Precalculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Factor ${(x + 1)}^{2}$ out of ${(x + 1)}^{3} (x - 2) + 3 {(x + 1)}^{2}$ .

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Step 1.1.1

Factor ${(x + 1)}^{2}$ out of ${(x + 1)}^{3} (x - 2)$ .

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

Step 1.1.2

Factor ${(x + 1)}^{2}$ out of $3 {(x + 1)}^{2}$ .

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

Step 1.1.3

Factor ${(x + 1)}^{2}$ out of ${(x + 1)}^{2} ((x + 1) (x - 2)) + {(x + 1)}^{2} \cdot 3$ .

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

Step 1.2

Expand $(x + 1) (x - 2)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.3.1.2

Move $- 2$ to the left of $x$ .

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

Step 1.3.1.3

Multiply $x$ by $1$ .

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

Step 1.3.1.4

Multiply $- 2$ by $1$ .

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

Step 1.3.2

Add $- 2 x$ and $x$ .

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

Step 1.4

Add $- 2$ and $3$ .

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

Step 2

Cancel the common factors.

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Step 2.1

Factor ${(x + 1)}^{2}$ out of ${(x + 1)}^{4}$ .

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

Step 2.2

Cancel the common factor.

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

Step 2.3

Rewrite the expression.

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