Examples

1

$1$ ,

3

$3$ ,

- 6

$- 6$

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 = 3

$x = 3$ was found by solving for

x

$x$ when

x - (3) = y

$x - (3) = y$ and

y = 0

$y = 0$ .

The factor is

x - 3

$x - 3$

Step 4

The root at

x = - 6

$x = - 6$ was found by solving for

x

$x$ when

x - (- 6) = y

$x - (- 6) = y$ and

y = 0

$y = 0$ .

The factor is

x + 6

$x + 6$

Step 5

Combine all the factors into a single equation.

y = (x - 1) (x - 3) (x + 6)

$y = (x - 1) (x - 3) (x + 6)$

Step 6

Multiply all the factors to simplify the equation

y = (x - 1) (x - 3) (x + 6)

$y = (x - 1) (x - 3) (x + 6)$ .

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Step 6.1

Expand

(x - 1) (x - 3)

$(x - 1) (x - 3)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

y = (x (x - 3) - 1 (x - 3)) (x + 6)

$y = (x (x - 3) - 1 (x - 3)) (x + 6)$

Step 6.1.2

Apply the distributive property.

y = (x \cdot x + x \cdot - 3 - 1 (x - 3)) (x + 6)

$y = (x \cdot x + x \cdot - 3 - 1 (x - 3)) (x + 6)$

Step 6.1.3

Apply the distributive property.

y = (x \cdot x + x \cdot - 3 - 1 x - 1 \cdot - 3) (x + 6)

$y = (x \cdot x + x \cdot - 3 - 1 x - 1 \cdot - 3) (x + 6)$

y = (x \cdot x + x \cdot - 3 - 1 x - 1 \cdot - 3) (x + 6)

$y = (x \cdot x + x \cdot - 3 - 1 x - 1 \cdot - 3) (x + 6)$

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

$x$ by

x

$x$ .

y = (x^{2} + x \cdot - 3 - 1 x - 1 \cdot - 3) (x + 6)

$y = (x^{2} + x \cdot - 3 - 1 x - 1 \cdot - 3) (x + 6)$

Step 6.2.1.2

Move

- 3

$- 3$ to the left of

x

$x$ .

y = (x^{2} - 3 \cdot x - 1 x - 1 \cdot - 3) (x + 6)

$y = (x^{2} - 3 \cdot x - 1 x - 1 \cdot - 3) (x + 6)$

Step 6.2.1.3

Rewrite

- 1 x

$- 1 x$ as

- x

$- x$ .

y = (x^{2} - 3 x - x - 1 \cdot - 3) (x + 6)

$y = (x^{2} - 3 x - x - 1 \cdot - 3) (x + 6)$

Step 6.2.1.4

Multiply

- 1

$- 1$ by

- 3

$- 3$ .

y = (x^{2} - 3 x - x + 3) (x + 6)

$y = (x^{2} - 3 x - x + 3) (x + 6)$

y = (x^{2} - 3 x - x + 3) (x + 6)

$y = (x^{2} - 3 x - x + 3) (x + 6)$

Step 6.2.2

Subtract

x

$x$ from

- 3 x

$- 3 x$ .

y = (x^{2} - 4 x + 3) (x + 6)

$y = (x^{2} - 4 x + 3) (x + 6)$

y = (x^{2} - 4 x + 3) (x + 6)

$y = (x^{2} - 4 x + 3) (x + 6)$

Step 6.3

Expand

(x^{2} - 4 x + 3) (x + 6)

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

y = x^{2} x + x^{2} \cdot 6 - 4 x \cdot x - 4 x \cdot 6 + 3 x + 3 \cdot 6

$y = x^{2} x + x^{2} \cdot 6 - 4 x \cdot x - 4 x \cdot 6 + 3 x + 3 \cdot 6$

Step 6.4

Simplify terms.

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Step 6.4.1

Simplify each term.

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Step 6.4.1.1

Multiply

x^{2}

$x^{2}$ by

x

$x$ by adding the exponents.

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Step 6.4.1.1.1

Multiply

x^{2}

$x^{2}$ by

x

$x$ .

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Step 6.4.1.1.1.1

Raise

x

$x$ to the power of

1

$1$ .

y = x^{2} x + x^{2} \cdot 6 - 4 x \cdot x - 4 x \cdot 6 + 3 x + 3 \cdot 6

$y = x^{2} x + x^{2} \cdot 6 - 4 x \cdot x - 4 x \cdot 6 + 3 x + 3 \cdot 6$

Step 6.4.1.1.1.2

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 6 - 4 x \cdot x - 4 x \cdot 6 + 3 x + 3 \cdot 6

$y = x^{2 + 1} + x^{2} \cdot 6 - 4 x \cdot x - 4 x \cdot 6 + 3 x + 3 \cdot 6$

y = x^{2 + 1} + x^{2} \cdot 6 - 4 x \cdot x - 4 x \cdot 6 + 3 x + 3 \cdot 6

$y = x^{2 + 1} + x^{2} \cdot 6 - 4 x \cdot x - 4 x \cdot 6 + 3 x + 3 \cdot 6$

Step 6.4.1.1.2

Add

2

$2$ and

1

$1$ .

y = x^{3} + x^{2} \cdot 6 - 4 x \cdot x - 4 x \cdot 6 + 3 x + 3 \cdot 6

$y = x^{3} + x^{2} \cdot 6 - 4 x \cdot x - 4 x \cdot 6 + 3 x + 3 \cdot 6$

y = x^{3} + x^{2} \cdot 6 - 4 x \cdot x - 4 x \cdot 6 + 3 x + 3 \cdot 6

$y = x^{3} + x^{2} \cdot 6 - 4 x \cdot x - 4 x \cdot 6 + 3 x + 3 \cdot 6$

Step 6.4.1.2

Move

6

$6$ to the left of

x^{2}

$x^{2}$ .

y = x^{3} + 6 \cdot x^{2} - 4 x \cdot x - 4 x \cdot 6 + 3 x + 3 \cdot 6

$y = x^{3} + 6 \cdot x^{2} - 4 x \cdot x - 4 x \cdot 6 + 3 x + 3 \cdot 6$

Step 6.4.1.3

Multiply

x

$x$ by

x

$x$ by adding the exponents.

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Step 6.4.1.3.1

Move

x

$x$ .

y = x^{3} + 6 x^{2} - 4 (x \cdot x) - 4 x \cdot 6 + 3 x + 3 \cdot 6

$y = x^{3} + 6 x^{2} - 4 (x \cdot x) - 4 x \cdot 6 + 3 x + 3 \cdot 6$

Step 6.4.1.3.2

Multiply

x

$x$ by

x

$x$ .

y = x^{3} + 6 x^{2} - 4 x^{2} - 4 x \cdot 6 + 3 x + 3 \cdot 6

$y = x^{3} + 6 x^{2} - 4 x^{2} - 4 x \cdot 6 + 3 x + 3 \cdot 6$

y = x^{3} + 6 x^{2} - 4 x^{2} - 4 x \cdot 6 + 3 x + 3 \cdot 6

$y = x^{3} + 6 x^{2} - 4 x^{2} - 4 x \cdot 6 + 3 x + 3 \cdot 6$

Step 6.4.1.4

Multiply

6

$6$ by

- 4

$- 4$ .

y = x^{3} + 6 x^{2} - 4 x^{2} - 24 x + 3 x + 3 \cdot 6

$y = x^{3} + 6 x^{2} - 4 x^{2} - 24 x + 3 x + 3 \cdot 6$

Step 6.4.1.5

Multiply

3

$3$ by

6

$6$ .

y = x^{3} + 6 x^{2} - 4 x^{2} - 24 x + 3 x + 18

$y = x^{3} + 6 x^{2} - 4 x^{2} - 24 x + 3 x + 18$

y = x^{3} + 6 x^{2} - 4 x^{2} - 24 x + 3 x + 18

$y = x^{3} + 6 x^{2} - 4 x^{2} - 24 x + 3 x + 18$

Step 6.4.2

Simplify by adding terms.

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Step 6.4.2.1

Subtract

4 x^{2}

$4 x^{2}$ from

6 x^{2}

$6 x^{2}$ .

y = x^{3} + 2 x^{2} - 24 x + 3 x + 18

$y = x^{3} + 2 x^{2} - 24 x + 3 x + 18$

Step 6.4.2.2

Add

- 24 x

$- 24 x$ and

3 x

$3 x$ .

y = x^{3} + 2 x^{2} - 21 x + 18

$y = x^{3} + 2 x^{2} - 21 x + 18$

y = x^{3} + 2 x^{2} - 21 x + 18

$y = x^{3} + 2 x^{2} - 21 x + 18$

y = x^{3} + 2 x^{2} - 21 x + 18

$y = x^{3} + 2 x^{2} - 21 x + 18$

y = x^{3} + 2 x^{2} - 21 x + 18

$y = x^{3} + 2 x^{2} - 21 x + 18$

Step 7

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