Examples

- 1

$- 1$ ,

0

$0$ ,

1

$1$

Step 1

Roots are the points where the graph intercepts with the x-axis

(y = 0)

$(y = 0)$ .

y = 0

$y = 0$ at the roots

Step 2

The root at

x = - 1

$x = - 1$ was found by solving for

x

$x$ when

x - (- 1) = y

$x - (- 1) = y$ and

y = 0

$y = 0$ .

The factor is

x + 1

$x + 1$

Step 3

The root at

x = 0

$x = 0$ was found by solving for

x

$x$ when

x - (0) = y

$x - (0) = y$ and

y = 0

$y = 0$ .

The factor is

x

$x$

Step 4

The root at

x = 1

$x = 1$ was found by solving for

x

$x$ when

x - (1) = y

$x - (1) = y$ and

y = 0

$y = 0$ .

The factor is

x - 1

$x - 1$

Step 5

Combine all the factors into a single equation.

y = (x + 1) (x) (x - 1)

$y = (x + 1) (x) (x - 1)$

Step 6

Multiply all the factors to simplify the equation

y = (x + 1) (x) (x - 1)

$y = (x + 1) (x) (x - 1)$ .

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Step 6.1

Simplify by multiplying through.

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Step 6.1.1

Apply the distributive property.

y = (x \cdot x + 1 x) (x - 1)

$y = (x \cdot x + 1 x) (x - 1)$

Step 6.1.2

Simplify the expression.

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Step 6.1.2.1

Multiply

x

$x$ by

x

$x$ .

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

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

Step 6.1.2.2

Multiply

x

$x$ by

1

$1$ .

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

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

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

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

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

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

Step 6.2

Expand

(x^{2} + x) (x - 1)

$(x^{2} + x) (x - 1)$ using the FOIL Method.

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Step 6.2.1

Apply the distributive property.

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

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

Step 6.2.2

Apply the distributive property.

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

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

Step 6.2.3

Apply the distributive property.

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

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

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

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

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}

$x^{2}$ by

x

$x$ by adding the exponents.

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Step 6.3.1.1.1

Multiply

x^{2}

$x^{2}$ by

x

$x$ .

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Step 6.3.1.1.1.1

Raise

x

$x$ to the power of

1

$1$ .

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

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

Step 6.3.1.1.1.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

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

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

Step 6.3.1.1.2

Add

2

$2$ and

1

$1$ .

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

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

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

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

Step 6.3.1.2

Move

- 1

$- 1$ to the left of

x^{2}

$x^{2}$ .

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

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

Step 6.3.1.3

Rewrite

- 1 x^{2}

$- 1 x^{2}$ as

- x^{2}

$- x^{2}$ .

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

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

Step 6.3.1.4

Multiply

x

$x$ by

x

$x$ .

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

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

Step 6.3.1.5

Move

- 1

$- 1$ to the left of

x

$x$ .

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

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

Step 6.3.1.6

Rewrite

- 1 x

$- 1 x$ as

- x

$- x$ .

y = x^{3} - x^{2} + x^{2} - x

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

y = x^{3} - x^{2} + x^{2} - x

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

Step 6.3.2

Add

- x^{2}

$- x^{2}$ and

x^{2}

$x^{2}$ .

y = x^{3} + 0 - x

$y = x^{3} + 0 - x$

Step 6.3.3

Add

x^{3}

$x^{3}$ and

0

$0$ .

y = x^{3} - x

$y = x^{3} - x$

y = x^{3} - x

$y = x^{3} - x$

y = x^{3} - x

$y = x^{3} - x$

Step 7

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