Precalculus Examples Rational Expressions and Equations

Find the Equation Given the Roots

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

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

The factor is $x - 3$

Step 4

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

The factor is $x + 6$

Step 5

Combine all the factors into a single equation.

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

Step 6

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

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Step 6.1

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

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Step 6.1.1

Apply the distributive property.

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

Step 6.1.2

Apply the distributive property.

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

Step 6.1.3

Apply the distributive property.

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

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 $x$ .

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

Step 6.2.1.2

Move $- 3$ to the left of $x$ .

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

Step 6.2.1.3

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

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

Step 6.2.1.4

Multiply $- 1$ by $- 3$ .

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

Step 6.2.2

Subtract $x$ from $- 3 x$ .

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

Step 6.3

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

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

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 $1$ .

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

Step 6.4.1.1.1.2

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

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

Step 6.4.1.1.2

Add $2$ and $1$ .

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

Step 6.4.1.2

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

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

Step 6.4.1.3

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

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Step 6.4.1.3.1

Move $x$ .

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

Step 6.4.1.3.2

Multiply $x$ by $x$ .

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

Step 6.4.1.4

Multiply $6$ by $- 4$ .

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

Step 6.4.1.5

Multiply $3$ by $6$ .

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

Step 6.4.2

Simplify by adding terms.

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Step 6.4.2.1

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

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

Step 6.4.2.2

Add $- 24 x$ and $3 x$ .

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

Step 7

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Find the Equation Given the Roots

Rational Expressions and Equations

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