Basic Math Examples

$\frac{5 b^{2} - 26 b - 24}{5 b^{2} + 4 b} \cdot \frac{b^{2} - 36 b}{b^{2} - 36}$

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 $a \cdot c = 5 \cdot - 24 = - 120$ and whose sum is $b = - 26$ .

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Step 1.1.1

Factor $- 26$ out of $- 26 b$ .

$\frac{5 b^{2} - 26 (b) - 24}{5 b^{2} + 4 b} \cdot \frac{b^{2} - 36 b}{b^{2} - 36}$

Step 1.1.2

Rewrite $- 26$ as $4$ plus $- 30$

$\frac{5 b^{2} + (4 - 30) b - 24}{5 b^{2} + 4 b} \cdot \frac{b^{2} - 36 b}{b^{2} - 36}$

Step 1.1.3

Apply the distributive property.

$\frac{5 b^{2} + 4 b - 30 b - 24}{5 b^{2} + 4 b} \cdot \frac{b^{2} - 36 b}{b^{2} - 36}$

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.

$\frac{(5 b^{2} + 4 b) - 30 b - 24}{5 b^{2} + 4 b} \cdot \frac{b^{2} - 36 b}{b^{2} - 36}$

Step 1.2.2

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

$\frac{b (5 b + 4) - 6 (5 b + 4)}{5 b^{2} + 4 b} \cdot \frac{b^{2} - 36 b}{b^{2} - 36}$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $5 b + 4$ .

$\frac{(5 b + 4) (b - 6)}{5 b^{2} + 4 b} \cdot \frac{b^{2} - 36 b}{b^{2} - 36}$

Step 2

Simplify with factoring out.

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Step 2.1

Factor $b$ out of $5 b^{2} + 4 b$ .

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Step 2.1.1

Factor $b$ out of $5 b^{2}$ .

$\frac{(5 b + 4) (b - 6)}{b (5 b) + 4 b} \cdot \frac{b^{2} - 36 b}{b^{2} - 36}$

Step 2.1.2

Factor $b$ out of $4 b$ .

$\frac{(5 b + 4) (b - 6)}{b (5 b) + b \cdot 4} \cdot \frac{b^{2} - 36 b}{b^{2} - 36}$

Step 2.1.3

Factor $b$ out of $b (5 b) + b \cdot 4$ .

$\frac{(5 b + 4) (b - 6)}{b (5 b + 4)} \cdot \frac{b^{2} - 36 b}{b^{2} - 36}$

Step 2.2

Factor $b$ out of $b^{2} - 36 b$ .

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Step 2.2.1

Factor $b$ out of $b^{2}$ .

$\frac{(5 b + 4) (b - 6)}{b (5 b + 4)} \cdot \frac{b \cdot b - 36 b}{b^{2} - 36}$

Step 2.2.2

Factor $b$ out of $- 36 b$ .

$\frac{(5 b + 4) (b - 6)}{b (5 b + 4)} \cdot \frac{b \cdot b + b \cdot - 36}{b^{2} - 36}$

Step 2.2.3

Factor $b$ out of $b \cdot b + b \cdot - 36$ .

$\frac{(5 b + 4) (b - 6)}{b (5 b + 4)} \cdot \frac{b (b - 36)}{b^{2} - 36}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $36$ as $6^{2}$ .

$\frac{(5 b + 4) (b - 6)}{b (5 b + 4)} \cdot \frac{b (b - 36)}{b^{2} - 6^{2}}$

Step 3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = b$ and $b = 6$ .

$\frac{(5 b + 4) (b - 6)}{b (5 b + 4)} \cdot \frac{b (b - 36)}{(b + 6) (b - 6)}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $b - 6$ .

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Step 4.1.1

Factor $b - 6$ out of $(5 b + 4) (b - 6)$ .

$\frac{(b - 6) (5 b + 4)}{b (5 b + 4)} \cdot \frac{b (b - 36)}{(b + 6) (b - 6)}$

Step 4.1.2

Factor $b - 6$ out of $(b + 6) (b - 6)$ .

$\frac{(b - 6) (5 b + 4)}{b (5 b + 4)} \cdot \frac{b (b - 36)}{(b - 6) (b + 6)}$

Step 4.1.3

Cancel the common factor.

$\frac{(b - 6) (5 b + 4)}{b (5 b + 4)} \cdot \frac{b (b - 36)}{(b - 6) (b + 6)}$

Step 4.1.4

Rewrite the expression.

$\frac{5 b + 4}{b (5 b + 4)} \cdot \frac{b (b - 36)}{b + 6}$

Step 4.2

Cancel the common factor of $b$ .

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Step 4.2.1

Cancel the common factor.

$\frac{5 b + 4}{b (5 b + 4)} \cdot \frac{b (b - 36)}{b + 6}$

Step 4.2.2

Rewrite the expression.

$\frac{5 b + 4}{5 b + 4} \cdot \frac{b - 36}{b + 6}$

Step 4.3

Multiply $\frac{5 b + 4}{5 b + 4}$ by $\frac{b - 36}{b + 6}$ .

$\frac{(5 b + 4) (b - 36)}{(5 b + 4) (b + 6)}$

Step 4.4

Cancel the common factor of $5 b + 4$ .

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Step 4.4.1

Cancel the common factor.

$\frac{(5 b + 4) (b - 36)}{(5 b + 4) (b + 6)}$

Step 4.4.2

Rewrite the expression.

$\frac{b - 36}{b + 6}$