Precalculus Examples

$x^{5} - 13 x^{3} + 36 x$

Step 1

Write $x^{5} - 13 x^{3} + 36 x$ as an equation.

$y = x^{5} - 13 x^{3} + 36 x$

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 $x$ .

$0 = x^{5} - 13 x^{3} + 36 x$

Step 2.2

Solve the equation.

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Step 2.2.1

Rewrite the equation as $x^{5} - 13 x^{3} + 36 x = 0$ .

$x^{5} - 13 x^{3} + 36 x = 0$

Step 2.2.2

Factor the left side of the equation.

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Step 2.2.2.1

Factor $x$ out of $x^{5} - 13 x^{3} + 36 x$ .

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Step 2.2.2.1.1

Factor $x$ out of $x^{5}$ .

$x \cdot x^{4} - 13 x^{3} + 36 x = 0$

Step 2.2.2.1.2

Factor $x$ out of $- 13 x^{3}$ .

$x \cdot x^{4} + x (- 13 x^{2}) + 36 x = 0$

Step 2.2.2.1.3

Factor $x$ out of $36 x$ .

$x \cdot x^{4} + x (- 13 x^{2}) + x \cdot 36 = 0$

Step 2.2.2.1.4

Factor $x$ out of $x \cdot x^{4} + x (- 13 x^{2})$ .

$x (x^{4} - 13 x^{2}) + x \cdot 36 = 0$

Step 2.2.2.1.5

Factor $x$ out of $x (x^{4} - 13 x^{2}) + x \cdot 36$ .

$x (x^{4} - 13 x^{2} + 36) = 0$

Step 2.2.2.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$x ({(x^{2})}^{2} - 13 x^{2} + 36) = 0$

Step 2.2.2.3

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$x (u^{2} - 13 u + 36) = 0$

Step 2.2.2.4

Factor $u^{2} - 13 u + 36$ using the AC method.

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Step 2.2.2.4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $36$ and whose sum is $- 13$ .

$- 9, - 4$

Step 2.2.2.4.2

Write the factored form using these integers.

$x ((u - 9) (u - 4)) = 0$

Step 2.2.2.5

Replace all occurrences of $u$ with $x^{2}$ .

$x ((x^{2} - 9) (x^{2} - 4)) = 0$

Step 2.2.2.6

Rewrite $9$ as $3^{2}$ .

$x ((x^{2} - 3^{2}) (x^{2} - 4)) = 0$

Step 2.2.2.7

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$x ((x + 3) (x - 3) (x^{2} - 4)) = 0$

Step 2.2.2.8

Rewrite $4$ as $2^{2}$ .

$x ((x + 3) (x - 3) (x^{2} - 2^{2})) = 0$

Step 2.2.2.9

Factor.

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Step 2.2.2.9.1

Factor.

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Step 2.2.2.9.1.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$x ((x + 3) (x - 3) ((x + 2) (x - 2))) = 0$

Step 2.2.2.9.1.2

Remove unnecessary parentheses.

$x ((x + 3) (x - 3) (x + 2) (x - 2)) = 0$

Step 2.2.2.9.2

Remove unnecessary parentheses.

$x (x + 3) (x - 3) (x + 2) (x - 2) = 0$

Step 2.2.3

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

$x = 0$

$x + 3 = 0$

$x - 3 = 0$

$x + 2 = 0$

$x - 2 = 0$

Step 2.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.2.5

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 2.2.5.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 2.2.5.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 2.2.6

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 2.2.6.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 2.2.6.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.2.7

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 2.2.7.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 2.2.7.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 2.2.8

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 2.2.8.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 2.2.8.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.2.9

The final solution is all the values that make $x (x + 3) (x - 3) (x + 2) (x - 2) = 0$ true.

$x = 0, - 3, 3, - 2, 2$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- 3, 0), (3, 0), (- 2, 0), (2, 0)$

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 $y$ .

$y = {(0)}^{5} - 13 {(0)}^{3} + 36 (0)$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = 0^{5} - 13 {(0)}^{3} + 36 (0)$

Step 3.2.2

Remove parentheses.

$y = 0^{5} - 13 \cdot 0^{3} + 36 (0)$

Step 3.2.3

Remove parentheses.

$y = {(0)}^{5} - 13 {(0)}^{3} + 36 (0)$

Step 3.2.4

Simplify ${(0)}^{5} - 13 {(0)}^{3} + 36 (0)$ .

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Step 3.2.4.1

Simplify each term.

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Step 3.2.4.1.1

Raising $0$ to any positive power yields $0$ .

$y = 0 - 13 {(0)}^{3} + 36 (0)$

Step 3.2.4.1.2

Raising $0$ to any positive power yields $0$ .

$y = 0 - 13 \cdot 0 + 36 (0)$

Step 3.2.4.1.3

Multiply $- 13$ by $0$ .

$y = 0 + 0 + 36 (0)$

Step 3.2.4.1.4

Multiply $36$ by $0$ .

$y = 0 + 0 + 0$

Step 3.2.4.2

Simplify by adding numbers.

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Step 3.2.4.2.1

Add $0$ and $0$ .

$y = 0 + 0$

Step 3.2.4.2.2

Add $0$ and $0$ .

$y = 0$

Step 3.3

y-intercept(s) in point form.

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

Step 4

List the intersections.

x-intercept(s): $(0, 0), (- 3, 0), (3, 0), (- 2, 0), (2, 0)$

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

Step 5