Basic Math Examples

$\frac{p^{2} - 4}{3 p^{2} + 3 p w} \cdot \frac{p^{2} + 6 p + p w + 6 w}{p^{2} + 8 p + 12}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $4$ as $2^{2}$ .

$\frac{p^{2} - 2^{2}}{3 p^{2} + 3 p w} \cdot \frac{p^{2} + 6 p + p w + 6 w}{p^{2} + 8 p + 12}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = p$ and $b = 2$ .

$\frac{(p + 2) (p - 2)}{3 p^{2} + 3 p w} \cdot \frac{p^{2} + 6 p + p w + 6 w}{p^{2} + 8 p + 12}$

Step 2

Factor $3 p$ out of $3 p^{2} + 3 p w$ .

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Step 2.1

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

$\frac{(p + 2) (p - 2)}{3 p (p) + 3 p w} \cdot \frac{p^{2} + 6 p + p w + 6 w}{p^{2} + 8 p + 12}$

Step 2.2

Factor $3 p$ out of $3 p w$ .

$\frac{(p + 2) (p - 2)}{3 p (p) + 3 p (w)} \cdot \frac{p^{2} + 6 p + p w + 6 w}{p^{2} + 8 p + 12}$

Step 2.3

Factor $3 p$ out of $3 p (p) + 3 p (w)$ .

$\frac{(p + 2) (p - 2)}{3 p (p + w)} \cdot \frac{p^{2} + 6 p + p w + 6 w}{p^{2} + 8 p + 12}$

Step 3

Simplify the numerator.

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Step 3.1

Factor out the greatest common factor from each group.

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Step 3.1.1

Group the first two terms and the last two terms.

$\frac{(p + 2) (p - 2)}{3 p (p + w)} \cdot \frac{(p^{2} + 6 p) + p w + 6 w}{p^{2} + 8 p + 12}$

Step 3.1.2

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

$\frac{(p + 2) (p - 2)}{3 p (p + w)} \cdot \frac{p (p + 6) + w (p + 6)}{p^{2} + 8 p + 12}$

Step 3.2

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

$\frac{(p + 2) (p - 2)}{3 p (p + w)} \cdot \frac{(p + 6) (p + w)}{p^{2} + 8 p + 12}$

Step 4

Factor $p^{2} + 8 p + 12$ 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 $12$ and whose sum is $8$ .

$2, 6$

Step 4.2

Write the factored form using these integers.

$\frac{(p + 2) (p - 2)}{3 p (p + w)} \cdot \frac{(p + 6) (p + w)}{(p + 2) (p + 6)}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $p + 2$ .

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Step 5.1.1

Cancel the common factor.

$\frac{(p + 2) (p - 2)}{3 p (p + w)} \cdot \frac{(p + 6) (p + w)}{(p + 2) (p + 6)}$

Step 5.1.2

Rewrite the expression.

$\frac{p - 2}{3 p (p + w)} \cdot \frac{(p + 6) (p + w)}{p + 6}$

Step 5.2

Cancel the common factor of $p + w$ .

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Step 5.2.1

Factor $p + w$ out of $3 p (p + w)$ .

$\frac{p - 2}{(p + w) (3 p)} \cdot \frac{(p + 6) (p + w)}{p + 6}$

Step 5.2.2

Factor $p + w$ out of $(p + 6) (p + w)$ .

$\frac{p - 2}{(p + w) (3 p)} \cdot \frac{(p + w) (p + 6)}{p + 6}$

Step 5.2.3

Cancel the common factor.

$\frac{p - 2}{(p + w) (3 p)} \cdot \frac{(p + w) (p + 6)}{p + 6}$

Step 5.2.4

Rewrite the expression.

$\frac{p - 2}{3 p} \cdot \frac{p + 6}{p + 6}$

Step 5.3

Multiply $\frac{p - 2}{3 p}$ by $\frac{p + 6}{p + 6}$ .

$\frac{(p - 2) (p + 6)}{3 p (p + 6)}$

Step 5.4

Cancel the common factor of $p + 6$ .

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Step 5.4.1

Cancel the common factor.

$\frac{(p - 2) (p + 6)}{3 p (p + 6)}$

Step 5.4.2

Rewrite the expression.

$\frac{p - 2}{3 p}$