Precalculus Examples

$10$ , $1$

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 $x = 10$ was found by solving for $x$ when $x - (10) = y$ and $y = 0$ ( $y = 0$ because the root is the x-intercept).

The factor is $x - 10$

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 $x - 1$

Combine all the factors into a single equation.

$y = (x - 10) (x - 1)$

Multiply all the factors to simplify the equation $y = (x - 10) (x - 1)$ .

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Expand $(x - 10) (x - 1)$ using the FOIL Method.

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Apply the distributive property.

$y = x (x - 1) - 10 (x - 1)$

Apply the distributive property.

$y = x x + x \cdot - 1 - 10 (x - 1)$

Apply the distributive property.

$y = x x + x \cdot - 1 + (- 10 x - 10 \cdot - 1)$

Remove parentheses.

$y = x x + x \cdot - 1 - 10 x - 10 \cdot - 1$

Simplify and combine like terms.

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Simplify each term.

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

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Combine $x$ and $x$

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

$y = x x + x \cdot - 1 - 10 x - 10 \cdot - 1$

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

$y = x x + x \cdot - 1 - 10 x - 10 \cdot - 1$

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

$y = x^{1 + 1} + x \cdot - 1 - 10 x - 10 \cdot - 1$

Add $1$ and $1$ .

$y = x^{2} + x \cdot - 1 - 10 x - 10 \cdot - 1$

Move $- 1$ to the left of the expression $x \cdot - 1$ .

$y = x^{2} - 1 \cdot x - 10 x - 10 \cdot - 1$

Multiply $- 1$ by $x$ .

$y = x^{2} - 1 x - 10 x - 10 \cdot - 1$

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

$y = x^{2} - x - 10 x - 10 \cdot - 1$

Multiply $- 10$ by $- 1$ .

$y = x^{2} - x - 10 x + 10$

Subtract $10 x$ from $- x$ .

$y = x^{2} - 11 x + 10$

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