} (- \frac{12}{x^{2}})}$

Step 3.1.4

Rewrite the expression.

$\frac{x^{2} \cdot 1 + x \cdot - 4 + x^{2} \frac{3}{x^{2}}}{x^{2} \cdot 1 + x^{2} \frac{1}{x} + x^{2} (- \frac{12}{x^{2}})}$

Step 3.2

Cancel the common factor of $x^{2}$ .

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Step 3.2.1

Cancel the common factor.

$\frac{x^{2} \cdot 1 + x \cdot - 4 + x^{2} \frac{3}{x^{2}}}{x^{2} \cdot 1 + x^{2} \frac{1}{x} + x^{2} (- \frac{12}{x^{2}})}$

Step 3.2.2

Rewrite the expression.

$\frac{x^{2} \cdot 1 + x \cdot - 4 + 3}{x^{2} \cdot 1 + x^{2} \frac{1}{x} + x^{2} (- \frac{12}{x^{2}})}$

Step 3.3

Cancel the common factor of $x$ .

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Step 3.3.1

Factor $x$ out of $x^{2}$ .

$\frac{x^{2} \cdot 1 + x \cdot - 4 + 3}{x^{2} \cdot 1 + x \cdot x \frac{1}{x} + x^{2} (- \frac{12}{x^{2}})}$

Step 3.3.2

Cancel the common factor.

$\frac{x^{2} \cdot 1 + x \cdot - 4 + 3}{x^{2} \cdot 1 + x \cdot x \frac{1}{x} + x^{2} (- \frac{12}{x^{2}})}$

Step 3.3.3

Rewrite the expression.

$\frac{x^{2} \cdot 1 + x \cdot - 4 + 3}{x^{2} \cdot 1 + x + x^{2} (- \frac{12}{x^{2}})}$

Step 3.4

Cancel the common factor of $x^{2}$ .

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Step 3.4.1

Move the leading negative in $- \frac{12}{x^{2}}$ into the numerator.

$\frac{x^{2} \cdot 1 + x \cdot - 4 + 3}{x^{2} \cdot 1 + x + x^{2} \frac{- 12}{x^{2}}}$

Step 3.4.2

Cancel the common factor.

$\frac{x^{2} \cdot 1 + x \cdot - 4 + 3}{x^{2} \cdot 1 + x + x^{2} \frac{- 12}{x^{2}}}$

Step 3.4.3

Rewrite the expression.

$\frac{x^{2} \cdot 1 + x \cdot - 4 + 3}{x^{2} \cdot 1 + x - 12}$

Step 4

Factor $x^{2} \cdot 1 + x \cdot - 4 + 3$ using the AC method.

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Step 4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $3$ and whose sum is $- 4$ .

$- 3, - 1$

Step 4.2

Write the factored form using these integers.

$\frac{(x - 3) (x - 1)}{x^{2} \cdot 1 + x - 12}$

Step 5

Factor $x^{2} \cdot 1 + x - 12$ using the AC method.

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Step 5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 12$ and whose sum is $1$ .

$- 3, 4$

Step 5.2

Write the factored form using these integers.

$\frac{(x - 3) (x - 1)}{(x - 3) (x + 4)}$

Step 6

Cancel the common factor of $x - 3$ .

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Step 6.1

Cancel the common factor.

$\frac{(x - 3) (x - 1)}{(x - 3) (x + 4)}$

Step 6.2

Rewrite the expression.

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