Algebra Examples

$\frac{- 9 x}{x^{2} - 8 x} \cdot \frac{9 x^{3} + 36 x^{2} - 189 x}{x^{2} - 10 x + 21} \div \frac{x + 7}{x^{2} - 15 x + 56}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{- 9 x}{x^{2} - 8 x} \cdot \frac{9 x^{3} + 36 x^{2} - 189 x}{x^{2} - 10 x + 21} \frac{x^{2} - 15 x + 56}{x + 7}$

Step 2

Factor $x$ out of $x^{2} - 8 x$ .

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Step 2.1

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

$\frac{- 9 x}{x \cdot x - 8 x} \cdot \frac{9 x^{3} + 36 x^{2} - 189 x}{x^{2} - 10 x + 21} \frac{x^{2} - 15 x + 56}{x + 7}$

Step 2.2

Factor $x$ out of $- 8 x$ .

$\frac{- 9 x}{x \cdot x + x \cdot - 8} \cdot \frac{9 x^{3} + 36 x^{2} - 189 x}{x^{2} - 10 x + 21} \frac{x^{2} - 15 x + 56}{x + 7}$

Step 2.3

Factor $x$ out of $x \cdot x + x \cdot - 8$ .

$\frac{- 9 x}{x (x - 8)} \cdot \frac{9 x^{3} + 36 x^{2} - 189 x}{x^{2} - 10 x + 21} \frac{x^{2} - 15 x + 56}{x + 7}$

Step 3

Simplify the numerator.

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Step 3.1

Factor $9 x$ out of $9 x^{3} + 36 x^{2} - 189 x$ .

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Step 3.1.1

Factor $9 x$ out of $9 x^{3}$ .

$\frac{- 9 x}{x (x - 8)} \cdot \frac{9 x (x^{2}) + 36 x^{2} - 189 x}{x^{2} - 10 x + 21} \frac{x^{2} - 15 x + 56}{x + 7}$

Step 3.1.2

Factor $9 x$ out of $36 x^{2}$ .

$\frac{- 9 x}{x (x - 8)} \cdot \frac{9 x (x^{2}) + 9 x (4 x) - 189 x}{x^{2} - 10 x + 21} \frac{x^{2} - 15 x + 56}{x + 7}$

Step 3.1.3

Factor $9 x$ out of $- 189 x$ .

$\frac{- 9 x}{x (x - 8)} \cdot \frac{9 x (x^{2}) + 9 x (4 x) + 9 x (- 21)}{x^{2} - 10 x + 21} \frac{x^{2} - 15 x + 56}{x + 7}$

Step 3.1.4

Factor $9 x$ out of $9 x (x^{2}) + 9 x (4 x)$ .

$\frac{- 9 x}{x (x - 8)} \cdot \frac{9 x (x^{2} + 4 x) + 9 x (- 21)}{x^{2} - 10 x + 21} \frac{x^{2} - 15 x + 56}{x + 7}$

Step 3.1.5

Factor $9 x$ out of $9 x (x^{2} + 4 x) + 9 x (- 21)$ .

$\frac{- 9 x}{x (x - 8)} \cdot \frac{9 x (x^{2} + 4 x - 21)}{x^{2} - 10 x + 21} \frac{x^{2} - 15 x + 56}{x + 7}$

Step 3.2

Factor $x^{2} + 4 x - 21$ 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 $- 21$ and whose sum is $4$ .

$- 3, 7$

Step 3.2.2

Write the factored form using these integers.

$\frac{- 9 x}{x (x - 8)} \cdot \frac{9 x ((x - 3) (x + 7))}{x^{2} - 10 x + 21} \frac{x^{2} - 15 x + 56}{x + 7}$

$\frac{- 9 x}{x (x - 8)} \cdot \frac{9 x (x - 3) (x + 7)}{x^{2} - 10 x + 21} \frac{x^{2} - 15 x + 56}{x + 7}$

Step 4

Factor $x^{2} - 10 x + 21$ 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 $21$ and whose sum is $- 10$ .

$- 7, - 3$

Step 4.2

Write the factored form using these integers.

$\frac{- 9 x}{x (x - 8)} \cdot \frac{9 x (x - 3) (x + 7)}{(x - 7) (x - 3)} \frac{x^{2} - 15 x + 56}{x + 7}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $x$ .

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Step 5.1.1

Factor $x$ out of $9 x (x - 3) (x + 7)$ .

$\frac{- 9 x}{x (x - 8)} \cdot \frac{x ((9 (x - 3)) (x + 7))}{(x - 7) (x - 3)} \frac{x^{2} - 15 x + 56}{x + 7}$

Step 5.1.2

Cancel the common factor.

$\frac{- 9 x}{x (x - 8)} \cdot \frac{x ((9 (x - 3)) (x + 7))}{(x - 7) (x - 3)} \frac{x^{2} - 15 x + 56}{x + 7}$

Step 5.1.3

Rewrite the expression.

$\frac{- 9 x}{x - 8} \cdot \frac{(9 (x - 3)) (x + 7)}{(x - 7) (x - 3)} \frac{x^{2} - 15 x + 56}{x + 7}$

Step 5.2

Multiply $\frac{- 9 x}{x - 8}$ by $\frac{(9 (x - 3)) (x + 7)}{(x - 7) (x - 3)}$ .

$\frac{- 9 x ((9 (x - 3)) (x + 7))}{(x - 8) ((x - 7) (x - 3))} \cdot \frac{x^{2} - 15 x + 56}{x + 7}$

Step 5.3

Multiply $9$ by $- 9$ .

$\frac{- 81 x ((x - 3) (x + 7))}{(x - 8) ((x - 7) (x - 3))} \cdot \frac{x^{2} - 15 x + 56}{x + 7}$

Step 5.4

Cancel the common factor of $x + 7$ .

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Step 5.4.1

Factor $x + 7$ out of $- 81 x ((x - 3) (x + 7))$ .

$\frac{(x + 7) (- 81 x (x - 3))}{(x - 8) ((x - 7) (x - 3))} \cdot \frac{x^{2} - 15 x + 56}{x + 7}$

Step 5.4.2

Cancel the common factor.

$\frac{(x + 7) (- 81 x (x - 3))}{(x - 8) ((x - 7) (x - 3))} \cdot \frac{x^{2} - 15 x + 56}{x + 7}$

Step 5.4.3

Rewrite the expression.

$\frac{- 81 x (x - 3)}{(x - 8) ((x - 7) (x - 3))} (x^{2} - 15 x + 56)$

Step 5.5

Cancel the common factor of $x - 3$ .

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Step 5.5.1

Cancel the common factor.

$\frac{- 81 x (x - 3)}{(x - 8) ((x - 7) (x - 3))} (x^{2} - 15 x + 56)$

Step 5.5.2

Rewrite the expression.

$\frac{- 81 x}{(x - 8) (x - 7)} (x^{2} - 15 x + 56)$

Step 5.6

Move the negative in front of the fraction.

$- \frac{81 x}{(x - 8) (x - 7)} (x^{2} - 15 x + 56)$

Step 5.7

Apply the distributive property.

$- \frac{81 x}{(x - 8) (x - 7)} x^{2} - \frac{81 x}{(x - 8) (x - 7)} (- 15 x) - \frac{81 x}{(x - 8) (x - 7)} \cdot 56$

Step 6

Simplify.

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Step 6.1

Multiply $- \frac{81 x}{(x - 8) (x - 7)} x^{2}$ .

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Step 6.1.1

Combine $x^{2}$ and $\frac{81 x}{(x - 8) (x - 7)}$ .

$- \frac{x^{2} (81 x)}{(x - 8) (x - 7)} - \frac{81 x}{(x - 8) (x - 7)} (- 15 x) - \frac{81 x}{(x - 8) (x - 7)} \cdot 56$

Step 6.1.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 6.1.2.1

Move $x$ .

$- \frac{x \cdot x^{2} \cdot 81}{(x - 8) (x - 7)} - \frac{81 x}{(x - 8) (x - 7)} (- 15 x) - \frac{81 x}{(x - 8) (x - 7)} \cdot 56$

Step 6.1.2.2

Multiply $x$ by $x^{2}$ .

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Step 6.1.2.2.1

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

$- \frac{x^{1} x^{2} \cdot 81}{(x - 8) (x - 7)} - \frac{81 x}{(x - 8) (x - 7)} (- 15 x) - \frac{81 x}{(x - 8) (x - 7)} \cdot 56$

Step 6.1.2.2.2

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

$- \frac{x^{1 + 2} \cdot 81}{(x - 8) (x - 7)} - \frac{81 x}{(x - 8) (x - 7)} (- 15 x) - \frac{81 x}{(x - 8) (x - 7)} \cdot 56$

Step 6.1.2.3

Add $1$ and $2$ .

$- \frac{x^{3} \cdot 81}{(x - 8) (x - 7)} - \frac{81 x}{(x - 8) (x - 7)} (- 15 x) - \frac{81 x}{(x - 8) (x - 7)} \cdot 56$

Step 6.2

Multiply $- \frac{81 x}{(x - 8) (x - 7)} (- 15 x)$ .

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Step 6.2.1

Multiply $- 15$ by $- 1$ .

$- \frac{x^{3} \cdot 81}{(x - 8) (x - 7)} + 15 \frac{81 x}{(x - 8) (x - 7)} x - \frac{81 x}{(x - 8) (x - 7)} \cdot 56$

Step 6.2.2

Combine $15$ and $\frac{81 x}{(x - 8) (x - 7)}$ .

$- \frac{x^{3} \cdot 81}{(x - 8) (x - 7)} + \frac{15 (81 x)}{(x - 8) (x - 7)} x - \frac{81 x}{(x - 8) (x - 7)} \cdot 56$

Step 6.2.3

Multiply $81$ by $15$ .

$- \frac{x^{3} \cdot 81}{(x - 8) (x - 7)} + \frac{1215 x}{(x - 8) (x - 7)} x - \frac{81 x}{(x - 8) (x - 7)} \cdot 56$

Step 6.2.4

Combine $\frac{1215 x}{(x - 8) (x - 7)}$ and $x$ .

$- \frac{x^{3} \cdot 81}{(x - 8) (x - 7)} + \frac{1215 x \cdot x}{(x - 8) (x - 7)} - \frac{81 x}{(x - 8) (x - 7)} \cdot 56$

Step 6.2.5

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

$- \frac{x^{3} \cdot 81}{(x - 8) (x - 7)} + \frac{1215 (x^{1} x)}{(x - 8) (x - 7)} - \frac{81 x}{(x - 8) (x - 7)} \cdot 56$

Step 6.2.6

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

$- \frac{x^{3} \cdot 81}{(x - 8) (x - 7)} + \frac{1215 (x^{1} x^{1})}{(x - 8) (x - 7)} - \frac{81 x}{(x - 8) (x - 7)} \cdot 56$

Step 6.2.7

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

$- \frac{x^{3} \cdot 81}{(x - 8) (x - 7)} + \frac{1215 x^{1 + 1}}{(x - 8) (x - 7)} - \frac{81 x}{(x - 8) (x - 7)} \cdot 56$

Step 6.2.8

Add $1$ and $1$ .

$- \frac{x^{3} \cdot 81}{(x - 8) (x - 7)} + \frac{1215 x^{2}}{(x - 8) (x - 7)} - \frac{81 x}{(x - 8) (x - 7)} \cdot 56$

Step 6.3

Multiply $- \frac{81 x}{(x - 8) (x - 7)} \cdot 56$ .

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Step 6.3.1

Multiply $56$ by $- 1$ .

$- \frac{x^{3} \cdot 81}{(x - 8) (x - 7)} + \frac{1215 x^{2}}{(x - 8) (x - 7)} - 56 \frac{81 x}{(x - 8) (x - 7)}$

Step 6.3.2

Combine $- 56$ and $\frac{81 x}{(x - 8) (x - 7)}$ .

$- \frac{x^{3} \cdot 81}{(x - 8) (x - 7)} + \frac{1215 x^{2}}{(x - 8) (x - 7)} + \frac{- 56 (81 x)}{(x - 8) (x - 7)}$

Step 6.3.3

Multiply $81$ by $- 56$ .

$- \frac{x^{3} \cdot 81}{(x - 8) (x - 7)} + \frac{1215 x^{2}}{(x - 8) (x - 7)} + \frac{- 4536 x}{(x - 8) (x - 7)}$

Step 7

Simplify each term.

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Step 7.1

Move $81$ to the left of $x^{3}$ .

$- \frac{81 \cdot x^{3}}{(x - 8) (x - 7)} + \frac{1215 x^{2}}{(x - 8) (x - 7)} + \frac{- 4536 x}{(x - 8) (x - 7)}$

Step 7.2

Move the negative in front of the fraction.

$- \frac{81 x^{3}}{(x - 8) (x - 7)} + \frac{1215 x^{2}}{(x - 8) (x - 7)} - \frac{4536 x}{(x - 8) (x - 7)}$

Step 8

Combine the numerators over the common denominator.

$\frac{- 81 x^{3} + 1215 x^{2} - 4536 x}{(x - 8) (x - 7)}$

Step 9

Simplify the numerator.

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Step 9.1

Factor $81 x$ out of $- 81 x^{3} + 1215 x^{2} - 4536 x$ .

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Step 9.1.1

Factor $81 x$ out of $- 81 x^{3}$ .

$\frac{81 x (- x^{2}) + 1215 x^{2} - 4536 x}{(x - 8) (x - 7)}$

Step 9.1.2

Factor $81 x$ out of $1215 x^{2}$ .

$\frac{81 x (- x^{2}) + 81 x (15 x) - 4536 x}{(x - 8) (x - 7)}$

Step 9.1.3

Factor $81 x$ out of $- 4536 x$ .

$\frac{81 x (- x^{2}) + 81 x (15 x) + 81 x (- 56)}{(x - 8) (x - 7)}$

Step 9.1.4

Factor $81 x$ out of $81 x (- x^{2}) + 81 x (15 x)$ .

$\frac{81 x (- x^{2} + 15 x) + 81 x (- 56)}{(x - 8) (x - 7)}$

Step 9.1.5

Factor $81 x$ out of $81 x (- x^{2} + 15 x) + 81 x (- 56)$ .

$\frac{81 x (- x^{2} + 15 x - 56)}{(x - 8) (x - 7)}$

Step 9.2

Factor by grouping.

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Step 9.2.1

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 $a \cdot c = - 1 \cdot - 56 = 56$ and whose sum is $b = 15$ .

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Step 9.2.1.1

Factor $15$ out of $15 x$ .

$\frac{81 x (- x^{2} + 15 (x) - 56)}{(x - 8) (x - 7)}$

Step 9.2.1.2

Rewrite $15$ as $7$ plus $8$

$\frac{81 x (- x^{2} + (7 + 8) x - 56)}{(x - 8) (x - 7)}$

Step 9.2.1.3

Apply the distributive property.

$\frac{81 x (- x^{2} + 7 x + 8 x - 56)}{(x - 8) (x - 7)}$

Step 9.2.2

Factor out the greatest common factor from each group.

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Step 9.2.2.1

Group the first two terms and the last two terms.

$\frac{81 x ((- x^{2} + 7 x) + 8 x - 56)}{(x - 8) (x - 7)}$

Step 9.2.2.2

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

$\frac{81 x (x (- x + 7) - 8 (- x + 7))}{(x - 8) (x - 7)}$

Step 9.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 7$ .

$\frac{81 x ((- x + 7) (x - 8))}{(x - 8) (x - 7)}$

$\frac{81 x (- x + 7) (x - 8)}{(x - 8) (x - 7)}$

Step 10

Reduce the expression by cancelling the common factors.

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Step 10.1

Cancel the common factor of $x - 8$ .

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Step 10.1.1

Cancel the common factor.

$\frac{81 x (- x + 7) (x - 8)}{(x - 8) (x - 7)}$

Step 10.1.2

Rewrite the expression.

$\frac{81 x (- x + 7)}{x - 7}$

Step 10.2

Cancel the common factor of $- x + 7$ and $x - 7$ .

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Step 10.2.1

Factor $- 1$ out of $- x$ .

$\frac{81 x (- (x) + 7)}{x - 7}$

Step 10.2.2

Rewrite $7$ as $- 1 (- 7)$ .

$\frac{81 x (- (x) - 1 (- 7))}{x - 7}$

Step 10.2.3

Factor $- 1$ out of $- (x) - 1 (- 7)$ .

$\frac{81 x (- (x - 7))}{x - 7}$

Step 10.2.4

Rewrite $- (x - 7)$ as $- 1 (x - 7)$ .

$\frac{81 x (- 1 (x - 7))}{x - 7}$

Step 10.2.5

Cancel the common factor.

$\frac{81 x (- 1 (x - 7))}{x - 7}$

Step 10.2.6

Divide $81 x \cdot (- 1)$ by $1$ .

$81 x \cdot (- 1)$

Step 10.3

Multiply $- 1$ by $81$ .

$- 81 x$