Algebra Examples

$\frac{x^{2} - x - 6}{x^{2} + 2 x - 15} \cdot \frac{x^{3} + 10 x^{2} + 25 x}{x^{2} + 7 x + 10}$

Step 1

Factor $x^{2} - x - 6$ 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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 1.2

Write the factored form using these integers.

$\frac{(x - 3) (x + 2)}{x^{2} + 2 x - 15} \cdot \frac{x^{3} + 10 x^{2} + 25 x}{x^{2} + 7 x + 10}$

Step 2

Factor $x^{2} + 2 x - 15$ using the AC method.

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Step 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 $- 15$ and whose sum is $2$ .

$- 3, 5$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 3) (x + 2)}{(x - 3) (x + 5)} \cdot \frac{x^{3} + 10 x^{2} + 25 x}{x^{2} + 7 x + 10}$

Step 3

Simplify the numerator.

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Step 3.1

Factor $x$ out of $x^{3} + 10 x^{2} + 25 x$ .

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Step 3.1.1

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

$\frac{(x - 3) (x + 2)}{(x - 3) (x + 5)} \cdot \frac{x \cdot x^{2} + 10 x^{2} + 25 x}{x^{2} + 7 x + 10}$

Step 3.1.2

Factor $x$ out of $10 x^{2}$ .

$\frac{(x - 3) (x + 2)}{(x - 3) (x + 5)} \cdot \frac{x \cdot x^{2} + x (10 x) + 25 x}{x^{2} + 7 x + 10}$

Step 3.1.3

Factor $x$ out of $25 x$ .

$\frac{(x - 3) (x + 2)}{(x - 3) (x + 5)} \cdot \frac{x \cdot x^{2} + x (10 x) + x \cdot 25}{x^{2} + 7 x + 10}$

Step 3.1.4

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

$\frac{(x - 3) (x + 2)}{(x - 3) (x + 5)} \cdot \frac{x (x^{2} + 10 x) + x \cdot 25}{x^{2} + 7 x + 10}$

Step 3.1.5

Factor $x$ out of $x (x^{2} + 10 x) + x \cdot 25$ .

$\frac{(x - 3) (x + 2)}{(x - 3) (x + 5)} \cdot \frac{x (x^{2} + 10 x + 25)}{x^{2} + 7 x + 10}$

Step 3.2

Factor using the perfect square rule.

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Step 3.2.1

Rewrite $25$ as $5^{2}$ .

$\frac{(x - 3) (x + 2)}{(x - 3) (x + 5)} \cdot \frac{x (x^{2} + 10 x + 5^{2})}{x^{2} + 7 x + 10}$

Step 3.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x = 2 \cdot x \cdot 5$

Step 3.2.3

Rewrite the polynomial.

$\frac{(x - 3) (x + 2)}{(x - 3) (x + 5)} \cdot \frac{x (x^{2} + 2 \cdot x \cdot 5 + 5^{2})}{x^{2} + 7 x + 10}$

Step 3.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 5$ .

$\frac{(x - 3) (x + 2)}{(x - 3) (x + 5)} \cdot \frac{x {(x + 5)}^{2}}{x^{2} + 7 x + 10}$

Step 4

Factor $x^{2} + 7 x + 10$ 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 $10$ and whose sum is $7$ .

$2, 5$

Step 4.2

Write the factored form using these integers.

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

Step 5

Simplify terms.

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Step 5.1

Combine.

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

Step 5.2

Cancel the common factor of $x - 3$ .

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Step 5.2.1

Cancel the common factor.

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

Step 5.2.2

Rewrite the expression.

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

Step 5.3

Cancel the common factor of $x + 2$ .

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Step 5.3.1

Cancel the common factor.

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

Step 5.3.2

Rewrite the expression.

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

Step 5.4

Cancel the common factor of ${(x + 5)}^{2}$ and $x + 5$ .

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Step 5.4.1

Factor $x + 5$ out of $x {(x + 5)}^{2}$ .

$\frac{(x + 5) (x (x + 5))}{(x + 5) (x + 5)}$

Step 5.4.2

Cancel the common factors.

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Step 5.4.2.1

Cancel the common factor.

$\frac{(x + 5) (x (x + 5))}{(x + 5) (x + 5)}$

Step 5.4.2.2

Rewrite the expression.

$\frac{x (x + 5)}{x + 5}$

Step 5.5

Cancel the common factor of $x + 5$ .

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Step 5.5.1

Cancel the common factor.

$\frac{x (x + 5)}{x + 5}$

Step 5.5.2

Divide $x$ by $1$ .

$x$