Algebra Examples

$- \frac{1}{2}$ , $0$ , $5$

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

The factor is $x$

Step 4

The root at $x = 5$ was found by solving for $x$ when $x - (5) = y$ and $y = 0$ .

The factor is $x - 5$

Step 5

Combine all the factors into a single equation.

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

Step 6

Multiply all the factors to simplify the equation $y = (x + \frac{1}{2}) (x) (x - 5)$ .

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Step 6.1

Simplify terms.

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Step 6.1.1

Apply the distributive property.

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

Step 6.1.2

Multiply $x$ by $x$ .

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

Step 6.1.3

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

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

Step 6.2

Expand $(x^{2} + \frac{x}{2}) (x - 5)$ using the FOIL Method.

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Step 6.2.1

Apply the distributive property.

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

Step 6.2.2

Apply the distributive property.

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

Step 6.2.3

Apply the distributive property.

$y = x^{2} x + x^{2} \cdot - 5 + \frac{x}{2} \cdot x + \frac{x}{2} \cdot - 5$

Step 6.3

Simplify and combine like terms.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

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

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Step 6.3.1.1.1

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

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Step 6.3.1.1.1.1

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

$y = x^{2} x + x^{2} \cdot - 5 + \frac{x}{2} \cdot x + \frac{x}{2} \cdot - 5$

Step 6.3.1.1.1.2

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

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

Step 6.3.1.1.2

Add $2$ and $1$ .

$y = x^{3} + x^{2} \cdot - 5 + \frac{x}{2} \cdot x + \frac{x}{2} \cdot - 5$

Step 6.3.1.2

Move $- 5$ to the left of $x^{2}$ .

$y = x^{3} - 5 \cdot x^{2} + \frac{x}{2} \cdot x + \frac{x}{2} \cdot - 5$

Step 6.3.1.3

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

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Step 6.3.1.3.1

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

$y = x^{3} - 5 x^{2} + \frac{x \cdot x}{2} + \frac{x}{2} \cdot - 5$

Step 6.3.1.3.2

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

$y = x^{3} - 5 x^{2} + \frac{x \cdot x}{2} + \frac{x}{2} \cdot - 5$

Step 6.3.1.3.3

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

$y = x^{3} - 5 x^{2} + \frac{x \cdot x}{2} + \frac{x}{2} \cdot - 5$

Step 6.3.1.3.4

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

$y = x^{3} - 5 x^{2} + \frac{x^{1 + 1}}{2} + \frac{x}{2} \cdot - 5$

Step 6.3.1.3.5

Add $1$ and $1$ .

$y = x^{3} - 5 x^{2} + \frac{x^{2}}{2} + \frac{x}{2} \cdot - 5$

Step 6.3.1.4

Combine $\frac{x}{2}$ and $- 5$ .

$y = x^{3} - 5 x^{2} + \frac{x^{2}}{2} + \frac{x \cdot - 5}{2}$

Step 6.3.1.5

Move $- 5$ to the left of $x$ .

$y = x^{3} - 5 x^{2} + \frac{x^{2}}{2} + \frac{- 5 \cdot x}{2}$

Step 6.3.1.6

Move the negative in front of the fraction.

$y = x^{3} - 5 x^{2} + \frac{x^{2}}{2} - \frac{5 x}{2}$

Step 6.3.2

To write $- 5 x^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = x^{3} - 5 x^{2} \cdot \frac{2}{2} + \frac{x^{2}}{2} - \frac{5 x}{2}$

Step 6.3.3

Combine $- 5 x^{2}$ and $\frac{2}{2}$ .

$y = x^{3} + \frac{- 5 x^{2} \cdot 2}{2} + \frac{x^{2}}{2} - \frac{5 x}{2}$

Step 6.3.4

Combine the numerators over the common denominator.

$y = x^{3} + \frac{- 5 x^{2} \cdot 2 + x^{2}}{2} - \frac{5 x}{2}$

Step 6.3.5

To write $x^{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = x^{3} \cdot \frac{2}{2} + \frac{- 5 x^{2} \cdot 2 + x^{2}}{2} - \frac{5 x}{2}$

Step 6.3.6

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

$y = \frac{x^{3} \cdot 2}{2} + \frac{- 5 x^{2} \cdot 2 + x^{2}}{2} - \frac{5 x}{2}$

Step 6.3.7

Combine the numerators over the common denominator.

$y = \frac{x^{3} \cdot 2 - 5 x^{2} \cdot 2 + x^{2}}{2} - \frac{5 x}{2}$

Step 6.3.8

Combine the numerators over the common denominator.

$y = \frac{x^{3} \cdot 2 - 5 x^{2} \cdot 2 + x^{2} - 5 x}{2}$

Step 6.4

Simplify the numerator.

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Step 6.4.1

Factor $x$ out of $x^{3} \cdot 2 - 5 x^{2} \cdot 2 + x^{2} - 5 x$ .

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Step 6.4.1.1

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

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

Step 6.4.1.2

Factor $x$ out of $- 5 x^{2} \cdot 2$ .

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

Step 6.4.1.3

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

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

Step 6.4.1.4

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

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

Step 6.4.1.5

Factor $x$ out of $x (x^{2} \cdot 2) + x (- 5 x \cdot 2)$ .

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

Step 6.4.1.6

Factor $x$ out of $x (x^{2} \cdot 2 - 5 x \cdot 2) + x \cdot x$ .

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

Step 6.4.1.7

Factor $x$ out of $x (x^{2} \cdot 2 - 5 x \cdot 2 + x) + x \cdot - 5$ .

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

Step 6.4.2

Move $2$ to the left of $x^{2}$ .

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

Step 6.4.3

Multiply $2$ by $- 5$ .

$y = \frac{x (2 x^{2} - 10 x + x - 5)}{2}$

Step 6.4.4

Add $- 10 x$ and $x$ .

$y = \frac{x (2 x^{2} - 9 x - 5)}{2}$

Step 6.4.5

Factor by grouping.

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Step 6.4.5.1

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 $a \cdot c = 2 \cdot - 5 = - 10$ and whose sum is $b = - 9$ .

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Step 6.4.5.1.1

Factor $- 9$ out of $- 9 x$ .

$y = \frac{x (2 x^{2} - 9 x - 5)}{2}$

Step 6.4.5.1.2

Rewrite $- 9$ as $1$ plus $- 10$

$y = \frac{x (2 x^{2} + (1 - 10) x - 5)}{2}$

Step 6.4.5.1.3

Apply the distributive property.

$y = \frac{x (2 x^{2} + 1 x - 10 x - 5)}{2}$

Step 6.4.5.1.4

Multiply $x$ by $1$ .

$y = \frac{x (2 x^{2} + x - 10 x - 5)}{2}$

Step 6.4.5.2

Factor out the greatest common factor from each group.

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Step 6.4.5.2.1

Group the first two terms and the last two terms.

$y = \frac{x ((2 x^{2} + x) - 10 x - 5)}{2}$

Step 6.4.5.2.2

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

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

Step 6.4.5.3

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

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

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

Step 6.5

Apply the distributive property.

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

Step 6.6

Rewrite using the commutative property of multiplication.

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

Step 6.7

Multiply $x$ by $1$ .

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

Step 6.8

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

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Step 6.8.1

Move $x$ .

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

Step 6.8.2

Multiply $x$ by $x$ .

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

Step 6.9

Expand $(2 x^{2} + x) (x - 5)$ using the FOIL Method.

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Step 6.9.1

Apply the distributive property.

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

Step 6.9.2

Apply the distributive property.

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

Step 6.9.3

Apply the distributive property.

$y = \frac{2 x^{2} x + 2 x^{2} \cdot - 5 + x \cdot x + x \cdot - 5}{2}$

Step 6.10

Simplify and combine like terms.

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Step 6.10.1

Simplify each term.

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Step 6.10.1.1

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

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Step 6.10.1.1.1

Move $x$ .

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

Step 6.10.1.1.2

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

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Step 6.10.1.1.2.1

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

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

Step 6.10.1.1.2.2

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

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

Step 6.10.1.1.3

Add $1$ and $2$ .

$y = \frac{2 x^{3} + 2 x^{2} \cdot - 5 + x \cdot x + x \cdot - 5}{2}$

Step 6.10.1.2

Multiply $- 5$ by $2$ .

$y = \frac{2 x^{3} - 10 x^{2} + x \cdot x + x \cdot - 5}{2}$

Step 6.10.1.3

Multiply $x$ by $x$ .

$y = \frac{2 x^{3} - 10 x^{2} + x^{2} + x \cdot - 5}{2}$

Step 6.10.1.4

Move $- 5$ to the left of $x$ .

$y = \frac{2 x^{3} - 10 x^{2} + x^{2} - 5 x}{2}$

Step 6.10.2

Add $- 10 x^{2}$ and $x^{2}$ .

$y = \frac{2 x^{3} - 9 x^{2} - 5 x}{2}$

Step 6.11

Split the fraction $\frac{2 x^{3} - 9 x^{2} - 5 x}{2}$ into two fractions.

$y = \frac{2 x^{3} - 9 x^{2}}{2} + \frac{- 5 x}{2}$

Step 6.12

Split the fraction $\frac{2 x^{3} - 9 x^{2}}{2}$ into two fractions.

$y = \frac{2 x^{3}}{2} + \frac{- 9 x^{2}}{2} + \frac{- 5 x}{2}$

Step 6.13

Cancel the common factor of $2$ .

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Step 6.13.1

Cancel the common factor.

$y = \frac{2 x^{3}}{2} + \frac{- 9 x^{2}}{2} + \frac{- 5 x}{2}$

Step 6.13.2

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

$y = x^{3} + \frac{- 9 x^{2}}{2} + \frac{- 5 x}{2}$

Step 6.14

Move the negative in front of the fraction.

$y = x^{3} - \frac{9 x^{2}}{2} + \frac{- 5 x}{2}$

Step 6.15

Move the negative in front of the fraction.

$y = x^{3} - \frac{9 x^{2}}{2} - \frac{5 x}{2}$

Step 7