Find the Equation Given the Roots -2 , 1 , 2 , 3

$- 2$ , $1$ , $2$ , $3$

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

The factor is $x + 2$

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

The factor is $x - 2$

Step 5

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

The factor is $x - 3$

Step 6

Combine all the factors into a single equation.

$y = (x + 2) (x - 1) (x - 2) (x - 3)$

Step 7

Multiply all the factors to simplify the equation $y = (x + 2) (x - 1) (x - 2) (x - 3)$ .

Expand $(x + 2) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$y = (x (x - 1) + 2 (x - 1)) (x - 2) (x - 3)$

Apply the distributive property.

$y = (x \cdot x + x \cdot - 1 + 2 (x - 1)) (x - 2) (x - 3)$

Apply the distributive property.

$y = (x \cdot x + x \cdot - 1 + 2 x + 2 \cdot - 1) (x - 2) (x - 3)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 1$ to the left of $x$ .

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

Rewrite $- 1 x$ as $- x$ .

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

Multiply $2$ by $- 1$ .

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

Add $- x$ and $2 x$ .

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

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

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

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 $1$ .

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

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

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

Add $2$ and $1$ .

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

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

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

Multiply $x$ by $x$ .

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

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

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

Multiply $- 2$ by $- 2$ .

$y = (x^{3} - 2 x^{2} + x^{2} - 2 x - 2 x + 4) (x - 3)$

Simplify by adding terms.

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

$y = (x^{3} - x^{2} - 2 x - 2 x + 4) (x - 3)$

Subtract $2 x$ from $- 2 x$ .

$y = (x^{3} - x^{2} - 4 x + 4) (x - 3)$

Expand $(x^{3} - x^{2} - 4 x + 4) (x - 3)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{3} x + x^{3} \cdot - 3 - x^{2} x - x^{2} \cdot - 3 - 4 x \cdot x - 4 x \cdot - 3 + 4 x + 4 \cdot - 3$

Simplify terms.

Simplify each term.

Multiply $x^{3}$ by $x$ by adding the exponents.

Multiply $x^{3}$ by $x$ .

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

$y = x^{3} x + x^{3} \cdot - 3 - x^{2} x - x^{2} \cdot - 3 - 4 x \cdot x - 4 x \cdot - 3 + 4 x + 4 \cdot - 3$

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

$y = x^{3 + 1} + x^{3} \cdot - 3 - x^{2} x - x^{2} \cdot - 3 - 4 x \cdot x - 4 x \cdot - 3 + 4 x + 4 \cdot - 3$

Add $3$ and $1$ .

$y = x^{4} + x^{3} \cdot - 3 - x^{2} x - x^{2} \cdot - 3 - 4 x \cdot x - 4 x \cdot - 3 + 4 x + 4 \cdot - 3$

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

$y = x^{4} - 3 \cdot x^{3} - x^{2} x - x^{2} \cdot - 3 - 4 x \cdot x - 4 x \cdot - 3 + 4 x + 4 \cdot - 3$

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

Move $x$ .

$y = x^{4} - 3 x^{3} - (x \cdot x^{2}) - x^{2} \cdot - 3 - 4 x \cdot x - 4 x \cdot - 3 + 4 x + 4 \cdot - 3$

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

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

$y = x^{4} - 3 x^{3} - (x \cdot x^{2}) - x^{2} \cdot - 3 - 4 x \cdot x - 4 x \cdot - 3 + 4 x + 4 \cdot - 3$

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

$y = x^{4} - 3 x^{3} - x^{1 + 2} - x^{2} \cdot - 3 - 4 x \cdot x - 4 x \cdot - 3 + 4 x + 4 \cdot - 3$

Add $1$ and $2$ .

$y = x^{4} - 3 x^{3} - x^{3} - x^{2} \cdot - 3 - 4 x \cdot x - 4 x \cdot - 3 + 4 x + 4 \cdot - 3$

Multiply $- 3$ by $- 1$ .

$y = x^{4} - 3 x^{3} - x^{3} + 3 x^{2} - 4 x \cdot x - 4 x \cdot - 3 + 4 x + 4 \cdot - 3$

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

Move $x$ .

$y = x^{4} - 3 x^{3} - x^{3} + 3 x^{2} - 4 (x \cdot x) - 4 x \cdot - 3 + 4 x + 4 \cdot - 3$

Multiply $x$ by $x$ .

$y = x^{4} - 3 x^{3} - x^{3} + 3 x^{2} - 4 x^{2} - 4 x \cdot - 3 + 4 x + 4 \cdot - 3$

Multiply $- 3$ by $- 4$ .

$y = x^{4} - 3 x^{3} - x^{3} + 3 x^{2} - 4 x^{2} + 12 x + 4 x + 4 \cdot - 3$

Multiply $4$ by $- 3$ .

$y = x^{4} - 3 x^{3} - x^{3} + 3 x^{2} - 4 x^{2} + 12 x + 4 x - 12$

Simplify by adding terms.

Subtract $x^{3}$ from $- 3 x^{3}$ .

$y = x^{4} - 4 x^{3} + 3 x^{2} - 4 x^{2} + 12 x + 4 x - 12$

Subtract $4 x^{2}$ from $3 x^{2}$ .

$y = x^{4} - 4 x^{3} - x^{2} + 12 x + 4 x - 12$

Add $12 x$ and $4 x$ .

$y = x^{4} - 4 x^{3} - x^{2} + 16 x - 12$

Step 8

Find the Equation Given the Roots -2 , 1 , 2 , 3

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