Precalculus Examples

$f (x) = 11 x^{4} + 10 x^{3} - 144 x^{2} - 130 x + 13$

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 $x$ .

$0 = 11 x^{4} + 10 x^{3} - 144 x^{2} - 130 x + 13$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $11 x^{4} + 10 x^{3} - 144 x^{2} - 130 x + 13 = 0$ .

$11 x^{4} + 10 x^{3} - 144 x^{2} - 130 x + 13 = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Regroup terms.

$10 x^{3} - 130 x + 11 x^{4} - 144 x^{2} + 13 = 0$

Step 1.2.2.2

Factor $10 x$ out of $10 x^{3} - 130 x$ .

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Step 1.2.2.2.1

Factor $10 x$ out of $10 x^{3}$ .

$10 x (x^{2}) - 130 x + 11 x^{4} - 144 x^{2} + 13 = 0$

Step 1.2.2.2.2

Factor $10 x$ out of $- 130 x$ .

$10 x (x^{2}) + 10 x (- 13) + 11 x^{4} - 144 x^{2} + 13 = 0$

Step 1.2.2.2.3

Factor $10 x$ out of $10 x (x^{2}) + 10 x (- 13)$ .

$10 x (x^{2} - 13) + 11 x^{4} - 144 x^{2} + 13 = 0$

Step 1.2.2.3

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

$10 x (x^{2} - 13) + 11 {(x^{2})}^{2} - 144 x^{2} + 13 = 0$

Step 1.2.2.4

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

$10 x (x^{2} - 13) + 11 u^{2} - 144 u + 13 = 0$

Step 1.2.2.5

Factor by grouping.

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Step 1.2.2.5.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 11 \cdot 13 = 143$ and whose sum is $b = - 144$ .

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Step 1.2.2.5.1.1

Factor $- 144$ out of $- 144 u$ .

$10 x (x^{2} - 13) + 11 u^{2} - 144 u + 13 = 0$

Step 1.2.2.5.1.2

Rewrite $- 144$ as $- 1$ plus $- 143$

$10 x (x^{2} - 13) + 11 u^{2} + (- 1 - 143) u + 13 = 0$

Step 1.2.2.5.1.3

Apply the distributive property.

$10 x (x^{2} - 13) + 11 u^{2} - 1 u - 143 u + 13 = 0$

Step 1.2.2.5.2

Factor out the greatest common factor from each group.

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Step 1.2.2.5.2.1

Group the first two terms and the last two terms.

$10 x (x^{2} - 13) + 11 u^{2} - 1 u - 143 u + 13 = 0$

Step 1.2.2.5.2.2

Factor out the greatest common factor (GCF) from each group.

$10 x (x^{2} - 13) + u (11 u - 1) - 13 (11 u - 1) = 0$

Step 1.2.2.5.3

Factor the polynomial by factoring out the greatest common factor, $11 u - 1$ .

$10 x (x^{2} - 13) + (11 u - 1) (u - 13) = 0$

Step 1.2.2.6

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

$10 x (x^{2} - 13) + (11 x^{2} - 1) (x^{2} - 13) = 0$

Step 1.2.2.7

Factor $x^{2} - 13$ out of $10 x (x^{2} - 13) + (11 x^{2} - 1) (x^{2} - 13)$ .

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Step 1.2.2.7.1

Factor $x^{2} - 13$ out of $10 x (x^{2} - 13)$ .

$(x^{2} - 13) (10 x) + (11 x^{2} - 1) (x^{2} - 13) = 0$

Step 1.2.2.7.2

Factor $x^{2} - 13$ out of $(11 x^{2} - 1) (x^{2} - 13)$ .

$(x^{2} - 13) (10 x) + (x^{2} - 13) (11 x^{2} - 1) = 0$

Step 1.2.2.7.3

Factor $x^{2} - 13$ out of $(x^{2} - 13) (10 x) + (x^{2} - 13) (11 x^{2} - 1)$ .

$(x^{2} - 13) (10 x + 11 x^{2} - 1) = 0$

Step 1.2.2.8

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

$(x^{2} - 13) (10 u + 11 u^{2} - 1) = 0$

Step 1.2.2.9

Factor by grouping.

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Step 1.2.2.9.1

Reorder terms.

$(x^{2} - 13) (11 u^{2} + 10 u - 1) = 0$

Step 1.2.2.9.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 11 \cdot - 1 = - 11$ and whose sum is $b = 10$ .

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Step 1.2.2.9.2.1

Factor $10$ out of $10 u$ .

$(x^{2} - 13) (11 u^{2} + 10 (u) - 1) = 0$

Step 1.2.2.9.2.2

Rewrite $10$ as $- 1$ plus $11$

$(x^{2} - 13) (11 u^{2} + (- 1 + 11) u - 1) = 0$

Step 1.2.2.9.2.3

Apply the distributive property.

$(x^{2} - 13) (11 u^{2} - 1 u + 11 u - 1) = 0$

Step 1.2.2.9.3

Factor out the greatest common factor from each group.

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Step 1.2.2.9.3.1

Group the first two terms and the last two terms.

$(x^{2} - 13) ((11 u^{2} - 1 u) + 11 u - 1) = 0$

Step 1.2.2.9.3.2

Factor out the greatest common factor (GCF) from each group.

$(x^{2} - 13) (u (11 u - 1) + 1 (11 u - 1)) = 0$

Step 1.2.2.9.4

Factor the polynomial by factoring out the greatest common factor, $11 u - 1$ .

$(x^{2} - 13) ((11 u - 1) (u + 1)) = 0$

Step 1.2.2.10

Factor.

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Step 1.2.2.10.1

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

$(x^{2} - 13) ((11 x - 1) (x + 1)) = 0$

Step 1.2.2.10.2

Remove unnecessary parentheses.

$(x^{2} - 13) (11 x - 1) (x + 1) = 0$

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 $0$ .

$x^{2} - 13 = 0$

$11 x - 1 = 0$

$x + 1 = 0$

Step 1.2.4

Set $x^{2} - 13$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $x^{2} - 13$ equal to $0$ .

$x^{2} - 13 = 0$

Step 1.2.4.2

Solve $x^{2} - 13 = 0$ for $x$ .

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Step 1.2.4.2.1

Add $13$ to both sides of the equation.

$x^{2} = 13$

Step 1.2.4.2.2

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

$x = \pm \sqrt{13}$

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.

$x = \sqrt{13}$

Step 1.2.4.2.3.2

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

$x = - \sqrt{13}$

Step 1.2.4.2.3.3

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

$x = \sqrt{13}, - \sqrt{13}$

Step 1.2.5

Set $11 x - 1$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $11 x - 1$ equal to $0$ .

$11 x - 1 = 0$

Step 1.2.5.2

Solve $11 x - 1 = 0$ for $x$ .

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Step 1.2.5.2.1

Add $1$ to both sides of the equation.

$11 x = 1$

Step 1.2.5.2.2

Divide each term in $11 x = 1$ by $11$ and simplify.

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Step 1.2.5.2.2.1

Divide each term in $11 x = 1$ by $11$ .

$\frac{11 x}{11} = \frac{1}{11}$

Step 1.2.5.2.2.2

Simplify the left side.

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Step 1.2.5.2.2.2.1

Cancel the common factor of $11$ .

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Step 1.2.5.2.2.2.1.1

Cancel the common factor.

$\frac{11 x}{11} = \frac{1}{11}$

Step 1.2.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{11}$

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 $(x^{2} - 13) (11 x - 1) (x + 1) = 0$ true.

$x = \sqrt{13}, - \sqrt{13}, \frac{1}{11}, - 1$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\sqrt{13}, 0), (- \sqrt{13}, 0), (\frac{1}{11}, 0), (- 1, 0)$

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 $y$ .

$y = 11 {(0)}^{4} + 10 {(0)}^{3} - 144 {(0)}^{2} - 130 \cdot 0 + 13$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 11 \cdot 0^{4} + 10 {(0)}^{3} - 144 {(0)}^{2} - 130 \cdot 0 + 13$

Step 2.2.2

Remove parentheses.

$y = 11 \cdot 0^{4} + 10 \cdot 0^{3} - 144 {(0)}^{2} - 130 \cdot 0 + 13$

Step 2.2.3

Remove parentheses.

$y = 11 \cdot 0^{4} + 10 \cdot 0^{3} - 144 \cdot 0^{2} - 130 \cdot 0 + 13$

Step 2.2.4

Remove parentheses.

$y = 11 {(0)}^{4} + 10 {(0)}^{3} - 144 {(0)}^{2} - 130 \cdot 0 + 13$

Step 2.2.5

Simplify $11 {(0)}^{4} + 10 {(0)}^{3} - 144 {(0)}^{2} - 130 \cdot 0 + 13$ .

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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 $0$ .

$y = 11 \cdot 0 + 10 {(0)}^{3} - 144 {(0)}^{2} - 130 \cdot 0 + 13$

Step 2.2.5.1.2

Multiply $11$ by $0$ .

$y = 0 + 10 {(0)}^{3} - 144 {(0)}^{2} - 130 \cdot 0 + 13$

Step 2.2.5.1.3

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

$y = 0 + 10 \cdot 0 - 144 {(0)}^{2} - 130 \cdot 0 + 13$

Step 2.2.5.1.4

Multiply $10$ by $0$ .

$y = 0 + 0 - 144 {(0)}^{2} - 130 \cdot 0 + 13$

Step 2.2.5.1.5

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

$y = 0 + 0 - 144 \cdot 0 - 130 \cdot 0 + 13$

Step 2.2.5.1.6

Multiply $- 144$ by $0$ .

$y = 0 + 0 + 0 - 130 \cdot 0 + 13$

Step 2.2.5.1.7

Multiply $- 130$ by $0$ .

$y = 0 + 0 + 0 + 0 + 13$

Step 2.2.5.2

Simplify by adding numbers.

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Step 2.2.5.2.1

Add $0$ and $0$ .

$y = 0 + 0 + 0 + 13$

Step 2.2.5.2.2

Add $0$ and $0$ .

$y = 0 + 0 + 13$

Step 2.2.5.2.3

Add $0$ and $0$ .

$y = 0 + 13$

Step 2.2.5.2.4

Add $0$ and $13$ .

$y = 13$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 13)$

Step 3

List the intersections.

x-intercept(s): $(\sqrt{13}, 0), (- \sqrt{13}, 0), (\frac{1}{11}, 0), (- 1, 0)$

y-intercept(s): $(0, 13)$

Step 4