Algebra Examples

$\frac{(x + 8) (x - 5)}{(x + 2) (x + 1) (x - 2)}$

Step 1

Write $\frac{(x + 8) (x - 5)}{(x + 2) (x + 1) (x - 2)}$ as an equation.

$y = \frac{(x + 8) (x - 5)}{(x + 2) (x + 1) (x - 2)}$

Step 2

Find the x-intercepts.

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Step 2.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \frac{(x + 8) (x - 5)}{(x + 2) (x + 1) (x - 2)}$

Step 2.2

Solve the equation.

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Step 2.2.1

Set the numerator equal to zero.

$(x + 8) (x - 5) = 0$

Step 2.2.2

Solve the equation for $x$ .

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Step 2.2.2.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 8 = 0$

$x - 5 = 0$

Step 2.2.2.2

Set $x + 8$ equal to $0$ and solve for $x$ .

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Step 2.2.2.2.1

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Step 2.2.2.2.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 2.2.2.3

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 2.2.2.3.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 2.2.2.3.2

Add $5$ to both sides of the equation.

$x = 5$

Step 2.2.2.4

The final solution is all the values that make $(x + 8) (x - 5) = 0$ true.

$x = - 8, 5$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(- 8, 0), (5, 0)$

Step 3

Find the y-intercepts.

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Step 3.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \frac{((0) + 8) ((0) - 5)}{((0) + 2) ((0) + 1) ((0) - 2)}$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = \frac{(0 + 8) ((0) - 5)}{((0) + 2) ((0) + 1) ((0) - 2)}$

Step 3.2.2

Remove parentheses.

$y = \frac{(0 + 8) (0 - 5)}{((0) + 2) ((0) + 1) ((0) - 2)}$

Step 3.2.3

Remove parentheses.

$y = \frac{(0 + 8) (0 - 5)}{(0 + 2) ((0) + 1) ((0) - 2)}$

Step 3.2.4

Remove parentheses.

$y = \frac{(0 + 8) (0 - 5)}{(0 + 2) (0 + 1) ((0) - 2)}$

Step 3.2.5

Remove parentheses.

$y = \frac{(0 + 8) (0 - 5)}{(0 + 2) (0 + 1) (0 - 2)}$

Step 3.2.6

Remove parentheses.

$y = \frac{((0) + 8) ((0) - 5)}{((0) + 2) ((0) + 1) ((0) - 2)}$

Step 3.2.7

Simplify $\frac{((0) + 8) ((0) - 5)}{((0) + 2) ((0) + 1) ((0) - 2)}$ .

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Step 3.2.7.1

Reduce the expression by cancelling the common factors.

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Step 3.2.7.1.1

Cancel the common factor of $(0) + 8$ and $(0) + 2$ .

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Step 3.2.7.1.1.1

Factor $2$ out of $((0) + 8) ((0) - 5)$ .

$y = \frac{2 ((0 + 4) ((0) - 5))}{((0) + 2) ((0) + 1) ((0) - 2)}$

Step 3.2.7.1.1.2

Cancel the common factors.

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Step 3.2.7.1.1.2.1

Factor $2$ out of $((0) + 2) ((0) + 1) ((0) - 2)$ .

$y = \frac{2 ((0 + 4) ((0) - 5))}{2 (((0 + 1) ((0) + 1)) ((0) - 2))}$

Step 3.2.7.1.1.2.2

Cancel the common factor.

$y = \frac{2 ((0 + 4) ((0) - 5))}{2 (((0 + 1) ((0) + 1)) ((0) - 2))}$

Step 3.2.7.1.1.2.3

Rewrite the expression.

$y = \frac{(0 + 4) ((0) - 5)}{((0 + 1) ((0) + 1)) ((0) - 2)}$

Step 3.2.7.1.2

Cancel the common factor of $0 + 4$ and $(0) - 2$ .

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Step 3.2.7.1.2.1

Factor $2$ out of $(0 + 4) ((0) - 5)$ .

$y = \frac{2 ((0 + 2) ((0) - 5))}{((0 + 1) ((0) + 1)) ((0) - 2)}$

Step 3.2.7.1.2.2

Cancel the common factors.

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Step 3.2.7.1.2.2.1

Factor $2$ out of $((0 + 1) ((0) + 1)) ((0) - 2)$ .

$y = \frac{2 ((0 + 2) ((0) - 5))}{2 ((0 + 1) ((0) + 1) (0 - 1))}$

Step 3.2.7.1.2.2.2

Cancel the common factor.

$y = \frac{2 ((0 + 2) ((0) - 5))}{2 ((0 + 1) ((0) + 1) (0 - 1))}$

Step 3.2.7.1.2.2.3

Rewrite the expression.

$y = \frac{(0 + 2) ((0) - 5)}{(0 + 1) ((0) + 1) (0 - 1)}$

Step 3.2.7.2

Simplify the numerator.

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Step 3.2.7.2.1

Add $0$ and $2$ .

$y = \frac{2 (0 - 5)}{(0 + 1) (0 + 1) (0 - 1)}$

Step 3.2.7.2.2

Subtract $5$ from $0$ .

$y = \frac{2 \cdot - 5}{(0 + 1) (0 + 1) (0 - 1)}$

Step 3.2.7.3

Simplify the denominator.

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Step 3.2.7.3.1

Raise $0 + 1$ to the power of $1$ .

$y = \frac{2 \cdot - 5}{{(0 + 1)}^{1} (0 + 1) (0 - 1)}$

Step 3.2.7.3.2

Raise $0 + 1$ to the power of $1$ .

$y = \frac{2 \cdot - 5}{{(0 + 1)}^{1} {(0 + 1)}^{1} (0 - 1)}$

Step 3.2.7.3.3

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

$y = \frac{2 \cdot - 5}{{(0 + 1)}^{1 + 1} (0 - 1)}$

Step 3.2.7.3.4

Add $1$ and $1$ .

$y = \frac{2 \cdot - 5}{{(0 + 1)}^{2} (0 - 1)}$

Step 3.2.7.3.5

Rewrite $0$ as $- 1 (0)$ .

$y = \frac{2 \cdot - 5}{{(- 1 (0) + 1)}^{2} (0 - 1)}$

Step 3.2.7.3.6

Rewrite $1$ as $- 1 (- 1)$ .

$y = \frac{2 \cdot - 5}{{(- 1 \cdot 0 - 1 \cdot - 1)}^{2} (0 - 1)}$

Step 3.2.7.3.7

Factor $- 1$ out of $- 1 \cdot 0 - 1 \cdot - 1$ .

$y = \frac{2 \cdot - 5}{{(- 1 \cdot (0 - 1))}^{2} (0 - 1)}$

Step 3.2.7.3.8

Apply the product rule to $- 1 \cdot (0 - 1)$ .

$y = \frac{2 \cdot - 5}{{(- 1)}^{2} \cdot {(0 - 1)}^{2} (0 - 1)}$

Step 3.2.7.3.9

Raise $- 1$ to the power of $2$ .

$y = \frac{2 \cdot - 5}{1 \cdot {(0 - 1)}^{2} (0 - 1)}$

Step 3.2.7.3.10

Multiply ${(0 - 1)}^{2}$ by $1$ .

$y = \frac{2 \cdot - 5}{{(0 - 1)}^{2} (0 - 1)}$

Step 3.2.7.3.11

Multiply ${(0 - 1)}^{2}$ by $0 - 1$ by adding the exponents.

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Step 3.2.7.3.11.1

Multiply ${(0 - 1)}^{2}$ by $0 - 1$ .

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Step 3.2.7.3.11.1.1

Raise $0 - 1$ to the power of $1$ .

$y = \frac{2 \cdot - 5}{{(0 - 1)}^{2} {(0 - 1)}^{1}}$

Step 3.2.7.3.11.1.2

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

$y = \frac{2 \cdot - 5}{{(0 - 1)}^{2 + 1}}$

Step 3.2.7.3.11.2

Add $2$ and $1$ .

$y = \frac{2 \cdot - 5}{{(0 - 1)}^{3}}$

Step 3.2.7.4

Multiply $2$ by $- 5$ .

$y = \frac{- 10}{{(0 - 1)}^{3}}$

Step 3.2.7.5

Simplify the denominator.

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Step 3.2.7.5.1

Subtract $1$ from $0$ .

$y = \frac{- 10}{{(- 1)}^{3}}$

Step 3.2.7.5.2

Raise $- 1$ to the power of $3$ .

$y = \frac{- 10}{- 1}$

Step 3.2.7.6

Divide $- 10$ by $- 1$ .

$y = 10$

Step 3.3

y-intercept(s) in point form.

y-intercept(s): $(0, 10)$

Step 4

List the intersections.

x-intercept(s): $(- 8, 0), (5, 0)$

y-intercept(s): $(0, 10)$

Step 5