Pre-Algebra Examples

$\frac{v^{2} - 3 v - 40}{v^{2} - 10 v + 25} \div \frac{v - 8}{v - 5}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{v^{2} - 3 v - 40}{v^{2} - 10 v + 25} \cdot \frac{v - 5}{v - 8}$

Step 2

Factor $v^{2} - 3 v - 40$ 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 $- 40$ and whose sum is $- 3$ .

$- 8, 5$

Step 2.2

Write the factored form using these integers.

$\frac{(v - 8) (v + 5)}{v^{2} - 10 v + 25} \cdot \frac{v - 5}{v - 8}$

Step 3

Factor using the perfect square rule.

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Step 3.1

Rewrite $25$ as $5^{2}$ .

$\frac{(v - 8) (v + 5)}{v^{2} - 10 v + 5^{2}} \cdot \frac{v - 5}{v - 8}$

Step 3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 v = 2 \cdot v \cdot 5$

Step 3.3

Rewrite the polynomial.

$\frac{(v - 8) (v + 5)}{v^{2} - 2 \cdot v \cdot 5 + 5^{2}} \cdot \frac{v - 5}{v - 8}$

Step 3.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = v$ and $b = 5$ .

$\frac{(v - 8) (v + 5)}{{(v - 5)}^{2}} \cdot \frac{v - 5}{v - 8}$

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $v - 8$ .

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Step 4.1.1

Cancel the common factor.

$\frac{(v - 8) (v + 5)}{{(v - 5)}^{2}} \cdot \frac{v - 5}{v - 8}$

Step 4.1.2

Rewrite the expression.

$\frac{v + 5}{{(v - 5)}^{2}} (v - 5)$

Step 4.2

Cancel the common factor of $v - 5$ .

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Step 4.2.1

Factor $v - 5$ out of ${(v - 5)}^{2}$ .

$\frac{v + 5}{(v - 5) (v - 5)} (v - 5)$

Step 4.2.2

Cancel the common factor.

$\frac{v + 5}{(v - 5) (v - 5)} (v - 5)$

Step 4.2.3

Rewrite the expression.

$\frac{v + 5}{v - 5}$