Find the Equation Given the Roots -8 , -1 , 3

$- 8$ , $- 1$ , $3$

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

The factor is $x + 8$

Step 3

The root at $x = - 1$ was found by solving for $x$ when $x - (- 1) = y$ and $y = 0$ .

The factor is $x + 1$

Step 4

The root at $x = 3$ was found by solving for $x$ when $x - (3) = y$ and $y = 0$ .

The factor is $x - 3$

Step 5

Combine all the factors into a single equation.

$y = (x + 8) (x + 1) (x - 3)$

Step 6

Multiply all the factors to simplify the equation $y = (x + 8) (x + 1) (x - 3)$ .

Expand $(x + 8) (x + 1)$ using the FOIL Method.

Apply the distributive property.

$y = (x (x + 1) + 8 (x + 1)) (x - 3)$

Apply the distributive property.

$y = (x \cdot x + x \cdot 1 + 8 (x + 1)) (x - 3)$

Apply the distributive property.

$y = (x \cdot x + x \cdot 1 + 8 x + 8 \cdot 1) (x - 3)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = (x^{2} + x \cdot 1 + 8 x + 8 \cdot 1) (x - 3)$

Multiply $x$ by $1$ .

$y = (x^{2} + x + 8 x + 8 \cdot 1) (x - 3)$

Multiply $8$ by $1$ .

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

Add $x$ and $8 x$ .

$y = (x^{2} + 9 x + 8) (x - 3)$

Expand $(x^{2} + 9 x + 8) (x - 3)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} \cdot - 3 + 9 x \cdot x + 9 x \cdot - 3 + 8 x + 8 \cdot - 3$

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 $1$ .

$y = x^{2} x + x^{2} \cdot - 3 + 9 x \cdot x + 9 x \cdot - 3 + 8 x + 8 \cdot - 3$

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

$y = x^{2 + 1} + x^{2} \cdot - 3 + 9 x \cdot x + 9 x \cdot - 3 + 8 x + 8 \cdot - 3$

Add $2$ and $1$ .

$y = x^{3} + x^{2} \cdot - 3 + 9 x \cdot x + 9 x \cdot - 3 + 8 x + 8 \cdot - 3$

Move $- 3$ to the left of $x^{2}$ .

$y = x^{3} - 3 \cdot x^{2} + 9 x \cdot x + 9 x \cdot - 3 + 8 x + 8 \cdot - 3$

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

Move $x$ .

$y = x^{3} - 3 x^{2} + 9 (x \cdot x) + 9 x \cdot - 3 + 8 x + 8 \cdot - 3$

Multiply $x$ by $x$ .

$y = x^{3} - 3 x^{2} + 9 x^{2} + 9 x \cdot - 3 + 8 x + 8 \cdot - 3$

Multiply $- 3$ by $9$ .

$y = x^{3} - 3 x^{2} + 9 x^{2} - 27 x + 8 x + 8 \cdot - 3$

Multiply $8$ by $- 3$ .

$y = x^{3} - 3 x^{2} + 9 x^{2} - 27 x + 8 x - 24$

Simplify by adding terms.

Add $- 3 x^{2}$ and $9 x^{2}$ .

$y = x^{3} + 6 x^{2} - 27 x + 8 x - 24$

Add $- 27 x$ and $8 x$ .

$y = x^{3} + 6 x^{2} - 19 x - 24$

Step 7

Find the Equation Given the Roots -8 , -1 , 3

Related Questions

Support LCMGCF.com by disabling your adblocker.