Precalculus Examples

$f (x) = - \frac{1}{4} \cdot ((x + 4) {(x + 1)}^{2} (x - 2^{2}))$

Step 1

Simplify and reorder the polynomial.

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Step 1.1

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

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

Step 1.2

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.3.1.2

Multiply $x$ by $1$ .

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

Step 1.3.1.3

Multiply $x$ by $1$ .

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

Step 1.3.1.4

Multiply $1$ by $1$ .

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

Step 1.3.2

Add $x$ and $x$ .

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

Step 1.4

Expand $(x + 4) (x^{2} + 2 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 1.5

Simplify terms.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

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

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Step 1.5.1.1.1

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

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Step 1.5.1.1.1.1

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

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

Step 1.5.1.1.1.2

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

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

Step 1.5.1.1.2

Add $1$ and $2$ .

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

Step 1.5.1.2

Rewrite using the commutative property of multiplication.

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

Step 1.5.1.3

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

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Step 1.5.1.3.1

Move $x$ .

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

Step 1.5.1.3.2

Multiply $x$ by $x$ .

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

Step 1.5.1.4

Multiply $x$ by $1$ .

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

Step 1.5.1.5

Multiply $2$ by $4$ .

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

Step 1.5.1.6

Multiply $4$ by $1$ .

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

Step 1.5.2

Simplify by adding terms.

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Step 1.5.2.1

Add $2 x^{2}$ and $4 x^{2}$ .

$- \frac{1}{4} \cdot ((x^{3} + 6 x^{2} + x + 8 x + 4) (x - 2^{2}))$

Step 1.5.2.2

Add $x$ and $8 x$ .

$- \frac{1}{4} \cdot ((x^{3} + 6 x^{2} + 9 x + 4) (x - 2^{2}))$

Step 1.5.3

Simplify each term.

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Step 1.5.3.1

Raise $2$ to the power of $2$ .

$- \frac{1}{4} \cdot ((x^{3} + 6 x^{2} + 9 x + 4) (x - 1 \cdot 4))$

Step 1.5.3.2

Multiply $- 1$ by $4$ .

$- \frac{1}{4} \cdot ((x^{3} + 6 x^{2} + 9 x + 4) (x - 4))$

Step 1.6

Expand $(x^{3} + 6 x^{2} + 9 x + 4) (x - 4)$ by multiplying each term in the first expression by each term in the second expression.

$- \frac{1}{4} \cdot (x^{3} x + x^{3} \cdot - 4 + 6 x^{2} x + 6 x^{2} \cdot - 4 + 9 x \cdot x + 9 x \cdot - 4 + 4 x + 4 \cdot - 4)$

Step 1.7

Simplify terms.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 1.7.1.1.1

Multiply $x^{3}$ by $x$ .

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Step 1.7.1.1.1.1

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

$- \frac{1}{4} \cdot (x^{3} x^{1} + x^{3} \cdot - 4 + 6 x^{2} x + 6 x^{2} \cdot - 4 + 9 x \cdot x + 9 x \cdot - 4 + 4 x + 4 \cdot - 4)$

Step 1.7.1.1.1.2

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

$- \frac{1}{4} \cdot (x^{3 + 1} + x^{3} \cdot - 4 + 6 x^{2} x + 6 x^{2} \cdot - 4 + 9 x \cdot x + 9 x \cdot - 4 + 4 x + 4 \cdot - 4)$

Step 1.7.1.1.2

Add $3$ and $1$ .

$- \frac{1}{4} \cdot (x^{4} + x^{3} \cdot - 4 + 6 x^{2} x + 6 x^{2} \cdot - 4 + 9 x \cdot x + 9 x \cdot - 4 + 4 x + 4 \cdot - 4)$

Step 1.7.1.2

Move $- 4$ to the left of $x^{3}$ .

$- \frac{1}{4} \cdot (x^{4} - 4 \cdot x^{3} + 6 x^{2} x + 6 x^{2} \cdot - 4 + 9 x \cdot x + 9 x \cdot - 4 + 4 x + 4 \cdot - 4)$

Step 1.7.1.3

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

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Step 1.7.1.3.1

Move $x$ .

$- \frac{1}{4} \cdot (x^{4} - 4 x^{3} + 6 (x \cdot x^{2}) + 6 x^{2} \cdot - 4 + 9 x \cdot x + 9 x \cdot - 4 + 4 x + 4 \cdot - 4)$

Step 1.7.1.3.2

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

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Step 1.7.1.3.2.1

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

$- \frac{1}{4} \cdot (x^{4} - 4 x^{3} + 6 (x^{1} x^{2}) + 6 x^{2} \cdot - 4 + 9 x \cdot x + 9 x \cdot - 4 + 4 x + 4 \cdot - 4)$

Step 1.7.1.3.2.2

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

$- \frac{1}{4} \cdot (x^{4} - 4 x^{3} + 6 x^{1 + 2} + 6 x^{2} \cdot - 4 + 9 x \cdot x + 9 x \cdot - 4 + 4 x + 4 \cdot - 4)$

Step 1.7.1.3.3

Add $1$ and $2$ .

$- \frac{1}{4} \cdot (x^{4} - 4 x^{3} + 6 x^{3} + 6 x^{2} \cdot - 4 + 9 x \cdot x + 9 x \cdot - 4 + 4 x + 4 \cdot - 4)$

Step 1.7.1.4

Multiply $- 4$ by $6$ .

$- \frac{1}{4} \cdot (x^{4} - 4 x^{3} + 6 x^{3} - 24 x^{2} + 9 x \cdot x + 9 x \cdot - 4 + 4 x + 4 \cdot - 4)$

Step 1.7.1.5

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

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Step 1.7.1.5.1

Move $x$ .

$- \frac{1}{4} \cdot (x^{4} - 4 x^{3} + 6 x^{3} - 24 x^{2} + 9 (x \cdot x) + 9 x \cdot - 4 + 4 x + 4 \cdot - 4)$

Step 1.7.1.5.2

Multiply $x$ by $x$ .

$- \frac{1}{4} \cdot (x^{4} - 4 x^{3} + 6 x^{3} - 24 x^{2} + 9 x^{2} + 9 x \cdot - 4 + 4 x + 4 \cdot - 4)$

Step 1.7.1.6

Multiply $- 4$ by $9$ .

$- \frac{1}{4} \cdot (x^{4} - 4 x^{3} + 6 x^{3} - 24 x^{2} + 9 x^{2} - 36 x + 4 x + 4 \cdot - 4)$

Step 1.7.1.7

Multiply $4$ by $- 4$ .

$- \frac{1}{4} \cdot (x^{4} - 4 x^{3} + 6 x^{3} - 24 x^{2} + 9 x^{2} - 36 x + 4 x - 16)$

Step 1.7.2

Simplify terms.

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Step 1.7.2.1

Add $- 4 x^{3}$ and $6 x^{3}$ .

$- \frac{1}{4} \cdot (x^{4} + 2 x^{3} - 24 x^{2} + 9 x^{2} - 36 x + 4 x - 16)$

Step 1.7.2.2

Add $- 24 x^{2}$ and $9 x^{2}$ .

$- \frac{1}{4} \cdot (x^{4} + 2 x^{3} - 15 x^{2} - 36 x + 4 x - 16)$

Step 1.7.2.3

Add $- 36 x$ and $4 x$ .

$- \frac{1}{4} \cdot (x^{4} + 2 x^{3} - 15 x^{2} - 32 x - 16)$

Step 1.7.2.4

Apply the distributive property.

$- \frac{1}{4} x^{4} - \frac{1}{4} (2 x^{3}) - \frac{1}{4} (- 15 x^{2}) - \frac{1}{4} (- 32 x) - \frac{1}{4} \cdot - 16$

Step 1.8

Simplify.

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Step 1.8.1

Combine $x^{4}$ and $\frac{1}{4}$ .

$- \frac{x^{4}}{4} - \frac{1}{4} (2 x^{3}) - \frac{1}{4} (- 15 x^{2}) - \frac{1}{4} (- 32 x) - \frac{1}{4} \cdot - 16$

Step 1.8.2

Cancel the common factor of $2$ .

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Step 1.8.2.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$- \frac{x^{4}}{4} + \frac{- 1}{4} (2 x^{3}) - \frac{1}{4} (- 15 x^{2}) - \frac{1}{4} (- 32 x) - \frac{1}{4} \cdot - 16$

Step 1.8.2.2

Factor $2$ out of $4$ .

$- \frac{x^{4}}{4} + \frac{- 1}{2 (2)} (2 x^{3}) - \frac{1}{4} (- 15 x^{2}) - \frac{1}{4} (- 32 x) - \frac{1}{4} \cdot - 16$

Step 1.8.2.3

Factor $2$ out of $2 x^{3}$ .

$- \frac{x^{4}}{4} + \frac{- 1}{2 (2)} (2 (x^{3})) - \frac{1}{4} (- 15 x^{2}) - \frac{1}{4} (- 32 x) - \frac{1}{4} \cdot - 16$

Step 1.8.2.4

Cancel the common factor.

$- \frac{x^{4}}{4} + \frac{- 1}{2 \cdot 2} (2 x^{3}) - \frac{1}{4} (- 15 x^{2}) - \frac{1}{4} (- 32 x) - \frac{1}{4} \cdot - 16$

Step 1.8.2.5

Rewrite the expression.

$- \frac{x^{4}}{4} + \frac{- 1}{2} x^{3} - \frac{1}{4} (- 15 x^{2}) - \frac{1}{4} (- 32 x) - \frac{1}{4} \cdot - 16$

Step 1.8.3

Combine $\frac{- 1}{2}$ and $x^{3}$ .

$- \frac{x^{4}}{4} + \frac{- x^{3}}{2} - \frac{1}{4} (- 15 x^{2}) - \frac{1}{4} (- 32 x) - \frac{1}{4} \cdot - 16$

Step 1.8.4

Multiply $- \frac{1}{4} (- 15 x^{2})$ .

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Step 1.8.4.1

Multiply $- 15$ by $- 1$ .

$- \frac{x^{4}}{4} + \frac{- x^{3}}{2} + 15 (\frac{1}{4}) x^{2} - \frac{1}{4} (- 32 x) - \frac{1}{4} \cdot - 16$

Step 1.8.4.2

Combine $15$ and $\frac{1}{4}$ .

$- \frac{x^{4}}{4} + \frac{- x^{3}}{2} + \frac{15}{4} x^{2} - \frac{1}{4} (- 32 x) - \frac{1}{4} \cdot - 16$

Step 1.8.4.3

Combine $\frac{15}{4}$ and $x^{2}$ .

$- \frac{x^{4}}{4} + \frac{- x^{3}}{2} + \frac{15 x^{2}}{4} - \frac{1}{4} (- 32 x) - \frac{1}{4} \cdot - 16$

Step 1.8.5

Cancel the common factor of $4$ .

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Step 1.8.5.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$- \frac{x^{4}}{4} + \frac{- x^{3}}{2} + \frac{15 x^{2}}{4} + \frac{- 1}{4} (- 32 x) - \frac{1}{4} \cdot - 16$

Step 1.8.5.2

Factor $4$ out of $- 32 x$ .

$- \frac{x^{4}}{4} + \frac{- x^{3}}{2} + \frac{15 x^{2}}{4} + \frac{- 1}{4} (4 (- 8 x)) - \frac{1}{4} \cdot - 16$

Step 1.8.5.3

Cancel the common factor.

$- \frac{x^{4}}{4} + \frac{- x^{3}}{2} + \frac{15 x^{2}}{4} + \frac{- 1}{4} (4 (- 8 x)) - \frac{1}{4} \cdot - 16$

Step 1.8.5.4

Rewrite the expression.

$- \frac{x^{4}}{4} + \frac{- x^{3}}{2} + \frac{15 x^{2}}{4} - 1 (- 8 x) - \frac{1}{4} \cdot - 16$

Step 1.8.6

Multiply $- 8$ by $- 1$ .

$- \frac{x^{4}}{4} + \frac{- x^{3}}{2} + \frac{15 x^{2}}{4} + 8 x - \frac{1}{4} \cdot - 16$

Step 1.8.7

Cancel the common factor of $4$ .

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Step 1.8.7.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$- \frac{x^{4}}{4} + \frac{- x^{3}}{2} + \frac{15 x^{2}}{4} + 8 x + \frac{- 1}{4} \cdot - 16$

Step 1.8.7.2

Factor $4$ out of $- 16$ .

$- \frac{x^{4}}{4} + \frac{- x^{3}}{2} + \frac{15 x^{2}}{4} + 8 x + \frac{- 1}{4} \cdot (4 (- 4))$

Step 1.8.7.3

Cancel the common factor.

$- \frac{x^{4}}{4} + \frac{- x^{3}}{2} + \frac{15 x^{2}}{4} + 8 x + \frac{- 1}{4} \cdot (4 \cdot - 4)$

Step 1.8.7.4

Rewrite the expression.

$- \frac{x^{4}}{4} + \frac{- x^{3}}{2} + \frac{15 x^{2}}{4} + 8 x - 1 \cdot - 4$

Step 1.8.8

Multiply $- 1$ by $- 4$ .

$- \frac{x^{4}}{4} + \frac{- x^{3}}{2} + \frac{15 x^{2}}{4} + 8 x + 4$

Step 1.9

Move the negative in front of the fraction.

$- \frac{x^{4}}{4} - \frac{x^{3}}{2} + \frac{15 x^{2}}{4} + 8 x + 4$

Step 2

The largest exponent is the degree of the polynomial.

$4$