Precalculus Examples

$3$ , $- 4$ , $7$

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

The factor is $x - 3$

Step 3

The root at $x = - 4$ was found by solving for $x$ when $x - (- 4) = y$ and $y = 0$ .

The factor is $x + 4$

Step 4

The root at $x = 7$ was found by solving for $x$ when $x - (7) = y$ and $y = 0$ .

The factor is $x - 7$

Step 5

Combine all the factors into a single equation.

$y = (x - 3) (x + 4) (x - 7)$

Step 6

Multiply all the factors to simplify the equation $y = (x - 3) (x + 4) (x - 7)$ .

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Step 6.1

Expand $(x - 3) (x + 4)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$y = (x (x + 4) - 3 (x + 4)) (x - 7)$

Step 6.1.2

Apply the distributive property.

$y = (x \cdot x + x \cdot 4 - 3 (x + 4)) (x - 7)$

Step 6.1.3

Apply the distributive property.

$y = (x \cdot x + x \cdot 4 - 3 x - 3 \cdot 4) (x - 7)$

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $x$ by $x$ .

$y = (x^{2} + x \cdot 4 - 3 x - 3 \cdot 4) (x - 7)$

Step 6.2.1.2

Move $4$ to the left of $x$ .

$y = (x^{2} + 4 \cdot x - 3 x - 3 \cdot 4) (x - 7)$

Step 6.2.1.3

Multiply $- 3$ by $4$ .

$y = (x^{2} + 4 x - 3 x - 12) (x - 7)$

Step 6.2.2

Subtract $3 x$ from $4 x$ .

$y = (x^{2} + x - 12) (x - 7)$

Step 6.3

Expand $(x^{2} + x - 12) (x - 7)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} \cdot - 7 + x \cdot x + x \cdot - 7 - 12 x - 12 \cdot - 7$

Step 6.4

Simplify terms.

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Step 6.4.1

Simplify each term.

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Step 6.4.1.1

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

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Step 6.4.1.1.1

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

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Step 6.4.1.1.1.1

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

$y = x^{2} x + x^{2} \cdot - 7 + x \cdot x + x \cdot - 7 - 12 x - 12 \cdot - 7$

Step 6.4.1.1.1.2

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

$y = x^{2 + 1} + x^{2} \cdot - 7 + x \cdot x + x \cdot - 7 - 12 x - 12 \cdot - 7$

Step 6.4.1.1.2

Add $2$ and $1$ .

$y = x^{3} + x^{2} \cdot - 7 + x \cdot x + x \cdot - 7 - 12 x - 12 \cdot - 7$

Step 6.4.1.2

Move $- 7$ to the left of $x^{2}$ .

$y = x^{3} - 7 \cdot x^{2} + x \cdot x + x \cdot - 7 - 12 x - 12 \cdot - 7$

Step 6.4.1.3

Multiply $x$ by $x$ .

$y = x^{3} - 7 x^{2} + x^{2} + x \cdot - 7 - 12 x - 12 \cdot - 7$

Step 6.4.1.4

Move $- 7$ to the left of $x$ .

$y = x^{3} - 7 x^{2} + x^{2} - 7 \cdot x - 12 x - 12 \cdot - 7$

Step 6.4.1.5

Multiply $- 12$ by $- 7$ .

$y = x^{3} - 7 x^{2} + x^{2} - 7 x - 12 x + 84$

Step 6.4.2

Simplify by adding terms.

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Step 6.4.2.1

Add $- 7 x^{2}$ and $x^{2}$ .

$y = x^{3} - 6 x^{2} - 7 x - 12 x + 84$

Step 6.4.2.2

Subtract $12 x$ from $- 7 x$ .

$y = x^{3} - 6 x^{2} - 19 x + 84$

Step 7