Pre-Algebra Examples

$(y^{2} - 9) \div \frac{y^{2} + 2 y - 3}{y^{2} + 1}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$(y^{2} - 9) \frac{y^{2} + 1}{y^{2} + 2 y - 3}$

Step 2

Factor $y^{2} + 2 y - 3$ 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 $- 3$ and whose sum is $2$ .

$- 1, 3$

Step 2.2

Write the factored form using these integers.

$(y^{2} - 9) \frac{y^{2} + 1}{(y - 1) (y + 3)}$

Step 3

Multiply $y^{2} - 9$ by $\frac{y^{2} + 1}{(y - 1) (y + 3)}$ .

$\frac{(y^{2} - 9) (y^{2} + 1)}{(y - 1) (y + 3)}$

Step 4

Simplify the numerator.

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Step 4.1

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

$\frac{(y^{2} - 3^{2}) (y^{2} + 1)}{(y - 1) (y + 3)}$

Step 4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 3$ .

$\frac{(y + 3) (y - 3) (y^{2} + 1)}{(y - 1) (y + 3)}$

Step 5

Cancel the common factor of $y + 3$ .

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Step 5.1

Cancel the common factor.

$\frac{(y + 3) (y - 3) (y^{2} + 1)}{(y - 1) (y + 3)}$

Step 5.2

Rewrite the expression.

$\frac{(y - 3) (y^{2} + 1)}{y - 1}$