Pre-Algebra Examples

$\frac{x^{2} - 12 x + 32}{8 x} - x^{2} - 8 x + 16$

Step 1

Factor $x^{2} - 12 x + 32$ using the AC method.

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Step 1.1

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 $32$ and whose sum is $- 12$ .

$- 8, - 4$

Step 1.2

Write the factored form using these integers.

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

Step 2

To write $- x^{2}$ as a fraction with a common denominator, multiply by $\frac{8 x}{8 x}$ .

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

Step 3

Simplify terms.

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Step 3.1

Combine $- x^{2}$ and $\frac{8 x}{8 x}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify each term.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Expand $(x - 8) (x - 4)$ using the FOIL Method.

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Step 4.1.1.1

Apply the distributive property.

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

Step 4.1.1.2

Apply the distributive property.

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

Step 4.1.1.3

Apply the distributive property.

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

Step 4.1.2

Simplify and combine like terms.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Multiply $x$ by $x$ .

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

Step 4.1.2.1.2

Move $- 4$ to the left of $x$ .

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

Step 4.1.2.1.3

Multiply $- 8$ by $- 4$ .

$\frac{x^{2} - 4 x - 8 x + 32 - x^{2} (8 x)}{8 x} - 8 x + 16$

Step 4.1.2.2

Subtract $8 x$ from $- 4 x$ .

$\frac{x^{2} - 12 x + 32 - x^{2} (8 x)}{8 x} - 8 x + 16$

Step 4.1.3

Rewrite using the commutative property of multiplication.

$\frac{x^{2} - 12 x + 32 - 1 \cdot 8 x^{2} x}{8 x} - 8 x + 16$

Step 4.1.4

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

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Step 4.1.4.1

Move $x$ .

$\frac{x^{2} - 12 x + 32 - 1 \cdot 8 (x \cdot x^{2})}{8 x} - 8 x + 16$

Step 4.1.4.2

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

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Step 4.1.4.2.1

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

$\frac{x^{2} - 12 x + 32 - 1 \cdot 8 (x^{1} x^{2})}{8 x} - 8 x + 16$

Step 4.1.4.2.2

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

$\frac{x^{2} - 12 x + 32 - 1 \cdot 8 x^{1 + 2}}{8 x} - 8 x + 16$

Step 4.1.4.3

Add $1$ and $2$ .

$\frac{x^{2} - 12 x + 32 - 1 \cdot 8 x^{3}}{8 x} - 8 x + 16$

Step 4.1.5

Multiply $- 1$ by $8$ .

$\frac{x^{2} - 12 x + 32 - 8 x^{3}}{8 x} - 8 x + 16$

Step 4.1.6

Reorder terms.

$\frac{- 8 x^{3} + x^{2} - 12 x + 32}{8 x} - 8 x + 16$

Step 4.1.7

Rewrite $- 8 x^{3} + x^{2} - 12 x + 32$ in a factored form.

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Step 4.1.7.1

Factor $x^{2} - 12 x + 32$ using the AC method.

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Step 4.1.7.1.1

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 $32$ and whose sum is $- 12$ .

$- 8, - 4$

Step 4.1.7.1.2

Write the factored form using these integers.

$\frac{- 8 x^{3} + (x - 8) (x - 4)}{8 x} - 8 x + 16$

Step 4.1.7.2

Factor $- 1$ out of $- 8 x^{3} + (x - 8) (x - 4)$ .

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Step 4.1.7.2.1

Factor $- 1$ out of $- 8 x^{3}$ .

$\frac{- (8 x^{3}) + (x - 8) (x - 4)}{8 x} - 8 x + 16$

Step 4.1.7.2.2

Factor $- 1$ out of $(x - 8) (x - 4)$ .

$\frac{- (8 x^{3}) - ((- x + 8) (x - 4))}{8 x} - 8 x + 16$

Step 4.1.7.2.3

Factor $- 1$ out of $- (8 x^{3}) - ((- x + 8) (x - 4))$ .

$\frac{- (8 x^{3} + (- x + 8) (x - 4))}{8 x} - 8 x + 16$

Step 4.1.7.3

Expand $(- x + 8) (x - 4)$ using the FOIL Method.

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Step 4.1.7.3.1

Apply the distributive property.

$\frac{- (8 x^{3} - x (x - 4) + 8 (x - 4))}{8 x} - 8 x + 16$

Step 4.1.7.3.2

Apply the distributive property.

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

Step 4.1.7.3.3

Apply the distributive property.

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

Step 4.1.7.4

Simplify and combine like terms.

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Step 4.1.7.4.1

Simplify each term.

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Step 4.1.7.4.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 4.1.7.4.1.1.1

Move $x$ .

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

Step 4.1.7.4.1.1.2

Multiply $x$ by $x$ .

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

Step 4.1.7.4.1.2

Multiply $- 4$ by $- 1$ .

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

Step 4.1.7.4.1.3

Multiply $8$ by $- 4$ .

$\frac{- (8 x^{3} - x^{2} + 4 x + 8 x - 32)}{8 x} - 8 x + 16$

Step 4.1.7.4.2

Add $4 x$ and $8 x$ .

$\frac{- (8 x^{3} - x^{2} + 12 x - 32)}{8 x} - 8 x + 16$

Step 4.2

Move the negative in front of the fraction.

$- \frac{8 x^{3} - x^{2} + 12 x - 32}{8 x} - 8 x + 16$

Step 5

To write $- 8 x$ as a fraction with a common denominator, multiply by $\frac{8 x}{8 x}$ .

$- \frac{8 x^{3} - x^{2} + 12 x - 32}{8 x} - 8 x \cdot \frac{8 x}{8 x} + 16$

Step 6

Simplify terms.

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Step 6.1

Combine $- 8 x$ and $\frac{8 x}{8 x}$ .

$- \frac{8 x^{3} - x^{2} + 12 x - 32}{8 x} + \frac{- 8 x (8 x)}{8 x} + 16$

Step 6.2

Combine the numerators over the common denominator.

$\frac{- (8 x^{3} - x^{2} + 12 x - 32) - 8 x (8 x)}{8 x} + 16$

Step 7

Simplify the numerator.

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Step 7.1

Apply the distributive property.

$\frac{- (8 x^{3}) - - x^{2} - (12 x) - - 32 - 8 x (8 x)}{8 x} + 16$

Step 7.2

Simplify.

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Step 7.2.1

Multiply $8$ by $- 1$ .

$\frac{- 8 x^{3} - - x^{2} - (12 x) - - 32 - 8 x (8 x)}{8 x} + 16$

Step 7.2.2

Multiply $- - x^{2}$ .

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Step 7.2.2.1

Multiply $- 1$ by $- 1$ .

$\frac{- 8 x^{3} + 1 x^{2} - (12 x) - - 32 - 8 x (8 x)}{8 x} + 16$

Step 7.2.2.2

Multiply $x^{2}$ by $1$ .

$\frac{- 8 x^{3} + x^{2} - (12 x) - - 32 - 8 x (8 x)}{8 x} + 16$

Step 7.2.3

Multiply $12$ by $- 1$ .

$\frac{- 8 x^{3} + x^{2} - 12 x - - 32 - 8 x (8 x)}{8 x} + 16$

Step 7.2.4

Multiply $- 1$ by $- 32$ .

$\frac{- 8 x^{3} + x^{2} - 12 x + 32 - 8 x (8 x)}{8 x} + 16$

Step 7.3

Rewrite using the commutative property of multiplication.

$\frac{- 8 x^{3} + x^{2} - 12 x + 32 - 8 \cdot 8 x \cdot x}{8 x} + 16$

Step 7.4

Multiply $x$ by $x$ by adding the exponents.

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Step 7.4.1

Move $x$ .

$\frac{- 8 x^{3} + x^{2} - 12 x + 32 - 8 \cdot 8 (x \cdot x)}{8 x} + 16$

Step 7.4.2

Multiply $x$ by $x$ .

$\frac{- 8 x^{3} + x^{2} - 12 x + 32 - 8 \cdot 8 x^{2}}{8 x} + 16$

Step 7.5

Multiply $- 8$ by $8$ .

$\frac{- 8 x^{3} + x^{2} - 12 x + 32 - 64 x^{2}}{8 x} + 16$

Step 7.6

Subtract $64 x^{2}$ from $x^{2}$ .

$\frac{- 8 x^{3} - 63 x^{2} - 12 x + 32}{8 x} + 16$

Step 8

To write $16$ as a fraction with a common denominator, multiply by $\frac{8 x}{8 x}$ .

$\frac{- 8 x^{3} - 63 x^{2} - 12 x + 32}{8 x} + 16 \cdot \frac{8 x}{8 x}$

Step 9

Simplify terms.

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Step 9.1

Combine $16$ and $\frac{8 x}{8 x}$ .

$\frac{- 8 x^{3} - 63 x^{2} - 12 x + 32}{8 x} + \frac{16 (8 x)}{8 x}$

Step 9.2

Combine the numerators over the common denominator.

$\frac{- 8 x^{3} - 63 x^{2} - 12 x + 32 + 16 (8 x)}{8 x}$

Step 10

Simplify the numerator.

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Step 10.1

Multiply $16$ by $8$ .

$\frac{- 8 x^{3} - 63 x^{2} - 12 x + 32 + 128 x}{8 x}$

Step 10.2

Add $- 12 x$ and $128 x$ .

$\frac{- 8 x^{3} - 63 x^{2} + 116 x + 32}{8 x}$

Step 11

Simplify with factoring out.

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Step 11.1

Factor $- 1$ out of $- 8 x^{3}$ .

$\frac{- (8 x^{3}) - 63 x^{2} + 116 x + 32}{8 x}$

Step 11.2

Factor $- 1$ out of $- 63 x^{2}$ .

$\frac{- (8 x^{3}) - (63 x^{2}) + 116 x + 32}{8 x}$

Step 11.3

Factor $- 1$ out of $- (8 x^{3}) - (63 x^{2})$ .

$\frac{- (8 x^{3} + 63 x^{2}) + 116 x + 32}{8 x}$

Step 11.4

Factor $- 1$ out of $116 x$ .

$\frac{- (8 x^{3} + 63 x^{2}) - (- 116 x) + 32}{8 x}$

Step 11.5

Factor $- 1$ out of $- (8 x^{3} + 63 x^{2}) - (- 116 x)$ .

$\frac{- (8 x^{3} + 63 x^{2} - 116 x) + 32}{8 x}$

Step 11.6

Rewrite $32$ as $- 1 (- 32)$ .

$\frac{- (8 x^{3} + 63 x^{2} - 116 x) - 1 (- 32)}{8 x}$

Step 11.7

Factor $- 1$ out of $- (8 x^{3} + 63 x^{2} - 116 x) - 1 (- 32)$ .

$\frac{- (8 x^{3} + 63 x^{2} - 116 x - 32)}{8 x}$

Step 11.8

Simplify the expression.

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Step 11.8.1

Rewrite $- (8 x^{3} + 63 x^{2} - 116 x - 32)$ as $- 1 (8 x^{3} + 63 x^{2} - 116 x - 32)$ .

$\frac{- 1 (8 x^{3} + 63 x^{2} - 116 x - 32)}{8 x}$

Step 11.8.2

Move the negative in front of the fraction.

$- \frac{8 x^{3} + 63 x^{2} - 116 x - 32}{8 x}$