Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Factor out the greatest common factor from each group.

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Step 1.1.1

Group the first two terms and the last two terms.

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

Step 1.1.2

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

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

Step 1.2

Factor the polynomial by factoring out the greatest common factor, $y - 3$ .

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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 $b = 3$ .

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

Step 3

Cancel the common factor of $y - 3$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

$\frac{y^{2} + 1}{y + 3}$