Algebra Examples

$\frac{- 4}{x - 4} \cdot \frac{x^{2} + 17 x + 70}{x^{2} + 3 x - 28} \div \frac{- x - 10}{x^{2} - 8 x + 16}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{- 4}{x - 4} \cdot \frac{x^{2} + 17 x + 70}{x^{2} + 3 x - 28} \frac{x^{2} - 8 x + 16}{- x - 10}$

Step 2

Factor $x^{2} + 17 x + 70$ 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 $70$ and whose sum is $17$ .

$7, 10$

Step 2.2

Write the factored form using these integers.

$\frac{- 4}{x - 4} \cdot \frac{(x + 7) (x + 10)}{x^{2} + 3 x - 28} \frac{x^{2} - 8 x + 16}{- x - 10}$

Step 3

Factor $x^{2} + 3 x - 28$ using the AC method.

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Step 3.1

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 $- 28$ and whose sum is $3$ .

$- 4, 7$

Step 3.2

Write the factored form using these integers.

$\frac{- 4}{x - 4} \cdot \frac{(x + 7) (x + 10)}{(x - 4) (x + 7)} \frac{x^{2} - 8 x + 16}{- x - 10}$

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Move the negative in front of the fraction.

$- \frac{4}{x - 4} \cdot \frac{(x + 7) (x + 10)}{(x - 4) (x + 7)} \frac{x^{2} - 8 x + 16}{- x - 10}$

Step 4.2

Cancel the common factor of $x + 7$ .

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Step 4.2.1

Cancel the common factor.

$- \frac{4}{x - 4} \cdot \frac{(x + 7) (x + 10)}{(x - 4) (x + 7)} \frac{x^{2} - 8 x + 16}{- x - 10}$

Step 4.2.2

Rewrite the expression.

$- \frac{4}{x - 4} \cdot \frac{x + 10}{x - 4} \frac{x^{2} - 8 x + 16}{- x - 10}$

Step 5

Multiply $- \frac{4}{x - 4} \cdot \frac{x + 10}{x - 4}$ .

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Step 5.1

Multiply $\frac{x + 10}{x - 4}$ by $\frac{4}{x - 4}$ .

$- \frac{(x + 10) \cdot 4}{(x - 4) (x - 4)} \cdot \frac{x^{2} - 8 x + 16}{- x - 10}$

Step 5.2

Raise $x - 4$ to the power of $1$ .

$- \frac{(x + 10) \cdot 4}{{(x - 4)}^{1} (x - 4)} \cdot \frac{x^{2} - 8 x + 16}{- x - 10}$

Step 5.3

Raise $x - 4$ to the power of $1$ .

$- \frac{(x + 10) \cdot 4}{{(x - 4)}^{1} {(x - 4)}^{1}} \cdot \frac{x^{2} - 8 x + 16}{- x - 10}$

Step 5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{(x + 10) \cdot 4}{{(x - 4)}^{1 + 1}} \cdot \frac{x^{2} - 8 x + 16}{- x - 10}$

Step 5.5

Add $1$ and $1$ .

$- \frac{(x + 10) \cdot 4}{{(x - 4)}^{2}} \cdot \frac{x^{2} - 8 x + 16}{- x - 10}$

Step 6

Move $4$ to the left of $x + 10$ .

$- \frac{4 \cdot (x + 10)}{{(x - 4)}^{2}} \cdot \frac{x^{2} - 8 x + 16}{- x - 10}$

Step 7

Factor using the perfect square rule.

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Step 7.1

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

$- \frac{4 (x + 10)}{{(x - 4)}^{2}} \cdot \frac{x^{2} - 8 x + 4^{2}}{- x - 10}$

Step 7.2

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

$8 x = 2 \cdot x \cdot 4$

Step 7.3

Rewrite the polynomial.

$- \frac{4 (x + 10)}{{(x - 4)}^{2}} \cdot \frac{x^{2} - 2 \cdot x \cdot 4 + 4^{2}}{- x - 10}$

Step 7.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 4$ .

$- \frac{4 (x + 10)}{{(x - 4)}^{2}} \cdot \frac{{(x - 4)}^{2}}{- x - 10}$

Step 8

Simplify terms.

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Step 8.1

Cancel the common factor of ${(x - 4)}^{2}$ .

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Step 8.1.1

Move the leading negative in $- \frac{4 (x + 10)}{{(x - 4)}^{2}}$ into the numerator.

$\frac{- 4 (x + 10)}{{(x - 4)}^{2}} \cdot \frac{{(x - 4)}^{2}}{- x - 10}$

Step 8.1.2

Cancel the common factor.

$\frac{- 4 (x + 10)}{{(x - 4)}^{2}} \cdot \frac{{(x - 4)}^{2}}{- x - 10}$

Step 8.1.3

Rewrite the expression.

$- 4 (x + 10) \frac{1}{- x - 10}$

Step 8.2

Combine $\frac{1}{- x - 10}$ and $- 4$ .

$\frac{- 4}{- x - 10} (x + 10)$

Step 8.3

Move the negative in front of the fraction.

$- \frac{4}{- x - 10} (x + 10)$

Step 8.4

Apply the distributive property.

$- \frac{4}{- x - 10} x - \frac{4}{- x - 10} \cdot 10$

Step 8.5

Combine $x$ and $\frac{4}{- x - 10}$ .

$- \frac{x \cdot 4}{- x - 10} - \frac{4}{- x - 10} \cdot 10$

Step 9

Multiply $- \frac{4}{- x - 10} \cdot 10$ .

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Step 9.1

Multiply $10$ by $- 1$ .

$- \frac{x \cdot 4}{- x - 10} - 10 \frac{4}{- x - 10}$

Step 9.2

Combine $- 10$ and $\frac{4}{- x - 10}$ .

$- \frac{x \cdot 4}{- x - 10} + \frac{- 10 \cdot 4}{- x - 10}$

Step 9.3

Multiply $- 10$ by $4$ .

$- \frac{x \cdot 4}{- x - 10} + \frac{- 40}{- x - 10}$

Step 10

Combine the numerators over the common denominator.

$\frac{- x \cdot 4 - 40}{- x - 10}$

Step 11

Multiply $4$ by $- 1$ .

$\frac{- 4 x - 40}{- x - 10}$

Step 12

Factor $4$ out of $- 4 x - 40$ .

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Step 12.1

Factor $4$ out of $- 4 x$ .

$\frac{4 (- x) - 40}{- x - 10}$

Step 12.2

Factor $4$ out of $- 40$ .

$\frac{4 (- x) + 4 (- 10)}{- x - 10}$

Step 12.3

Factor $4$ out of $4 (- x) + 4 (- 10)$ .

$\frac{4 (- x - 10)}{- x - 10}$

Step 13

Cancel the common factor of $- x - 10$ .

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Step 13.1

Cancel the common factor.

$\frac{4 (- x - 10)}{- x - 10}$

Step 13.2

Divide $4$ by $1$ .

$4$