Find the Equation Given the Roots -12 , 0 , 1

$- 12$ , $0$ , $1$

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 $x = - 12$ was found by solving for $x$ when $x - (- 12) = y$ and $y = 0$ .

The factor is $x + 12$

Step 3

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

The factor is $x$

Step 4

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

The factor is $x - 1$

Step 5

Combine all the factors into a single equation.

$y = (x + 12) (x) (x - 1)$

Step 6

Multiply all the factors to simplify the equation $y = (x + 12) (x) (x - 1)$ .

Simplify by multiplying through.

Apply the distributive property.

$y = (x \cdot x + 12 x) (x - 1)$

Multiply $x$ by $x$ .

$y = (x^{2} + 12 x) (x - 1)$

Expand $(x^{2} + 12 x) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$y = x^{2} (x - 1) + 12 x (x - 1)$

Apply the distributive property.

$y = x^{2} x + x^{2} \cdot - 1 + 12 x (x - 1)$

Apply the distributive property.

$y = x^{2} x + x^{2} \cdot - 1 + 12 x \cdot x + 12 x \cdot - 1$

Simplify and combine like terms.

Simplify each term.

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

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

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

$y = x^{2} x + x^{2} \cdot - 1 + 12 x \cdot x + 12 x \cdot - 1$

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

$y = x^{2 + 1} + x^{2} \cdot - 1 + 12 x \cdot x + 12 x \cdot - 1$

Add $2$ and $1$ .

$y = x^{3} + x^{2} \cdot - 1 + 12 x \cdot x + 12 x \cdot - 1$

Move $- 1$ to the left of $x^{2}$ .

$y = x^{3} - 1 \cdot x^{2} + 12 x \cdot x + 12 x \cdot - 1$

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$y = x^{3} - x^{2} + 12 x \cdot x + 12 x \cdot - 1$

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

Move $x$ .

$y = x^{3} - x^{2} + 12 (x \cdot x) + 12 x \cdot - 1$

Multiply $x$ by $x$ .

$y = x^{3} - x^{2} + 12 x^{2} + 12 x \cdot - 1$

Multiply $- 1$ by $12$ .

$y = x^{3} - x^{2} + 12 x^{2} - 12 x$

Add $- x^{2}$ and $12 x^{2}$ .

$y = x^{3} + 11 x^{2} - 12 x$

Step 7

Find the Equation Given the Roots -12 , 0 , 1

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