Algebra Examples

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

Step 1

Simplify and reorder the polynomial.

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Step 1.1

Simplify terms.

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Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

Combine $x$ and $\frac{1}{2}$ .

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

Step 1.1.3

Cancel the common factor of $2$ .

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Step 1.1.3.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 1.1.3.2

Factor $2$ out of $- 2$ .

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

Step 1.1.3.3

Cancel the common factor.

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

Step 1.1.3.4

Rewrite the expression.

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

Step 1.1.4

Simplify the expression.

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Step 1.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 1.1.4.2

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

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

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 $x$ .

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

Step 1.3.1.2

Multiply $x$ by $1$ .

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

Step 1.3.1.3

Multiply $x$ by $1$ .

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

Step 1.3.1.4

Multiply $1$ by $1$ .

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

Step 1.3.2

Add $x$ and $x$ .

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

Step 1.4

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

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

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 $- \frac{x}{2} x^{2}$ .

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Step 1.5.1.1.1

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

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

Step 1.5.1.1.2

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

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Step 1.5.1.1.2.1

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

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Step 1.5.1.1.2.1.1

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

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

Step 1.5.1.1.2.1.2

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

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

Step 1.5.1.1.2.2

Add $2$ and $1$ .

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

Step 1.5.1.2

Cancel the common factor of $2$ .

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Step 1.5.1.2.1

Move the leading negative in $- \frac{x}{2}$ into the numerator.

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

Step 1.5.1.2.2

Factor $2$ out of $2 x$ .

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

Step 1.5.1.2.3

Cancel the common factor.

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

Step 1.5.1.2.4

Rewrite the expression.

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

Step 1.5.1.3

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

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

Step 1.5.1.4

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

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

Step 1.5.1.5

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

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

Step 1.5.1.6

Add $1$ and $1$ .

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

Step 1.5.1.7

Multiply $- 1$ by $1$ .

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

Step 1.5.1.8

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

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

Step 1.5.1.9

Multiply $2 x$ by $1$ .

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

Step 1.5.1.10

Multiply $1$ by $1$ .

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

Step 1.5.2

Combine the opposite terms in $- \frac{x^{3}}{2} - x^{2} - \frac{x}{2} + x^{2} + 2 x + 1$ .

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Step 1.5.2.1

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

$(- \frac{x^{3}}{2} - \frac{x}{2} + 0 + 2 x + 1) (x - 1)$

Step 1.5.2.2

Add $- \frac{x^{3}}{2} - \frac{x}{2}$ and $0$ .

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

Step 1.6

Expand $(- \frac{x^{3}}{2} - \frac{x}{2} + 2 x + 1) (x - 1)$ by multiplying each term in the first expression by each term in the second expression.

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

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 $- \frac{x^{3}}{2} x$ .

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Step 1.7.1.1.1

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

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

Step 1.7.1.1.2

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

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Step 1.7.1.1.2.1

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

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Step 1.7.1.1.2.1.1

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

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

Step 1.7.1.1.2.1.2

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

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

Step 1.7.1.1.2.2

Add $1$ and $3$ .

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

Step 1.7.1.2

Multiply $- \frac{x^{3}}{2} \cdot - 1$ .

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Step 1.7.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 1.7.1.2.2

Multiply $\frac{x^{3}}{2}$ by $1$ .

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

Step 1.7.1.3

Multiply $- \frac{x}{2} x$ .

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Step 1.7.1.3.1

Combine $x$ and $\frac{x}{2}$ .

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

Step 1.7.1.3.2

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

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

Step 1.7.1.3.3

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

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

Step 1.7.1.3.4

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

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

Step 1.7.1.3.5

Add $1$ and $1$ .

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

Step 1.7.1.4

Multiply $- \frac{x}{2} \cdot - 1$ .

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Step 1.7.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 1.7.1.4.2

Multiply $\frac{x}{2}$ by $1$ .

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

Step 1.7.1.5

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

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Step 1.7.1.5.1

Move $x$ .

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

Step 1.7.1.5.2

Multiply $x$ by $x$ .

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

Step 1.7.1.6

Multiply $- 1$ by $2$ .

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

Step 1.7.1.7

Multiply $x$ by $1$ .

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

Step 1.7.1.8

Multiply $- 1$ by $1$ .

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

Step 1.7.2

Add $- 2 x$ and $x$ .

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

Step 2

The largest exponent is the degree of the polynomial.

$4$