Basic Math Examples

$\frac{p^{2} + 12 p + 32}{p^{2} - 2 p - 24} \div (6 p^{2} + 47 p - 8)$

Step 1

Rewrite the division as a fraction.

$\frac{\frac{p^{2} + 12 p + 32}{p^{2} - 2 p - 24}}{6 p^{2} + 47 p - 8}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{p^{2} + 12 p + 32}{p^{2} - 2 p - 24} \cdot \frac{1}{6 p^{2} + 47 p - 8}$

Step 3

Factor $p^{2} + 12 p + 32$ 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 $32$ and whose sum is $12$ .

$4, 8$

Step 3.2

Write the factored form using these integers.

$\frac{(p + 4) (p + 8)}{p^{2} - 2 p - 24} \cdot \frac{1}{6 p^{2} + 47 p - 8}$

Step 4

Factor $p^{2} - 2 p - 24$ 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 $- 24$ and whose sum is $- 2$ .

$- 6, 4$

Step 4.2

Write the factored form using these integers.

$\frac{(p + 4) (p + 8)}{(p - 6) (p + 4)} \cdot \frac{1}{6 p^{2} + 47 p - 8}$

Step 5

Cancel the common factor of $p + 4$ .

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Step 5.1

Cancel the common factor.

$\frac{(p + 4) (p + 8)}{(p - 6) (p + 4)} \cdot \frac{1}{6 p^{2} + 47 p - 8}$

Step 5.2

Rewrite the expression.

$\frac{p + 8}{p - 6} \cdot \frac{1}{6 p^{2} + 47 p - 8}$

Step 6

Factor by grouping.

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Step 6.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 6 \cdot - 8 = - 48$ and whose sum is $b = 47$ .

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Step 6.1.1

Factor $47$ out of $47 p$ .

$\frac{p + 8}{p - 6} \cdot \frac{1}{6 p^{2} + 47 (p) - 8}$

Step 6.1.2

Rewrite $47$ as $- 1$ plus $48$

$\frac{p + 8}{p - 6} \cdot \frac{1}{6 p^{2} + (- 1 + 48) p - 8}$

Step 6.1.3

Apply the distributive property.

$\frac{p + 8}{p - 6} \cdot \frac{1}{6 p^{2} - 1 p + 48 p - 8}$

Step 6.2

Factor out the greatest common factor from each group.

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Step 6.2.1

Group the first two terms and the last two terms.

$\frac{p + 8}{p - 6} \cdot \frac{1}{(6 p^{2} - 1 p) + 48 p - 8}$

Step 6.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{p + 8}{p - 6} \cdot \frac{1}{p (6 p - 1) + 8 (6 p - 1)}$

Step 6.3

Factor the polynomial by factoring out the greatest common factor, $6 p - 1$ .

$\frac{p + 8}{p - 6} \cdot \frac{1}{(6 p - 1) (p + 8)}$

Step 7

Cancel the common factor of $p + 8$ .

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Step 7.1

Factor $p + 8$ out of $(6 p - 1) (p + 8)$ .

$\frac{p + 8}{p - 6} \cdot \frac{1}{(p + 8) (6 p - 1)}$

Step 7.2

Cancel the common factor.

$\frac{p + 8}{p - 6} \cdot \frac{1}{(p + 8) (6 p - 1)}$

Step 7.3

Rewrite the expression.

$\frac{1}{p - 6} \cdot \frac{1}{6 p - 1}$

Step 8

Multiply $\frac{1}{p - 6}$ by $\frac{1}{6 p - 1}$ .

$\frac{1}{(p - 6) (6 p - 1)}$