Precalculus Examples

$f (x) = x^{4} - x^{3} - 14 x^{2} + 2 x + 24$

Step 1

Find the x-intercepts.

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Step 1.1

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

$0 = x^{4} - x^{3} - 14 x^{2} + 2 x + 24$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $x^{4} - x^{3} - 14 x^{2} + 2 x + 24 = 0$ .

$x^{4} - x^{3} - 14 x^{2} + 2 x + 24 = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Regroup terms.

$- x^{3} + 2 x + x^{4} - 14 x^{2} + 24 = 0$

Step 1.2.2.2

Factor $- x$ out of $- x^{3} + 2 x$ .

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Step 1.2.2.2.1

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

$- x \cdot x^{2} + 2 x + x^{4} - 14 x^{2} + 24 = 0$

Step 1.2.2.2.2

Factor $- x$ out of $2 x$ .

$- x \cdot x^{2} - x \cdot - 2 + x^{4} - 14 x^{2} + 24 = 0$

Step 1.2.2.2.3

Factor $- x$ out of $- x (x^{2}) - x (- 2)$ .

$- x (x^{2} - 2) + x^{4} - 14 x^{2} + 24 = 0$

Step 1.2.2.3

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

$- x (x^{2} - 2) + {(x^{2})}^{2} - 14 x^{2} + 24 = 0$

Step 1.2.2.4

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

$- x (x^{2} - 2) + u^{2} - 14 u + 24 = 0$

Step 1.2.2.5

Factor $u^{2} - 14 u + 24$ using the AC method.

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Step 1.2.2.5.1

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 $24$ and whose sum is $- 14$ .

$- 12, - 2$

Step 1.2.2.5.2

Write the factored form using these integers.

$- x (x^{2} - 2) + (u - 12) (u - 2) = 0$

Step 1.2.2.6

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

$- x (x^{2} - 2) + (x^{2} - 12) (x^{2} - 2) = 0$

Step 1.2.2.7

Factor $x^{2} - 2$ out of $- x (x^{2} - 2) + (x^{2} - 12) (x^{2} - 2)$ .

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Step 1.2.2.7.1

Factor $x^{2} - 2$ out of $- x (x^{2} - 2)$ .

$(x^{2} - 2) (- x) + (x^{2} - 12) (x^{2} - 2) = 0$

Step 1.2.2.7.2

Factor $x^{2} - 2$ out of $(x^{2} - 12) (x^{2} - 2)$ .

$(x^{2} - 2) (- x) + (x^{2} - 2) (x^{2} - 12) = 0$

Step 1.2.2.7.3

Factor $x^{2} - 2$ out of $(x^{2} - 2) (- x) + (x^{2} - 2) (x^{2} - 12)$ .

$(x^{2} - 2) (- x + x^{2} - 12) = 0$

Step 1.2.2.8

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$(x^{2} - 2) (- u + u^{2} - 12) = 0$

Step 1.2.2.9

Factor $- u + u^{2} - 12$ using the AC method.

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Step 1.2.2.9.1

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 $- 12$ and whose sum is $- 1$ .

$- 4, 3$

Step 1.2.2.9.2

Write the factored form using these integers.

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

Step 1.2.2.10

Factor.

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Step 1.2.2.10.1

Replace all occurrences of $u$ with $x$ .

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

Step 1.2.2.10.2

Remove unnecessary parentheses.

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

Step 1.2.3

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

$x^{2} - 2 = 0$

$x - 4 = 0$

$x + 3 = 0$

Step 1.2.4

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

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Step 1.2.4.1

Set $x^{2} - 2$ equal to $0$ .

$x^{2} - 2 = 0$

Step 1.2.4.2

Solve $x^{2} - 2 = 0$ for $x$ .

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Step 1.2.4.2.1

Add $2$ to both sides of the equation.

$x^{2} = 2$

Step 1.2.4.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{2}$

Step 1.2.4.2.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.4.2.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{2}$

Step 1.2.4.2.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{2}$

Step 1.2.4.2.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{2}, - \sqrt{2}$

Step 1.2.5

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 1.2.5.2

Add $4$ to both sides of the equation.

$x = 4$

Step 1.2.6

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

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Step 1.2.6.1

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

$x + 3 = 0$

Step 1.2.6.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 1.2.7

The final solution is all the values that make $(x^{2} - 2) (x - 4) (x + 3) = 0$ true.

$x = \sqrt{2}, - \sqrt{2}, 4, - 3$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\sqrt{2}, 0), (- \sqrt{2}, 0), (4, 0), (- 3, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

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

$y = {(0)}^{4} - {(0)}^{3} - 14 {(0)}^{2} + 2 (0) + 24$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 0^{4} - {(0)}^{3} - 14 {(0)}^{2} + 2 (0) + 24$

Step 2.2.2

Remove parentheses.

$y = 0^{4} - 0^{3} - 14 {(0)}^{2} + 2 (0) + 24$

Step 2.2.3

Remove parentheses.

$y = 0^{4} - 0^{3} - 14 \cdot 0^{2} + 2 (0) + 24$

Step 2.2.4

Remove parentheses.

$y = {(0)}^{4} - {(0)}^{3} - 14 {(0)}^{2} + 2 (0) + 24$

Step 2.2.5

Simplify ${(0)}^{4} - {(0)}^{3} - 14 {(0)}^{2} + 2 (0) + 24$ .

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Step 2.2.5.1

Simplify each term.

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Step 2.2.5.1.1

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

$y = 0 - {(0)}^{3} - 14 {(0)}^{2} + 2 (0) + 24$

Step 2.2.5.1.2

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

$y = 0 - 0 - 14 {(0)}^{2} + 2 (0) + 24$

Step 2.2.5.1.3

Multiply $- 1$ by $0$ .

$y = 0 + 0 - 14 {(0)}^{2} + 2 (0) + 24$

Step 2.2.5.1.4

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

$y = 0 + 0 - 14 \cdot 0 + 2 (0) + 24$

Step 2.2.5.1.5

Multiply $- 14$ by $0$ .

$y = 0 + 0 + 0 + 2 (0) + 24$

Step 2.2.5.1.6

Multiply $2$ by $0$ .

$y = 0 + 0 + 0 + 0 + 24$

Step 2.2.5.2

Simplify by adding numbers.

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Step 2.2.5.2.1

Add $0$ and $0$ .

$y = 0 + 0 + 0 + 24$

Step 2.2.5.2.2

Add $0$ and $0$ .

$y = 0 + 0 + 24$

Step 2.2.5.2.3

Add $0$ and $0$ .

$y = 0 + 24$

Step 2.2.5.2.4

Add $0$ and $24$ .

$y = 24$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 24)$

Step 3

List the intersections.

x-intercept(s): $(\sqrt{2}, 0), (- \sqrt{2}, 0), (4, 0), (- 3, 0)$

y-intercept(s): $(0, 24)$

Step 4