Find the Equation Given the Roots -4 , -2 , 6

$- 4$ , $- 2$ , $6$

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

The factor is $x + 4$

Step 3

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

The factor is $x + 2$

Step 4

The root at $x = 6$ was found by solving for $x$ when $x - (6) = y$ and $y = 0$ .

The factor is $x - 6$

Step 5

Combine all the factors into a single equation.

$y = (x + 4) (x + 2) (x - 6)$

Step 6

Multiply all the factors to simplify the equation $y = (x + 4) (x + 2) (x - 6)$ .

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

Apply the distributive property.

$y = (x (x + 2) + 4 (x + 2)) (x - 6)$

Apply the distributive property.

$y = (x \cdot x + x \cdot 2 + 4 (x + 2)) (x - 6)$

Apply the distributive property.

$y = (x \cdot x + x \cdot 2 + 4 x + 4 \cdot 2) (x - 6)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

Multiply $4$ by $2$ .

$y = (x^{2} + 2 x + 4 x + 8) (x - 6)$

Add $2 x$ and $4 x$ .

$y = (x^{2} + 6 x + 8) (x - 6)$

Expand $(x^{2} + 6 x + 8) (x - 6)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} \cdot - 6 + 6 x \cdot x + 6 x \cdot - 6 + 8 x + 8 \cdot - 6$

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 $1$ .

$y = x^{2} x + x^{2} \cdot - 6 + 6 x \cdot x + 6 x \cdot - 6 + 8 x + 8 \cdot - 6$

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

$y = x^{2 + 1} + x^{2} \cdot - 6 + 6 x \cdot x + 6 x \cdot - 6 + 8 x + 8 \cdot - 6$

Add $2$ and $1$ .

$y = x^{3} + x^{2} \cdot - 6 + 6 x \cdot x + 6 x \cdot - 6 + 8 x + 8 \cdot - 6$

Move $- 6$ to the left of $x^{2}$ .

$y = x^{3} - 6 \cdot x^{2} + 6 x \cdot x + 6 x \cdot - 6 + 8 x + 8 \cdot - 6$

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

Move $x$ .

$y = x^{3} - 6 x^{2} + 6 (x \cdot x) + 6 x \cdot - 6 + 8 x + 8 \cdot - 6$

Multiply $x$ by $x$ .

$y = x^{3} - 6 x^{2} + 6 x^{2} + 6 x \cdot - 6 + 8 x + 8 \cdot - 6$

Multiply $- 6$ by $6$ .

$y = x^{3} - 6 x^{2} + 6 x^{2} - 36 x + 8 x + 8 \cdot - 6$

Multiply $8$ by $- 6$ .

$y = x^{3} - 6 x^{2} + 6 x^{2} - 36 x + 8 x - 48$

Simplify by adding terms.

Combine the opposite terms in $x^{3} - 6 x^{2} + 6 x^{2} - 36 x + 8 x - 48$ .

Add $- 6 x^{2}$ and $6 x^{2}$ .

$y = x^{3} + 0 - 36 x + 8 x - 48$

Add $x^{3}$ and $0$ .

$y = x^{3} - 36 x + 8 x - 48$

Add $- 36 x$ and $8 x$ .

$y = x^{3} - 28 x - 48$

Step 7

Find the Equation Given the Roots -4 , -2 , 6

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