Precalculus Examples

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

Step 1

Find the common denominator.

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Step 1.1

Write $x^{2}$ as a fraction with denominator $1$ .

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

Step 1.2

Multiply $\frac{x^{2}}{1}$ by $\frac{x - 1}{x - 1}$ .

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

Step 1.3

Multiply $\frac{x^{2}}{1}$ by $\frac{x - 1}{x - 1}$ .

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

Step 1.4

Write $- 5 x$ as a fraction with denominator $1$ .

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

Step 1.5

Multiply $\frac{- 5 x}{1}$ by $\frac{x - 1}{x - 1}$ .

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

Step 1.6

Multiply $\frac{- 5 x}{1}$ by $\frac{x - 1}{x - 1}$ .

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 3.2.1

Multiply $x^{2}$ by $x$ .

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Step 3.2.1.1

Raise $x$ to the power of $1$ .

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

Step 3.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.2.2

Add $2$ and $1$ .

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

Step 3.3

Move $- 1$ to the left of $x^{2}$ .

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

Step 3.4

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

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

Step 3.5

Apply the distributive property.

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

Step 3.6

Multiply $x$ by $x$ by adding the exponents.

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Step 3.6.1

Move $x$ .

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

Step 3.6.2

Multiply $x$ by $x$ .

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

Step 3.7

Multiply $- 1$ by $- 5$ .

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

Step 4

Subtract $5 x^{2}$ from $- x^{2}$ .

$\frac{x^{3} - 6 x^{2} + 5 x + 4}{x - 1}$