Algebra Examples

$\frac{x^{3} + 7 x^{2} + 10 x}{x^{4} (x + 2)}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $x$ out of $x^{3} + 7 x^{2} + 10 x$ .

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Step 1.1.1

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

$\frac{x \cdot x^{2} + 7 x^{2} + 10 x}{x^{4} (x + 2)}$

Step 1.1.2

Factor $x$ out of $7 x^{2}$ .

$\frac{x \cdot x^{2} + x (7 x) + 10 x}{x^{4} (x + 2)}$

Step 1.1.3

Factor $x$ out of $10 x$ .

$\frac{x \cdot x^{2} + x (7 x) + x \cdot 10}{x^{4} (x + 2)}$

Step 1.1.4

Factor $x$ out of $x \cdot x^{2} + x (7 x)$ .

$\frac{x (x^{2} + 7 x) + x \cdot 10}{x^{4} (x + 2)}$

Step 1.1.5

Factor $x$ out of $x (x^{2} + 7 x) + x \cdot 10$ .

$\frac{x (x^{2} + 7 x + 10)}{x^{4} (x + 2)}$

Step 1.2

Factor $x^{2} + 7 x + 10$ using the AC method.

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Step 1.2.1

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 $10$ and whose sum is $7$ .

$2, 5$

Step 1.2.2

Write the factored form using these integers.

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

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

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $x$ and $x^{4}$ .

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Step 2.1.1

Factor $x$ out of $x (x + 2) (x + 5)$ .

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

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

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

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

Step 2.1.2.2

Cancel the common factor.

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

Step 2.1.2.3

Rewrite the expression.

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

Step 2.2

Cancel the common factor of $x + 2$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

$\frac{x + 5}{x^{3}}$