Precalculus Examples

$\frac{2}{4 p^{2} + 16 p} - \frac{p - 1}{2 p - 3}$

Step 1

Simplify each term.

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Step 1.1

Factor $4 p$ out of $4 p^{2} + 16 p$ .

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Step 1.1.1

Factor $4 p$ out of $4 p^{2}$ .

$\frac{2}{4 p (p) + 16 p} - \frac{p - 1}{2 p - 3}$

Step 1.1.2

Factor $4 p$ out of $16 p$ .

$\frac{2}{4 p (p) + 4 p (4)} - \frac{p - 1}{2 p - 3}$

Step 1.1.3

Factor $4 p$ out of $4 p (p) + 4 p (4)$ .

$\frac{2}{4 p (p + 4)} - \frac{p - 1}{2 p - 3}$

Step 1.2

Cancel the common factor of $2$ and $4$ .

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Step 1.2.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{4 p (p + 4)} - \frac{p - 1}{2 p - 3}$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Factor $2$ out of $4 p (p + 4)$ .

$\frac{2 (1)}{2 (2 p (p + 4))} - \frac{p - 1}{2 p - 3}$

Step 1.2.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 (2 p (p + 4))} - \frac{p - 1}{2 p - 3}$

Step 1.2.2.3

Rewrite the expression.

$\frac{1}{2 p (p + 4)} - \frac{p - 1}{2 p - 3}$

Step 2

To write $\frac{1}{2 p (p + 4)}$ as a fraction with a common denominator, multiply by $\frac{2 p - 3}{2 p - 3}$ .

$\frac{1}{2 p (p + 4)} \cdot \frac{2 p - 3}{2 p - 3} - \frac{p - 1}{2 p - 3}$

Step 3

To write $- \frac{p - 1}{2 p - 3}$ as a fraction with a common denominator, multiply by $\frac{2 p (p + 4)}{2 p (p + 4)}$ .

$\frac{1}{2 p (p + 4)} \cdot \frac{2 p - 3}{2 p - 3} - \frac{p - 1}{2 p - 3} \cdot \frac{2 p (p + 4)}{2 p (p + 4)}$

Step 4

Write each expression with a common denominator of $2 p (p + 4) (2 p - 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{1}{2 p (p + 4)}$ by $\frac{2 p - 3}{2 p - 3}$ .

$\frac{2 p - 3}{2 p (p + 4) (2 p - 3)} - \frac{p - 1}{2 p - 3} \cdot \frac{2 p (p + 4)}{2 p (p + 4)}$

Step 4.2

Multiply $\frac{p - 1}{2 p - 3}$ by $\frac{2 p (p + 4)}{2 p (p + 4)}$ .

$\frac{2 p - 3}{2 p (p + 4) (2 p - 3)} - \frac{(p - 1) (2 p (p + 4))}{(2 p - 3) (2 p (p + 4))}$

Step 4.3

Reorder the factors of $2 p (p + 4) (2 p - 3)$ .

$\frac{2 p - 3}{2 p (2 p - 3) (p + 4)} - \frac{(p - 1) (2 p (p + 4))}{(2 p - 3) (2 p (p + 4))}$

Step 4.4

Reorder the factors of $(2 p - 3) (2 p (p + 4))$ .

$\frac{2 p - 3}{2 p (2 p - 3) (p + 4)} - \frac{(p - 1) (2 p (p + 4))}{2 p (2 p - 3) (p + 4)}$

Step 5

Combine the numerators over the common denominator.

$\frac{2 p - 3 - (p - 1) (2 p (p + 4))}{2 p (2 p - 3) (p + 4)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{2 p - 3 + (- p - - 1) (2 p (p + 4))}{2 p (2 p - 3) (p + 4)}$

Step 6.2

Multiply $- 1$ by $- 1$ .

$\frac{2 p - 3 + (- p + 1) (2 p (p + 4))}{2 p (2 p - 3) (p + 4)}$

Step 6.3

Apply the distributive property.

$\frac{2 p - 3 + (- p + 1) (2 p \cdot p + 2 p \cdot 4)}{2 p (2 p - 3) (p + 4)}$

Step 6.4

Multiply $p$ by $p$ by adding the exponents.

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Step 6.4.1

Move $p$ .

$\frac{2 p - 3 + (- p + 1) (2 (p \cdot p) + 2 p \cdot 4)}{2 p (2 p - 3) (p + 4)}$

Step 6.4.2

Multiply $p$ by $p$ .

$\frac{2 p - 3 + (- p + 1) (2 p^{2} + 2 p \cdot 4)}{2 p (2 p - 3) (p + 4)}$

Step 6.5

Multiply $4$ by $2$ .

$\frac{2 p - 3 + (- p + 1) (2 p^{2} + 8 p)}{2 p (2 p - 3) (p + 4)}$

Step 6.6

Expand $(- p + 1) (2 p^{2} + 8 p)$ using the FOIL Method.

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Step 6.6.1

Apply the distributive property.

$\frac{2 p - 3 - p (2 p^{2} + 8 p) + 1 (2 p^{2} + 8 p)}{2 p (2 p - 3) (p + 4)}$

Step 6.6.2

Apply the distributive property.

$\frac{2 p - 3 - p (2 p^{2}) - p (8 p) + 1 (2 p^{2} + 8 p)}{2 p (2 p - 3) (p + 4)}$

Step 6.6.3

Apply the distributive property.

$\frac{2 p - 3 - p (2 p^{2}) - p (8 p) + 1 (2 p^{2}) + 1 (8 p)}{2 p (2 p - 3) (p + 4)}$

Step 6.7

Simplify and combine like terms.

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Step 6.7.1

Simplify each term.

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Step 6.7.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 p - 3 - 1 \cdot 2 p \cdot p^{2} - p (8 p) + 1 (2 p^{2}) + 1 (8 p)}{2 p (2 p - 3) (p + 4)}$

Step 6.7.1.2

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

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Step 6.7.1.2.1

Move $p^{2}$ .

$\frac{2 p - 3 - 1 \cdot 2 (p^{2} p) - p (8 p) + 1 (2 p^{2}) + 1 (8 p)}{2 p (2 p - 3) (p + 4)}$

Step 6.7.1.2.2

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

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Step 6.7.1.2.2.1

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

$\frac{2 p - 3 - 1 \cdot 2 (p^{2} p^{1}) - p (8 p) + 1 (2 p^{2}) + 1 (8 p)}{2 p (2 p - 3) (p + 4)}$

Step 6.7.1.2.2.2

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

$\frac{2 p - 3 - 1 \cdot 2 p^{2 + 1} - p (8 p) + 1 (2 p^{2}) + 1 (8 p)}{2 p (2 p - 3) (p + 4)}$

Step 6.7.1.2.3

Add $2$ and $1$ .

$\frac{2 p - 3 - 1 \cdot 2 p^{3} - p (8 p) + 1 (2 p^{2}) + 1 (8 p)}{2 p (2 p - 3) (p + 4)}$

Step 6.7.1.3

Multiply $- 1$ by $2$ .

$\frac{2 p - 3 - 2 p^{3} - p (8 p) + 1 (2 p^{2}) + 1 (8 p)}{2 p (2 p - 3) (p + 4)}$

Step 6.7.1.4

Rewrite using the commutative property of multiplication.

$\frac{2 p - 3 - 2 p^{3} - 1 \cdot 8 p \cdot p + 1 (2 p^{2}) + 1 (8 p)}{2 p (2 p - 3) (p + 4)}$

Step 6.7.1.5

Multiply $p$ by $p$ by adding the exponents.

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Step 6.7.1.5.1

Move $p$ .

$\frac{2 p - 3 - 2 p^{3} - 1 \cdot 8 (p \cdot p) + 1 (2 p^{2}) + 1 (8 p)}{2 p (2 p - 3) (p + 4)}$

Step 6.7.1.5.2

Multiply $p$ by $p$ .

$\frac{2 p - 3 - 2 p^{3} - 1 \cdot 8 p^{2} + 1 (2 p^{2}) + 1 (8 p)}{2 p (2 p - 3) (p + 4)}$

Step 6.7.1.6

Multiply $- 1$ by $8$ .

$\frac{2 p - 3 - 2 p^{3} - 8 p^{2} + 1 (2 p^{2}) + 1 (8 p)}{2 p (2 p - 3) (p + 4)}$

Step 6.7.1.7

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

$\frac{2 p - 3 - 2 p^{3} - 8 p^{2} + 2 p^{2} + 1 (8 p)}{2 p (2 p - 3) (p + 4)}$

Step 6.7.1.8

Multiply $8 p$ by $1$ .

$\frac{2 p - 3 - 2 p^{3} - 8 p^{2} + 2 p^{2} + 8 p}{2 p (2 p - 3) (p + 4)}$

Step 6.7.2

Add $- 8 p^{2}$ and $2 p^{2}$ .

$\frac{2 p - 3 - 2 p^{3} - 6 p^{2} + 8 p}{2 p (2 p - 3) (p + 4)}$

Step 6.8

Add $2 p$ and $8 p$ .

$\frac{- 3 - 2 p^{3} - 6 p^{2} + 10 p}{2 p (2 p - 3) (p + 4)}$

Step 6.9

Reorder terms.

$\frac{- 2 p^{3} - 6 p^{2} + 10 p - 3}{2 p (2 p - 3) (p + 4)}$

Step 7

Simplify with factoring out.

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Step 7.1

Factor $- 1$ out of $- 2 p^{3}$ .

$\frac{- (2 p^{3}) - 6 p^{2} + 10 p - 3}{2 p (2 p - 3) (p + 4)}$

Step 7.2

Factor $- 1$ out of $- 6 p^{2}$ .

$\frac{- (2 p^{3}) - (6 p^{2}) + 10 p - 3}{2 p (2 p - 3) (p + 4)}$

Step 7.3

Factor $- 1$ out of $- (2 p^{3}) - (6 p^{2})$ .

$\frac{- (2 p^{3} + 6 p^{2}) + 10 p - 3}{2 p (2 p - 3) (p + 4)}$

Step 7.4

Factor $- 1$ out of $10 p$ .

$\frac{- (2 p^{3} + 6 p^{2}) - (- 10 p) - 3}{2 p (2 p - 3) (p + 4)}$

Step 7.5

Factor $- 1$ out of $- (2 p^{3} + 6 p^{2}) - (- 10 p)$ .

$\frac{- (2 p^{3} + 6 p^{2} - 10 p) - 3}{2 p (2 p - 3) (p + 4)}$

Step 7.6

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{- (2 p^{3} + 6 p^{2} - 10 p) - 1 (3)}{2 p (2 p - 3) (p + 4)}$

Step 7.7

Factor $- 1$ out of $- (2 p^{3} + 6 p^{2} - 10 p) - 1 (3)$ .

$\frac{- (2 p^{3} + 6 p^{2} - 10 p + 3)}{2 p (2 p - 3) (p + 4)}$

Step 7.8

Simplify the expression.

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Step 7.8.1

Rewrite $- (2 p^{3} + 6 p^{2} - 10 p + 3)$ as $- 1 (2 p^{3} + 6 p^{2} - 10 p + 3)$ .

$\frac{- 1 (2 p^{3} + 6 p^{2} - 10 p + 3)}{2 p (2 p - 3) (p + 4)}$

Step 7.8.2

Move the negative in front of the fraction.

$- \frac{2 p^{3} + 6 p^{2} - 10 p + 3}{2 p (2 p - 3) (p + 4)}$