Precalculus Examples

$f (x) = x^{4} - 3 x^{3} - 9 x^{2} + 15 x + 20$

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 $x$ .

$0 = x^{4} - 3 x^{3} - 9 x^{2} + 15 x + 20$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $x^{4} - 3 x^{3} - 9 x^{2} + 15 x + 20 = 0$ .

$x^{4} - 3 x^{3} - 9 x^{2} + 15 x + 20 = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Regroup terms.

$- 3 x^{3} + 15 x + x^{4} - 9 x^{2} + 20 = 0$

Step 1.2.2.2

Factor $- 3 x$ out of $- 3 x^{3} + 15 x$ .

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Step 1.2.2.2.1

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

$- 3 x \cdot x^{2} + 15 x + x^{4} - 9 x^{2} + 20 = 0$

Step 1.2.2.2.2

Factor $- 3 x$ out of $15 x$ .

$- 3 x \cdot x^{2} - 3 x \cdot - 5 + x^{4} - 9 x^{2} + 20 = 0$

Step 1.2.2.2.3

Factor $- 3 x$ out of $- 3 x (x^{2}) - 3 x (- 5)$ .

$- 3 x (x^{2} - 5) + x^{4} - 9 x^{2} + 20 = 0$

Step 1.2.2.3

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

$- 3 x (x^{2} - 5) + {(x^{2})}^{2} - 9 x^{2} + 20 = 0$

Step 1.2.2.4

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

$- 3 x (x^{2} - 5) + u^{2} - 9 u + 20 = 0$

Step 1.2.2.5

Factor $u^{2} - 9 u + 20$ 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 $20$ and whose sum is $- 9$ .

$- 5, - 4$

Step 1.2.2.5.2

Write the factored form using these integers.

$- 3 x (x^{2} - 5) + (u - 5) (u - 4) = 0$

Step 1.2.2.6

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

$- 3 x (x^{2} - 5) + (x^{2} - 5) (x^{2} - 4) = 0$

Step 1.2.2.7

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

$- 3 x (x^{2} - 5) + (x^{2} - 5) (x^{2} - 2^{2}) = 0$

Step 1.2.2.8

Factor.

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Step 1.2.2.8.1

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 $b = 2$ .

$- 3 x (x^{2} - 5) + (x^{2} - 5) ((x + 2) (x - 2)) = 0$

Step 1.2.2.8.2

Remove unnecessary parentheses.

$- 3 x (x^{2} - 5) + (x^{2} - 5) (x + 2) (x - 2) = 0$

Step 1.2.2.9

Factor $x^{2} - 5$ out of $- 3 x (x^{2} - 5) + (x^{2} - 5) (x + 2) (x - 2)$ .

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Step 1.2.2.9.1

Factor $x^{2} - 5$ out of $- 3 x (x^{2} - 5)$ .

$(x^{2} - 5) (- 3 x) + (x^{2} - 5) (x + 2) (x - 2) = 0$

Step 1.2.2.9.2

Factor $x^{2} - 5$ out of $(x^{2} - 5) (x + 2) (x - 2)$ .

$(x^{2} - 5) (- 3 x) + (x^{2} - 5) ((x + 2) (x - 2)) = 0$

Step 1.2.2.9.3

Factor $x^{2} - 5$ out of $(x^{2} - 5) (- 3 x) + (x^{2} - 5) ((x + 2) (x - 2))$ .

$(x^{2} - 5) (- 3 x + (x + 2) (x - 2)) = 0$

Step 1.2.2.10

Expand $(x + 2) (x - 2)$ using the FOIL Method.

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Step 1.2.2.10.1

Apply the distributive property.

$(x^{2} - 5) (- 3 x + x (x - 2) + 2 (x - 2)) = 0$

Step 1.2.2.10.2

Apply the distributive property.

$(x^{2} - 5) (- 3 x + x \cdot x + x \cdot - 2 + 2 (x - 2)) = 0$

Step 1.2.2.10.3

Apply the distributive property.

$(x^{2} - 5) (- 3 x + x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2) = 0$

Step 1.2.2.11

Combine the opposite terms in $x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2$ .

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Step 1.2.2.11.1

Reorder the factors in the terms $x \cdot - 2$ and $2 x$ .

$(x^{2} - 5) (- 3 x + x \cdot x - 2 x + 2 x + 2 \cdot - 2) = 0$

Step 1.2.2.11.2

Add $- 2 x$ and $2 x$ .

$(x^{2} - 5) (- 3 x + x \cdot x + 0 + 2 \cdot - 2) = 0$

Step 1.2.2.11.3

Add $x \cdot x$ and $0$ .

$(x^{2} - 5) (- 3 x + x \cdot x + 2 \cdot - 2) = 0$

Step 1.2.2.12

Simplify each term.

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Step 1.2.2.12.1

Multiply $x$ by $x$ .

$(x^{2} - 5) (- 3 x + x^{2} + 2 \cdot - 2) = 0$

Step 1.2.2.12.2

Multiply $2$ by $- 2$ .

$(x^{2} - 5) (- 3 x + x^{2} - 4) = 0$

Step 1.2.2.13

Reorder terms.

$(x^{2} - 5) (x^{2} - 3 x - 4) = 0$

Step 1.2.2.14

Factor.

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Step 1.2.2.14.1

Factor $x^{2} - 3 x - 4$ using the AC method.

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Step 1.2.2.14.1.1

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 $- 4$ and whose sum is $- 3$ .

$- 4, 1$

Step 1.2.2.14.1.2

Write the factored form using these integers.

$(x^{2} - 5) ((x - 4) (x + 1)) = 0$

Step 1.2.2.14.2

Remove unnecessary parentheses.

$(x^{2} - 5) (x - 4) (x + 1) = 0$

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 $0$ .

$x^{2} - 5 = 0$

$x - 4 = 0$

$x + 1 = 0$

Step 1.2.4

Set $x^{2} - 5$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $x^{2} - 5$ equal to $0$ .

$x^{2} - 5 = 0$

Step 1.2.4.2

Solve $x^{2} - 5 = 0$ for $x$ .

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Step 1.2.4.2.1

Add $5$ to both sides of the equation.

$x^{2} = 5$

Step 1.2.4.2.2

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

$x = \pm \sqrt{5}$

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.

$x = \sqrt{5}$

Step 1.2.4.2.3.2

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

$x = - \sqrt{5}$

Step 1.2.4.2.3.3

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

$x = \sqrt{5}, - \sqrt{5}$

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 $x + 1$ equal to $0$ and solve for $x$ .

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Step 1.2.6.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 1.2.6.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 1.2.7

The final solution is all the values that make $(x^{2} - 5) (x - 4) (x + 1) = 0$ true.

$x = \sqrt{5}, - \sqrt{5}, 4, - 1$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\sqrt{5}, 0), (- \sqrt{5}, 0), (4, 0), (- 1, 0)$

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 $y$ .

$y = {(0)}^{4} - 3 {(0)}^{3} - 9 {(0)}^{2} + 15 (0) + 20$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 0^{4} - 3 {(0)}^{3} - 9 {(0)}^{2} + 15 (0) + 20$

Step 2.2.2

Remove parentheses.

$y = 0^{4} - 3 \cdot 0^{3} - 9 {(0)}^{2} + 15 (0) + 20$

Step 2.2.3

Remove parentheses.

$y = 0^{4} - 3 \cdot 0^{3} - 9 \cdot 0^{2} + 15 (0) + 20$

Step 2.2.4

Remove parentheses.

$y = {(0)}^{4} - 3 {(0)}^{3} - 9 {(0)}^{2} + 15 (0) + 20$

Step 2.2.5

Simplify ${(0)}^{4} - 3 {(0)}^{3} - 9 {(0)}^{2} + 15 (0) + 20$ .

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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 $0$ .

$y = 0 - 3 {(0)}^{3} - 9 {(0)}^{2} + 15 (0) + 20$

Step 2.2.5.1.2

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

$y = 0 - 3 \cdot 0 - 9 {(0)}^{2} + 15 (0) + 20$

Step 2.2.5.1.3

Multiply $- 3$ by $0$ .

$y = 0 + 0 - 9 {(0)}^{2} + 15 (0) + 20$

Step 2.2.5.1.4

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

$y = 0 + 0 - 9 \cdot 0 + 15 (0) + 20$

Step 2.2.5.1.5

Multiply $- 9$ by $0$ .

$y = 0 + 0 + 0 + 15 (0) + 20$

Step 2.2.5.1.6

Multiply $15$ by $0$ .

$y = 0 + 0 + 0 + 0 + 20$

Step 2.2.5.2

Simplify by adding numbers.

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Step 2.2.5.2.1

Add $0$ and $0$ .

$y = 0 + 0 + 0 + 20$

Step 2.2.5.2.2

Add $0$ and $0$ .

$y = 0 + 0 + 20$

Step 2.2.5.2.3

Add $0$ and $0$ .

$y = 0 + 20$

Step 2.2.5.2.4

Add $0$ and $20$ .

$y = 20$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 20)$

Step 3

List the intersections.

x-intercept(s): $(\sqrt{5}, 0), (- \sqrt{5}, 0), (4, 0), (- 1, 0)$

y-intercept(s): $(0, 20)$

Step 4