Algebra Examples

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

Step 1

Factor $x^{2} + 2 x - 3$ using the AC method.

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Step 1.1

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 $2$ .

$- 1, 3$

Step 1.2

Write the factored form using these integers.

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

Step 2

Simplify terms.

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Step 2.1

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

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Step 2.1.1

Factor $x^{2}$ out of $x^{3}$ .

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

Step 2.1.2

Multiply by $1$ .

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

Step 2.1.3

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot 1$ .

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

Step 2.2

Cancel the common factor of $x + 3$ .

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Step 2.2.1

Factor $x + 3$ out of $(x - 1) (x + 3)$ .

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

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

Step 2.3

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

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Step 2.3.1

Cancel the common factor.

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

Step 2.3.2

Rewrite the expression.

$(x - 1) \cdot (x + 1)$

Step 3

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

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Step 3.1

Apply the distributive property.

$x (x + 1) - 1 (x + 1)$

Step 3.2

Apply the distributive property.

$x \cdot x + x \cdot 1 - 1 (x + 1)$

Step 3.3

Apply the distributive property.

$x \cdot x + x \cdot 1 - 1 x - 1 \cdot 1$

Step 4

Simplify terms.

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Step 4.1

Combine the opposite terms in $x \cdot x + x \cdot 1 - 1 x - 1 \cdot 1$ .

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Step 4.1.1

Reorder the factors in the terms $x \cdot 1$ and $- 1 x$ .

$x \cdot x + 1 x - 1 x - 1 \cdot 1$

Step 4.1.2

Subtract $1 x$ from $1 x$ .

$x \cdot x + 0 - 1 \cdot 1$

Step 4.1.3

Add $x \cdot x$ and $0$ .

$x \cdot x - 1 \cdot 1$

Step 4.2

Simplify each term.

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Step 4.2.1

Multiply $x$ by $x$ .

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

Step 4.2.2

Multiply $- 1$ by $1$ .

$x^{2} - 1$