Precalculus Examples

$1$ , $3$ , $- 6$

Roots (also called zeros) are the points where the graph intercepts with the x-axis $(y = 0)$ .

$y = 0$ at the roots

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

The factor is $x - 1$

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

The factor is $x - 3$

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

The factor is $x + 6$

Combine all the factors into a single equation.

$y = (x - 1) (x - 3) (x + 6)$

Multiply all the factors to simplify the equation $y = (x - 1) (x - 3) (x + 6)$ .

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Expand $(x - 1) (x - 3)$ using the FOIL Method.

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Apply the distributive property.

$y = (x (x - 3) - 1 (x - 3)) (x + 6)$

Apply the distributive property.

$y = (x x + x \cdot - 3 - 1 (x - 3)) (x + 6)$

Apply the distributive property.

$y = (x x + x \cdot - 3 + (- 1 x - 1 \cdot - 3)) (x + 6)$

Remove parentheses.

$y = (x x + x \cdot - 3 - 1 x - 1 \cdot - 3) (x + 6)$

Simplify and combine like terms.

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Simplify each term.

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

$y = (x^{1 + 1} + x \cdot - 3 - 1 x - 1 \cdot - 3) (x + 6)$

Add $1$ and $1$ to get $2$ .

$y = (x^{2} + x \cdot - 3 - 1 x - 1 \cdot - 3) (x + 6)$

Move $- 3$ to the left of the expression $x \cdot - 3$ .

$y = (x^{2} - 3 \cdot x - 1 x - 1 \cdot - 3) (x + 6)$

Multiply $- 3$ by $x$ to get $- 3 x$ .

$y = (x^{2} - 3 x - 1 x - 1 \cdot - 3) (x + 6)$

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

$y = (x^{2} - 3 x - x - 1 \cdot - 3) (x + 6)$

Multiply $- 1$ by $- 3$ to get $3$ .

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

Subtract $x$ from $- 3 x$ to get $- 4 x$ .

$y = (x^{2} - 4 x + 3) (x + 6)$

Expand $(x^{2} - 4 x + 3) (x + 6)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} \cdot 6 + (- 4 x x - 4 x \cdot 6) + (3 x + 3 \cdot 6)$

Simplify terms.

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Remove unnecessary parentheses.

$y = x^{2} x + x^{2} \cdot 6 - 4 x x - 4 x \cdot 6 + 3 x + 3 \cdot 6$

Simplify each term.

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

$y = x^{2 + 1} + x^{2} \cdot 6 - 4 x x - 4 x \cdot 6 + 3 x + 3 \cdot 6$

Add $2$ and $1$ to get $3$ .

$y = x^{3} + x^{2} \cdot 6 - 4 x x - 4 x \cdot 6 + 3 x + 3 \cdot 6$

Move $6$ to the left of the expression $x^{2} \cdot 6$ .

$y = x^{3} + 6 \cdot x^{2} - 4 x x - 4 x \cdot 6 + 3 x + 3 \cdot 6$

Multiply $6$ by $x^{2}$ to get $6 x^{2}$ .

$y = x^{3} + 6 x^{2} - 4 x x - 4 x \cdot 6 + 3 x + 3 \cdot 6$

Move $x$ .

$y = x^{3} + 6 x^{2} - 4 (x x) - 4 x \cdot 6 + 3 x + 3 \cdot 6$

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

$y = x^{3} + 6 x^{2} - 4 x^{1 + 1} - 4 x \cdot 6 + 3 x + 3 \cdot 6$

Add $1$ and $1$ to get $2$ .

$y = x^{3} + 6 x^{2} - 4 x^{2} - 4 x \cdot 6 + 3 x + 3 \cdot 6$

Multiply $6$ by $- 4$ to get $- 24$ .

$y = x^{3} + 6 x^{2} - 4 x^{2} - 24 x + 3 x + 3 \cdot 6$

Multiply $3$ by $6$ to get $18$ .

$y = x^{3} + 6 x^{2} - 4 x^{2} - 24 x + 3 x + 18$

Simplify by adding terms.

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Subtract $4 x^{2}$ from $6 x^{2}$ to get $2 x^{2}$ .

$y = x^{3} + 2 x^{2} - 24 x + 3 x + 18$

Add $- 24 x$ and $3 x$ to get $- 21 x$ .

$y = x^{3} + 2 x^{2} - 21 x + 18$

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