Basic Math Examples

$\frac{s^{3} - s^{2} - 6 s}{s - 5} \div (s^{2} - s - 6)$

Step 1

Rewrite the division as a fraction.

$\frac{\frac{s^{3} - s^{2} - 6 s}{s - 5}}{s^{2} - s - 6}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{s^{3} - s^{2} - 6 s}{s - 5} \cdot \frac{1}{s^{2} - s - 6}$

Step 3

Simplify the numerator.

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Step 3.1

Factor $s$ out of $s^{3} - s^{2} - 6 s$ .

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Step 3.1.1

Factor $s$ out of $s^{3}$ .

$\frac{s \cdot s^{2} - s^{2} - 6 s}{s - 5} \cdot \frac{1}{s^{2} - s - 6}$

Step 3.1.2

Factor $s$ out of $- s^{2}$ .

$\frac{s \cdot s^{2} + s (- s) - 6 s}{s - 5} \cdot \frac{1}{s^{2} - s - 6}$

Step 3.1.3

Factor $s$ out of $- 6 s$ .

$\frac{s \cdot s^{2} + s (- s) + s \cdot - 6}{s - 5} \cdot \frac{1}{s^{2} - s - 6}$

Step 3.1.4

Factor $s$ out of $s \cdot s^{2} + s (- s)$ .

$\frac{s (s^{2} - s) + s \cdot - 6}{s - 5} \cdot \frac{1}{s^{2} - s - 6}$

Step 3.1.5

Factor $s$ out of $s (s^{2} - s) + s \cdot - 6$ .

$\frac{s (s^{2} - s - 6)}{s - 5} \cdot \frac{1}{s^{2} - s - 6}$

Step 3.2

Factor $s^{2} - s - 6$ using the AC method.

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Step 3.2.1

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 3.2.2

Write the factored form using these integers.

$\frac{s ((s - 3) (s + 2))}{s - 5} \cdot \frac{1}{s^{2} - s - 6}$

$\frac{s (s - 3) (s + 2)}{s - 5} \cdot \frac{1}{s^{2} - s - 6}$

Step 4

Factor $s^{2} - s - 6$ using the AC method.

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Step 4.1

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 4.2

Write the factored form using these integers.

$\frac{s (s - 3) (s + 2)}{s - 5} \cdot \frac{1}{(s - 3) (s + 2)}$

Step 5

Cancel the common factor of $(s - 3) (s + 2)$ .

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Step 5.1

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

$\frac{(s - 3) (s + 2) (s)}{s - 5} \cdot \frac{1}{(s - 3) (s + 2)}$

Step 5.2

Cancel the common factor.

$\frac{(s - 3) (s + 2) s}{s - 5} \cdot \frac{1}{(s - 3) (s + 2)}$

Step 5.3

Rewrite the expression.

$\frac{s}{s - 5}$