Algebra Examples

$\frac{1}{2}$ , $- 1$ , $2$

Step 1

Roots are the points where the graph intercepts with the x-axis $(y = 0)$ .

$y = 0$ at the roots

Step 2

The root at $x = \frac{1}{2}$ was found by solving for $x$ when $x - (\frac{1}{2}) = y$ and $y = 0$ .

The factor is $x - \frac{1}{2}$

Step 3

The root at $x = - 1$ was found by solving for $x$ when $x - (- 1) = y$ and $y = 0$ .

The factor is $x + 1$

Step 4

The root at $x = 2$ was found by solving for $x$ when $x - (2) = y$ and $y = 0$ .

The factor is $x - 2$

Step 5

Combine all the factors into a single equation.

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

Step 6

Multiply all the factors to simplify the equation $y = (x - \frac{1}{2}) (x + 1) (x - 2)$ .

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Step 6.1

Expand $(x - \frac{1}{2}) (x + 1)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

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

Step 6.1.2

Apply the distributive property.

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

Step 6.1.3

Apply the distributive property.

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

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $x$ by $x$ .

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

Step 6.2.1.2

Multiply $x$ by $1$ .

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

Step 6.2.1.3

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

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

Step 6.2.1.4

Multiply $- 1$ by $1$ .

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

Step 6.2.2

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

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

Step 6.2.3

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

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

Step 6.2.4

Combine the numerators over the common denominator.

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

Step 6.3

Combine the numerators over the common denominator.

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

Step 6.4

Move $2$ to the left of $x$ .

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

Step 6.5

Subtract $x$ from $2 x$ .

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

Step 6.6

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

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

Step 6.7

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

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

Step 6.8

Combine the numerators over the common denominator.

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

Step 6.9

Factor by grouping.

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Step 6.9.1

Reorder terms.

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

Step 6.9.2

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 $a \cdot c = 2 \cdot - 1 = - 2$ and whose sum is $b = 1$ .

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Step 6.9.2.1

Multiply by $1$ .

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

Step 6.9.2.2

Rewrite $1$ as $- 1$ plus $2$

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

Step 6.9.2.3

Apply the distributive property.

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

Step 6.9.3

Factor out the greatest common factor from each group.

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Step 6.9.3.1

Group the first two terms and the last two terms.

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

Step 6.9.3.2

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

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

Step 6.9.4

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

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

Step 6.10

Simplify terms.

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Step 6.10.1

Apply the distributive property.

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

Step 6.10.2

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

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

Step 6.10.3

Cancel the common factor of $2$ .

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Step 6.10.3.1

Factor $2$ out of $- 2$ .

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

Step 6.10.3.2

Cancel the common factor.

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

Step 6.10.3.3

Rewrite the expression.

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

Step 6.11

Simplify each term.

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Step 6.11.1

Expand $(2 x - 1) (x + 1)$ using the FOIL Method.

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Step 6.11.1.1

Apply the distributive property.

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

Step 6.11.1.2

Apply the distributive property.

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

Step 6.11.1.3

Apply the distributive property.

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

Step 6.11.2

Simplify and combine like terms.

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Step 6.11.2.1

Simplify each term.

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Step 6.11.2.1.1

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

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Step 6.11.2.1.1.1

Move $x$ .

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

Step 6.11.2.1.1.2

Multiply $x$ by $x$ .

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

Step 6.11.2.1.2

Multiply $2$ by $1$ .

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

Step 6.11.2.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 6.11.2.1.4

Multiply $- 1$ by $1$ .

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

Step 6.11.2.2

Subtract $x$ from $2 x$ .

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

Step 6.11.3

Apply the distributive property.

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

Step 6.11.4

Simplify.

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Step 6.11.4.1

Multiply $- 1$ by $2$ .

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

Step 6.11.4.2

Move $- 1$ to the left of $x$ .

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

Step 6.11.4.3

Multiply $- 1$ by $- 1$ .

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

Step 6.11.5

Rewrite $- 1 x$ as $- x$ .

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

Step 6.12

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

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

Step 6.13

Simplify terms.

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Step 6.13.1

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

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

Step 6.13.2

Combine the numerators over the common denominator.

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

Step 6.14

Simplify the numerator.

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Step 6.14.1

Factor $x$ out of $(2 x - 1) (x + 1) x - 2 x^{2} \cdot 2$ .

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Step 6.14.1.1

Factor $x$ out of $(2 x - 1) (x + 1) x$ .

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

Step 6.14.1.2

Factor $x$ out of $- 2 x^{2} \cdot 2$ .

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

Step 6.14.1.3

Factor $x$ out of $x ((2 x - 1) (x + 1)) + x (- 2 x \cdot 2)$ .

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

Step 6.14.2

Expand $(2 x - 1) (x + 1)$ using the FOIL Method.

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Step 6.14.2.1

Apply the distributive property.

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

Step 6.14.2.2

Apply the distributive property.

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

Step 6.14.2.3

Apply the distributive property.

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

Step 6.14.3

Simplify and combine like terms.

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Step 6.14.3.1

Simplify each term.

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Step 6.14.3.1.1

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

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Step 6.14.3.1.1.1

Move $x$ .

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

Step 6.14.3.1.1.2

Multiply $x$ by $x$ .

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

Step 6.14.3.1.2

Multiply $2$ by $1$ .

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

Step 6.14.3.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 6.14.3.1.4

Multiply $- 1$ by $1$ .

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

Step 6.14.3.2

Subtract $x$ from $2 x$ .

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

Step 6.14.4

Multiply $2$ by $- 2$ .

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

Step 6.14.5

Subtract $4 x$ from $x$ .

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

Step 6.15

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

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

Step 6.16

Simplify terms.

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Step 6.16.1

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

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

Step 6.16.2

Combine the numerators over the common denominator.

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

Step 6.17

Simplify the numerator.

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Step 6.17.1

Factor $x$ out of $x (2 x^{2} - 3 x - 1) - x \cdot 2$ .

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Step 6.17.1.1

Factor $x$ out of $- x \cdot 2$ .

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

Step 6.17.1.2

Factor $x$ out of $x (2 x^{2} - 3 x - 1) + x (- 1 \cdot 2)$ .

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

Step 6.17.2

Multiply $- 1$ by $2$ .

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

Step 6.17.3

Subtract $2$ from $- 1$ .

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

Step 6.18

Combine into one fraction.

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Step 6.18.1

Write $1$ as a fraction with a common denominator.

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

Step 6.18.2

Combine the numerators over the common denominator.

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

Step 6.19

Simplify the numerator.

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Step 6.19.1

Apply the distributive property.

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

Step 6.19.2

Simplify.

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Step 6.19.2.1

Rewrite using the commutative property of multiplication.

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

Step 6.19.2.2

Rewrite using the commutative property of multiplication.

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

Step 6.19.2.3

Move $- 3$ to the left of $x$ .

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

Step 6.19.3

Simplify each term.

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Step 6.19.3.1

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

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Step 6.19.3.1.1

Move $x^{2}$ .

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

Step 6.19.3.1.2

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

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Step 6.19.3.1.2.1

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

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

Step 6.19.3.1.2.2

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

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

Step 6.19.3.1.3

Add $2$ and $1$ .

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

Step 6.19.3.2

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

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Step 6.19.3.2.1

Move $x$ .

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

Step 6.19.3.2.2

Multiply $x$ by $x$ .

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

$y = \frac{2 x^{3} - 3 x^{2} - 3 x + 2}{2}$

Step 6.19.4

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

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Step 6.19.4.1

Factor $2 x^{3} - 3 x^{2} - 3 x + 2$ using the rational roots test.

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Step 6.19.4.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 2$

$q = \pm 1, \pm 2$

Step 6.19.4.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.5, \pm 2$

Step 6.19.4.1.3

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

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Step 6.19.4.1.3.1

Substitute $- 1$ into the polynomial.

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

Step 6.19.4.1.3.2

Raise $- 1$ to the power of $3$ .

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

Step 6.19.4.1.3.3

Multiply $2$ by $- 1$ .

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

Step 6.19.4.1.3.4

Raise $- 1$ to the power of $2$ .

$- 2 - 3 \cdot 1 - 3 \cdot - 1 + 2$

Step 6.19.4.1.3.5

Multiply $- 3$ by $1$ .

$- 2 - 3 - 3 \cdot - 1 + 2$

Step 6.19.4.1.3.6

Subtract $3$ from $- 2$ .

$- 5 - 3 \cdot - 1 + 2$

Step 6.19.4.1.3.7

Multiply $- 3$ by $- 1$ .

$- 5 + 3 + 2$

Step 6.19.4.1.3.8

Add $- 5$ and $3$ .

$- 2 + 2$

Step 6.19.4.1.3.9

Add $- 2$ and $2$ .

$0$

Step 6.19.4.1.4

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

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

Step 6.19.4.1.5

Divide $2 x^{3} - 3 x^{2} - 3 x + 2$ by $x + 1$ .

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Step 6.19.4.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$1$

$2 x^{3}$

$3 x^{2}$

$3 x$

$2$

Step 6.19.4.1.5.2

Divide the highest order term in the dividend $2 x^{3}$ by the highest order term in divisor $x$ .

					$2 x^{2}$
	$x$	+	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$3 x$	+	$2$

Step 6.19.4.1.5.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$3 x$	+	$2$
			+	$2 x^{3}$	+	$2 x^{2}$

Step 6.19.4.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{3} + 2 x^{2}$

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$3 x$	+	$2$
			-	$2 x^{3}$	-	$2 x^{2}$

Step 6.19.4.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$3 x$	+	$2$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$5 x^{2}$

Step 6.19.4.1.5.6

Pull the next terms from the original dividend down into the current dividend.

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$3 x$	+	$2$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$5 x^{2}$	-	$3 x$

Step 6.19.4.1.5.7

Divide the highest order term in the dividend $- 5 x^{2}$ by the highest order term in divisor $x$ .

				$2 x^{2}$	-	$5 x$
$x$	+	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$3 x$	+	$2$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$5 x^{2}$	-	$3 x$

Step 6.19.4.1.5.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$5 x$
$x$	+	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$3 x$	+	$2$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$5 x^{2}$	-	$3 x$
					-	$5 x^{2}$	-	$5 x$

Step 6.19.4.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x^{2} - 5 x$

				$2 x^{2}$	-	$5 x$
$x$	+	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$3 x$	+	$2$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$5 x^{2}$	-	$3 x$
					+	$5 x^{2}$	+	$5 x$

Step 6.19.4.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$2 x^{2}$	-	$5 x$
$x$	+	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$3 x$	+	$2$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$5 x^{2}$	-	$3 x$
					+	$5 x^{2}$	+	$5 x$
							+	$2 x$

Step 6.19.4.1.5.11

Pull the next terms from the original dividend down into the current dividend.

				$2 x^{2}$	-	$5 x$
$x$	+	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$3 x$	+	$2$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$5 x^{2}$	-	$3 x$
					+	$5 x^{2}$	+	$5 x$
							+	$2 x$	+	$2$

Step 6.19.4.1.5.12

Divide the highest order term in the dividend $2 x$ by the highest order term in divisor $x$ .

				$2 x^{2}$	-	$5 x$	+	$2$
$x$	+	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$3 x$	+	$2$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$5 x^{2}$	-	$3 x$
					+	$5 x^{2}$	+	$5 x$
							+	$2 x$	+	$2$

Step 6.19.4.1.5.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$5 x$	+	$2$
$x$	+	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$3 x$	+	$2$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$5 x^{2}$	-	$3 x$
					+	$5 x^{2}$	+	$5 x$
							+	$2 x$	+	$2$
							+	$2 x$	+	$2$

Step 6.19.4.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $2 x + 2$

				$2 x^{2}$	-	$5 x$	+	$2$
$x$	+	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$3 x$	+	$2$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$5 x^{2}$	-	$3 x$
					+	$5 x^{2}$	+	$5 x$
							+	$2 x$	+	$2$
							-	$2 x$	-	$2$

Step 6.19.4.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$2 x^{2}$	-	$5 x$	+	$2$
$x$	+	$1$		$2 x^{3}$	-	$3 x^{2}$	-	$3 x$	+	$2$
			-	$2 x^{3}$	-	$2 x^{2}$
					-	$5 x^{2}$	-	$3 x$
					+	$5 x^{2}$	+	$5 x$
							+	$2 x$	+	$2$
							-	$2 x$	-	$2$
										$0$

Step 6.19.4.1.5.16

Since the remander is $0$ , the final answer is the quotient.

$2 x^{2} - 5 x + 2$

Step 6.19.4.1.6

Write $2 x^{3} - 3 x^{2} - 3 x + 2$ as a set of factors.

$y = \frac{(x + 1) (2 x^{2} - 5 x + 2)}{2}$

Step 6.19.4.2

Factor by grouping.

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Step 6.19.4.2.1

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 $a \cdot c = 2 \cdot 2 = 4$ and whose sum is $b = - 5$ .

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Step 6.19.4.2.1.1

Factor $- 5$ out of $- 5 x$ .

$y = \frac{(x + 1) (2 x^{2} - 5 x + 2)}{2}$

Step 6.19.4.2.1.2

Rewrite $- 5$ as $- 1$ plus $- 4$

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

Step 6.19.4.2.1.3

Apply the distributive property.

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

Step 6.19.4.2.2

Factor out the greatest common factor from each group.

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Step 6.19.4.2.2.1

Group the first two terms and the last two terms.

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

Step 6.19.4.2.2.2

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

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

Step 6.19.4.2.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

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

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

Step 6.20

Expand $(x + 1) (2 x - 1)$ using the FOIL Method.

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Step 6.20.1

Apply the distributive property.

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

Step 6.20.2

Apply the distributive property.

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

Step 6.20.3

Apply the distributive property.

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

Step 6.21

Simplify and combine like terms.

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Step 6.21.1

Simplify each term.

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Step 6.21.1.1

Rewrite using the commutative property of multiplication.

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

Step 6.21.1.2

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

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Step 6.21.1.2.1

Move $x$ .

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

Step 6.21.1.2.2

Multiply $x$ by $x$ .

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

Step 6.21.1.3

Move $- 1$ to the left of $x$ .

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

Step 6.21.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 6.21.1.5

Multiply $2 x$ by $1$ .

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

Step 6.21.1.6

Multiply $- 1$ by $1$ .

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

Step 6.21.2

Add $- x$ and $2 x$ .

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

Step 6.22

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

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

Step 6.23

Simplify each term.

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Step 6.23.1

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

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Step 6.23.1.1

Move $x$ .

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

Step 6.23.1.2

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

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Step 6.23.1.2.1

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

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

Step 6.23.1.2.2

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

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

Step 6.23.1.3

Add $1$ and $2$ .

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

Step 6.23.2

Multiply $- 2$ by $2$ .

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

Step 6.23.3

Multiply $x$ by $x$ .

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

Step 6.23.4

Move $- 2$ to the left of $x$ .

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

Step 6.23.5

Rewrite $- 1 x$ as $- x$ .

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

Step 6.23.6

Multiply $- 1$ by $- 2$ .

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

Step 6.24

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

$y = \frac{2 x^{3} - 3 x^{2} - 2 x - x + 2}{2}$

Step 6.25

Subtract $x$ from $- 2 x$ .

$y = \frac{2 x^{3} - 3 x^{2} - 3 x + 2}{2}$

Step 6.26

Split the fraction $\frac{2 x^{3} - 3 x^{2} - 3 x + 2}{2}$ into two fractions.

$y = \frac{2 x^{3} - 3 x^{2} - 3 x}{2} + \frac{2}{2}$

Step 6.27

Split the fraction $\frac{2 x^{3} - 3 x^{2} - 3 x}{2}$ into two fractions.

$y = \frac{2 x^{3} - 3 x^{2}}{2} + \frac{- 3 x}{2} + \frac{2}{2}$

Step 6.28

Split the fraction $\frac{2 x^{3} - 3 x^{2}}{2}$ into two fractions.

$y = \frac{2 x^{3}}{2} + \frac{- 3 x^{2}}{2} + \frac{- 3 x}{2} + \frac{2}{2}$

Step 6.29

Cancel the common factor of $2$ .

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Step 6.29.1

Cancel the common factor.

$y = \frac{2 x^{3}}{2} + \frac{- 3 x^{2}}{2} + \frac{- 3 x}{2} + \frac{2}{2}$

Step 6.29.2

Divide $x^{3}$ by $1$ .

$y = x^{3} + \frac{- 3 x^{2}}{2} + \frac{- 3 x}{2} + \frac{2}{2}$

Step 6.30

Move the negative in front of the fraction.

$y = x^{3} - \frac{3 x^{2}}{2} + \frac{- 3 x}{2} + \frac{2}{2}$

Step 6.31

Move the negative in front of the fraction.

$y = x^{3} - \frac{3 x^{2}}{2} - \frac{3 x}{2} + \frac{2}{2}$

Step 6.32

Divide $2$ by $2$ .

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

Step 7