Pre-Algebra Examples

$\frac{x^{2} - 3 x - 10}{x^{2} - 11 x + 5} \div \frac{x^{2} - 5 x - 6}{2 x^{2} - 7 x + 3}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 3 x - 10}{x^{2} - 11 x + 5} \cdot \frac{2 x^{2} - 7 x + 3}{x^{2} - 5 x - 6}$

Step 2

Factor $x^{2} - 3 x - 10$ 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 $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 5) (x + 2)}{x^{2} - 11 x + 5} \cdot \frac{2 x^{2} - 7 x + 3}{x^{2} - 5 x - 6}$

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = 2 \cdot 3 = 6$ and whose sum is $b = - 7$ .

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Step 3.1.1

Factor $- 7$ out of $- 7 x$ .

$\frac{(x - 5) (x + 2)}{x^{2} - 11 x + 5} \cdot \frac{2 x^{2} - 7 (x) + 3}{x^{2} - 5 x - 6}$

Step 3.1.2

Rewrite $- 7$ as $- 1$ plus $- 6$

$\frac{(x - 5) (x + 2)}{x^{2} - 11 x + 5} \cdot \frac{2 x^{2} + (- 1 - 6) x + 3}{x^{2} - 5 x - 6}$

Step 3.1.3

Apply the distributive property.

$\frac{(x - 5) (x + 2)}{x^{2} - 11 x + 5} \cdot \frac{2 x^{2} - 1 x - 6 x + 3}{x^{2} - 5 x - 6}$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$\frac{(x - 5) (x + 2)}{x^{2} - 11 x + 5} \cdot \frac{(2 x^{2} - 1 x) - 6 x + 3}{x^{2} - 5 x - 6}$

Step 3.2.2

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

$\frac{(x - 5) (x + 2)}{x^{2} - 11 x + 5} \cdot \frac{x (2 x - 1) - 3 (2 x - 1)}{x^{2} - 5 x - 6}$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$\frac{(x - 5) (x + 2)}{x^{2} - 11 x + 5} \cdot \frac{(2 x - 1) (x - 3)}{x^{2} - 5 x - 6}$

Step 4

Factor $x^{2} - 5 x - 6$ 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 $- 5$ .

$- 6, 1$

Step 4.2

Write the factored form using these integers.

$\frac{(x - 5) (x + 2)}{x^{2} - 11 x + 5} \cdot \frac{(2 x - 1) (x - 3)}{(x - 6) (x + 1)}$

Step 5

Multiply $\frac{(x - 5) (x + 2)}{x^{2} - 11 x + 5}$ by $\frac{(2 x - 1) (x - 3)}{(x - 6) (x + 1)}$ .

$\frac{(x - 5) (x + 2) (2 x - 1) (x - 3)}{(x^{2} - 11 x + 5) (x - 6) (x + 1)}$