Algebra Examples Rational Expressions and Equations

Find the Equation Given the Roots

$- 1$ , $0$ , $1$

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

The factor is $x$

Step 4

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

The factor is $x - 1$

Step 5

Combine all the factors into a single equation.

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

Step 6

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

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Step 6.1

Simplify by multiplying through.

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Step 6.1.1

Apply the distributive property.

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

Step 6.1.2

Simplify the expression.

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Step 6.1.2.1

Multiply $x$ by $x$ .

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

Step 6.1.2.2

Multiply $x$ by $1$ .

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

Step 6.2

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

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Step 6.2.1

Apply the distributive property.

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

Step 6.2.2

Apply the distributive property.

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

Step 6.2.3

Apply the distributive property.

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

Step 6.3

Simplify and combine like terms.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

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

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Step 6.3.1.1.1

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

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Step 6.3.1.1.1.1

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

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

Step 6.3.1.1.1.2

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

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

Step 6.3.1.1.2

Add $2$ and $1$ .

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

Step 6.3.1.2

Move $- 1$ to the left of $x^{2}$ .

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

Step 6.3.1.3

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

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

Step 6.3.1.4

Multiply $x$ by $x$ .

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

Step 6.3.1.5

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

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

Step 6.3.1.6

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

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

Step 6.3.2

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

$y = x^{3} + 0 - x$

Step 6.3.3

Add $x^{3}$ and $0$ .

$y = x^{3} - x$

Step 7

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Find the Equation Given the Roots

Rational Expressions and Equations

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