Find the Equation Given the Roots -4 , 2 , 8

$- 4$ , $2$ , $8$

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

$y = 0$ at the roots

The root at $x = - 4$ was found by solving for $x$ when $x - (- 4) = y$ and $y = 0$ .

The factor is $x + 4$

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

The factor is $x - 2$

The root at $x = 8$ was found by solving for $x$ when $x - (8) = y$ and $y = 0$ .

The factor is $x - 8$

Combine all the factors into a single equation.

$y = (x + 4) (x - 2) (x - 8)$

Multiply all the factors to simplify the equation $y = (x + 4) (x - 2) (x - 8)$ .

Expand $(x + 4) (x - 2)$ using the FOIL Method.

Apply the distributive property.

$y = (x (x - 2) + 4 (x - 2)) (x - 8)$

Apply the distributive property.

$y = (x \cdot x + x \cdot - 2 + 4 (x - 2)) (x - 8)$

Apply the distributive property.

$y = (x \cdot x + x \cdot - 2 + 4 x + 4 \cdot - 2) (x - 8)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = (x^{2} + x \cdot - 2 + 4 x + 4 \cdot - 2) (x - 8)$

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

$y = (x^{2} - 2 \cdot x + 4 x + 4 \cdot - 2) (x - 8)$

Multiply $4$ by $- 2$ .

$y = (x^{2} - 2 x + 4 x - 8) (x - 8)$

Add $- 2 x$ and $4 x$ .

$y = (x^{2} + 2 x - 8) (x - 8)$

Expand $(x^{2} + 2 x - 8) (x - 8)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} \cdot - 8 + 2 x \cdot x + 2 x \cdot - 8 - 8 x - 8 \cdot - 8$

Simplify terms.

Simplify each term.

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

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

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

$y = x^{2} x + x^{2} \cdot - 8 + 2 x \cdot x + 2 x \cdot - 8 - 8 x - 8 \cdot - 8$

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

$y = x^{2 + 1} + x^{2} \cdot - 8 + 2 x \cdot x + 2 x \cdot - 8 - 8 x - 8 \cdot - 8$

Add $2$ and $1$ .

$y = x^{3} + x^{2} \cdot - 8 + 2 x \cdot x + 2 x \cdot - 8 - 8 x - 8 \cdot - 8$

Move $- 8$ to the left of $x^{2}$ .

$y = x^{3} - 8 \cdot x^{2} + 2 x \cdot x + 2 x \cdot - 8 - 8 x - 8 \cdot - 8$

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

Move $x$ .

$y = x^{3} - 8 x^{2} + 2 (x \cdot x) + 2 x \cdot - 8 - 8 x - 8 \cdot - 8$

Multiply $x$ by $x$ .

$y = x^{3} - 8 x^{2} + 2 x^{2} + 2 x \cdot - 8 - 8 x - 8 \cdot - 8$

Multiply $- 8$ by $2$ .

$y = x^{3} - 8 x^{2} + 2 x^{2} - 16 x - 8 x - 8 \cdot - 8$

Multiply $- 8$ by $- 8$ .

$y = x^{3} - 8 x^{2} + 2 x^{2} - 16 x - 8 x + 64$

Simplify by adding terms.

Add $- 8 x^{2}$ and $2 x^{2}$ .

$y = x^{3} - 6 x^{2} - 16 x - 8 x + 64$

Subtract $8 x$ from $- 16 x$ .

$y = x^{3} - 6 x^{2} - 24 x + 64$

Find the Equation Given the Roots -4 , 2 , 8

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