Precalculus Examples

$x^{4} + 6 x^{3} + 11 x^{2} + 12 x + 18$

Step 1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18$

$q = \pm 1$

Step 2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18$

Step 3

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

${(- 3)}^{4} + 6 {(- 3)}^{3} + 11 {(- 3)}^{2} + 12 (- 3) + 18$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 3$ is a root of the polynomial.

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Step 4.1

Simplify each term.

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Step 4.1.1

Raise $- 3$ to the power of $4$ .

$81 + 6 {(- 3)}^{3} + 11 {(- 3)}^{2} + 12 (- 3) + 18$

Step 4.1.2

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

$81 + 6 \cdot - 27 + 11 {(- 3)}^{2} + 12 (- 3) + 18$

Step 4.1.3

Multiply $6$ by $- 27$ .

$81 - 162 + 11 {(- 3)}^{2} + 12 (- 3) + 18$

Step 4.1.4

Raise $- 3$ to the power of $2$ .

$81 - 162 + 11 \cdot 9 + 12 (- 3) + 18$

Step 4.1.5

Multiply $11$ by $9$ .

$81 - 162 + 99 + 12 (- 3) + 18$

Step 4.1.6

Multiply $12$ by $- 3$ .

$81 - 162 + 99 - 36 + 18$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Subtract $162$ from $81$ .

$- 81 + 99 - 36 + 18$

Step 4.2.2

Add $- 81$ and $99$ .

$18 - 36 + 18$

Step 4.2.3

Subtract $36$ from $18$ .

$- 18 + 18$

Step 4.2.4

Add $- 18$ and $18$ .

$0$

Step 5

Since $- 3$ is a known root, divide the polynomial by $x + 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} + 6 x^{3} + 11 x^{2} + 12 x + 18}{x + 3}$

Step 6

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

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Step 6.1

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- 3$	$1$	$6$	$11$	$12$	$18$

Step 6.2

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$- 3$	$1$	$6$	$11$	$12$	$18$

	$1$

Step 6.3

Multiply the newest entry in the result $(1)$ by the divisor $(- 3)$ and place the result of $(- 3)$ under the next term in the dividend $(6)$ .

$- 3$	$1$	$6$	$11$	$12$	$18$
		$- 3$
	$1$

Step 6.4

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 3$	$1$	$6$	$11$	$12$	$18$
		$- 3$
	$1$	$3$

Step 6.5

Multiply the newest entry in the result $(3)$ by the divisor $(- 3)$ and place the result of $(- 9)$ under the next term in the dividend $(11)$ .

$- 3$	$1$	$6$	$11$	$12$	$18$
		$- 3$	$- 9$
	$1$	$3$

Step 6.6

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 3$	$1$	$6$	$11$	$12$	$18$
		$- 3$	$- 9$
	$1$	$3$	$2$

Step 6.7

Multiply the newest entry in the result $(2)$ by the divisor $(- 3)$ and place the result of $(- 6)$ under the next term in the dividend $(12)$ .

$- 3$	$1$	$6$	$11$	$12$	$18$
		$- 3$	$- 9$	$- 6$
	$1$	$3$	$2$

Step 6.8

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 3$	$1$	$6$	$11$	$12$	$18$
		$- 3$	$- 9$	$- 6$
	$1$	$3$	$2$	$6$

Step 6.9

Multiply the newest entry in the result $(6)$ by the divisor $(- 3)$ and place the result of $(- 18)$ under the next term in the dividend $(18)$ .

$- 3$	$1$	$6$	$11$	$12$	$18$
		$- 3$	$- 9$	$- 6$	$- 18$
	$1$	$3$	$2$	$6$

Step 6.10

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 3$	$1$	$6$	$11$	$12$	$18$
		$- 3$	$- 9$	$- 6$	$- 18$
	$1$	$3$	$2$	$6$	$0$

Step 6.11

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + 3 x^{2} + (2) x + 6$

Step 6.12

Simplify the quotient polynomial.

$x^{3} + 3 x^{2} + 2 x + 6$

Step 7

Factor out the greatest common factor from each group.

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Step 7.1

Group the first two terms and the last two terms.

$(x + 3) (x^{3} + 3 x^{2}) + 2 x + 6$

Step 7.2

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

$(x + 3) + x^{2} (x + 3) + 2 (x + 3)$

$(x + 3) x^{2} (x + 3) + 2 (x + 3)$

Step 8

Factor the polynomial by factoring out the greatest common factor, $x + 3$ .

$(x + 3) (x + 3) (x^{2} + 2)$

Step 9

Factor the left side of the equation.

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Step 9.1

Regroup terms.

$6 x^{3} + 12 x + x^{4} + 11 x^{2} + 18 = 0$

Step 9.2

Factor $6 x$ out of $6 x^{3} + 12 x$ .

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Step 9.2.1

Factor $6 x$ out of $6 x^{3}$ .

$6 x (x^{2}) + 12 x + x^{4} + 11 x^{2} + 18 = 0$

Step 9.2.2

Factor $6 x$ out of $12 x$ .

$6 x (x^{2}) + 6 x (2) + x^{4} + 11 x^{2} + 18 = 0$

Step 9.2.3

Factor $6 x$ out of $6 x (x^{2}) + 6 x (2)$ .

$6 x (x^{2} + 2) + x^{4} + 11 x^{2} + 18 = 0$

Step 9.3

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

$6 x (x^{2} + 2) + {(x^{2})}^{2} + 11 x^{2} + 18 = 0$

Step 9.4

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

$6 x (x^{2} + 2) + u^{2} + 11 u + 18 = 0$

Step 9.5

Factor $u^{2} + 11 u + 18$ using the AC method.

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Step 9.5.1

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 $18$ and whose sum is $11$ .

$2, 9$

Step 9.5.2

Write the factored form using these integers.

$6 x (x^{2} + 2) + (u + 2) (u + 9) = 0$

Step 9.6

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

$6 x (x^{2} + 2) + (x^{2} + 2) (x^{2} + 9) = 0$

Step 9.7

Factor $x^{2} + 2$ out of $6 x (x^{2} + 2) + (x^{2} + 2) (x^{2} + 9)$ .

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Step 9.7.1

Factor $x^{2} + 2$ out of $6 x (x^{2} + 2)$ .

$(x^{2} + 2) (6 x) + (x^{2} + 2) (x^{2} + 9) = 0$

Step 9.7.2

Factor $x^{2} + 2$ out of $(x^{2} + 2) (6 x) + (x^{2} + 2) (x^{2} + 9)$ .

$(x^{2} + 2) (6 x + x^{2} + 9) = 0$

Step 9.8

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

$(x^{2} + 2) (6 u + u^{2} + 9) = 0$

Step 9.9

Factor using the perfect square rule.

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Step 9.9.1

Rearrange terms.

$(x^{2} + 2) (u^{2} + 6 u + 9) = 0$

Step 9.9.2

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

$(x^{2} + 2) (u^{2} + 6 u + 3^{2}) = 0$

Step 9.9.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 u = 2 \cdot u \cdot 3$

Step 9.9.4

Rewrite the polynomial.

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

Step 9.9.5

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = u$ and $b = 3$ .

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

Step 9.10

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

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

Step 10

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

$x^{2} + 2 = 0$

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

Step 11

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

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Step 11.1

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

$x^{2} + 2 = 0$

Step 11.2

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

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Step 11.2.1

Subtract $2$ from both sides of the equation.

$x^{2} = - 2$

Step 11.2.2

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

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

Step 11.2.3

Simplify $\pm \sqrt{- 2}$ .

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Step 11.2.3.1

Rewrite $- 2$ as $- 1 (2)$ .

$x = \pm \sqrt{- 1 (2)}$

Step 11.2.3.2

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{2}$

Step 11.2.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \sqrt{2}$

Step 11.2.4

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

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Step 11.2.4.1

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

$x = i \sqrt{2}$

Step 11.2.4.2

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

$x = - i \sqrt{2}$

Step 11.2.4.3

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

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

Step 12

Set ${(x + 3)}^{2}$ equal to $0$ and solve for $x$ .

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Step 12.1

Set ${(x + 3)}^{2}$ equal to $0$ .

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

Step 12.2

Solve ${(x + 3)}^{2} = 0$ for $x$ .

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Step 12.2.1

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

$x + 3 = 0$

Step 12.2.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 13

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

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

Step 14