Pre-Algebra Examples

$\frac{c^{2} + 11 c + 24}{c^{2} - 2 c - 15} \div (c^{2} + 3 c - 40)$

Step 1

Rewrite the division as a fraction.

$\frac{\frac{c^{2} + 11 c + 24}{c^{2} - 2 c - 15}}{c^{2} + 3 c - 40}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{c^{2} + 11 c + 24}{c^{2} - 2 c - 15} \cdot \frac{1}{c^{2} + 3 c - 40}$

Step 3

Factor $c^{2} + 11 c + 24$ using the AC method.

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Step 3.1

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 $24$ and whose sum is $11$ .

$3, 8$

Step 3.2

Write the factored form using these integers.

$\frac{(c + 3) (c + 8)}{c^{2} - 2 c - 15} \cdot \frac{1}{c^{2} + 3 c - 40}$

Step 4

Factor $c^{2} - 2 c - 15$ 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 $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Step 4.2

Write the factored form using these integers.

$\frac{(c + 3) (c + 8)}{(c - 5) (c + 3)} \cdot \frac{1}{c^{2} + 3 c - 40}$

Step 5

Cancel the common factor of $c + 3$ .

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Step 5.1

Cancel the common factor.

$\frac{(c + 3) (c + 8)}{(c - 5) (c + 3)} \cdot \frac{1}{c^{2} + 3 c - 40}$

Step 5.2

Rewrite the expression.

$\frac{c + 8}{c - 5} \cdot \frac{1}{c^{2} + 3 c - 40}$

Step 6

Factor $c^{2} + 3 c - 40$ using the AC method.

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Step 6.1

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 $- 40$ and whose sum is $3$ .

$- 5, 8$

Step 6.2

Write the factored form using these integers.

$\frac{c + 8}{c - 5} \cdot \frac{1}{(c - 5) (c + 8)}$

Step 7

Simplify terms.

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Step 7.1

Cancel the common factor of $c + 8$ .

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Step 7.1.1

Factor $c + 8$ out of $(c - 5) (c + 8)$ .

$\frac{c + 8}{c - 5} \cdot \frac{1}{(c + 8) (c - 5)}$

Step 7.1.2

Cancel the common factor.

$\frac{c + 8}{c - 5} \cdot \frac{1}{(c + 8) (c - 5)}$

Step 7.1.3

Rewrite the expression.

$\frac{1}{c - 5} \cdot \frac{1}{c - 5}$

Step 7.2

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

$\frac{1}{(c - 5) (c - 5)}$

Step 8

Raise $c - 5$ to the power of $1$ .

$\frac{1}{{(c - 5)}^{1} (c - 5)}$

Step 9

Raise $c - 5$ to the power of $1$ .

$\frac{1}{{(c - 5)}^{1} {(c - 5)}^{1}}$

Step 10

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

$\frac{1}{{(c - 5)}^{1 + 1}}$

Step 11

Add $1$ and $1$ .

$\frac{1}{{(c - 5)}^{2}}$