Basic Math Examples

$\frac{p^{2} - 9}{2} \cdot \frac{p^{2} - p - 6}{p^{2} - 6 p + 9}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $9$ as $3^{2}$ .

$\frac{p^{2} - 3^{2}}{2} \cdot \frac{p^{2} - p - 6}{p^{2} - 6 p + 9}$

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 $b = 3$ .

$\frac{(p + 3) (p - 3)}{2} \cdot \frac{p^{2} - p - 6}{p^{2} - 6 p + 9}$

Step 2

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

$- 3, 2$

Step 2.2

Write the factored form using these integers.

$\frac{(p + 3) (p - 3)}{2} \cdot \frac{(p - 3) (p + 2)}{p^{2} - 6 p + 9}$

Step 3

Factor using the perfect square rule.

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Step 3.1

Rewrite $9$ as $3^{2}$ .

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

Step 3.2

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

$6 p = 2 \cdot p \cdot 3$

Step 3.3

Rewrite the polynomial.

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

Step 3.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = p$ and $b = 3$ .

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

Step 4

Simplify terms.

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Step 4.1

Combine.

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

Step 4.2

Cancel the common factor of $p - 3$ and ${(p - 3)}^{2}$ .

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Step 4.2.1

Factor $p - 3$ out of $(p + 3) (p - 3) ((p - 3) (p + 2))$ .

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

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $p - 3$ out of $2 {(p - 3)}^{2}$ .

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

Step 4.2.2.2

Cancel the common factor.

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

Step 4.2.2.3

Rewrite the expression.

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

Step 4.3

Cancel the common factor of $p - 3$ .

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Step 4.3.1

Cancel the common factor.

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

Step 4.3.2

Rewrite the expression.

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