Calculus Examples

$\frac{3 x^{2} + 10 x - 8}{x^{2} - 16}$

Step 1

Factor by grouping.

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Step 1.1

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 $a \cdot c = 3 \cdot - 8 = - 24$ and whose sum is $b = 10$ .

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Step 1.1.1

Factor $10$ out of $10 x$ .

$\frac{3 x^{2} + 10 (x) - 8}{x^{2} - 16}$

Step 1.1.2

Rewrite $10$ as $- 2$ plus $12$

$\frac{3 x^{2} + (- 2 + 12) x - 8}{x^{2} - 16}$

Step 1.1.3

Apply the distributive property.

$\frac{3 x^{2} - 2 x + 12 x - 8}{x^{2} - 16}$

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

$\frac{(3 x^{2} - 2 x) + 12 x - 8}{x^{2} - 16}$

Step 1.2.2

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

$\frac{x (3 x - 2) + 4 (3 x - 2)}{x^{2} - 16}$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 2$ .

$\frac{(3 x - 2) (x + 4)}{x^{2} - 16}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $16$ as $4^{2}$ .

$\frac{(3 x - 2) (x + 4)}{x^{2} - 4^{2}}$

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 $a = x$ and $b = 4$ .

$\frac{(3 x - 2) (x + 4)}{(x + 4) (x - 4)}$

Step 3

Cancel the common factor of $x + 4$ .

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Step 3.1

Cancel the common factor.

$\frac{(3 x - 2) (x + 4)}{(x + 4) (x - 4)}$

Step 3.2

Rewrite the expression.

$\frac{3 x - 2}{x - 4}$