Find the Equation Given the Roots -1 , 0 , 2

$- 1$ , $0$ , $2$

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

The factor is $x + 1$

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

The factor is $x - 2$

Step 5

Combine all the factors into a single equation.

$y = (x + 1) (x) (x - 2)$

Step 6

Multiply all the factors to simplify the equation $y = (x + 1) (x) (x - 2)$ .

Simplify by multiplying through.

Apply the distributive property.

$y = (x \cdot x + 1 x) (x - 2)$

Simplify the expression.

Multiply $x$ by $x$ .

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

Multiply $x$ by $1$ .

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

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 $1$ .

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

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

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

Add $2$ and $1$ .

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

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

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

Multiply $x$ by $x$ .

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

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

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

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

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

Step 7

Find the Equation Given the Roots -1 , 0 , 2

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