Precalculus Examples

$9 x^{4} + 10 x^{3} + 19 x^{2} + 20 x + 2$

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.

$p = \pm 1, \pm 2$

$q = \pm 1, \pm 3, \pm 9$

Step 2

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

$\pm 1, \pm \frac{1}{3}, \pm \frac{1}{9}, \pm 2, \pm \frac{2}{3}, \pm \frac{2}{9}$

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.

$9 {(- 1)}^{4} + 10 {(- 1)}^{3} + 19 {(- 1)}^{2} + 20 (- 1) + 2$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 1$ 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 $- 1$ to the power of $4$ .

$9 \cdot 1 + 10 {(- 1)}^{3} + 19 {(- 1)}^{2} + 20 (- 1) + 2$

Step 4.1.2

Multiply $9$ by $1$ .

$9 + 10 {(- 1)}^{3} + 19 {(- 1)}^{2} + 20 (- 1) + 2$

Step 4.1.3

Raise $- 1$ to the power of $3$ .

$9 + 10 \cdot - 1 + 19 {(- 1)}^{2} + 20 (- 1) + 2$

Step 4.1.4

Multiply $10$ by $- 1$ .

$9 - 10 + 19 {(- 1)}^{2} + 20 (- 1) + 2$

Step 4.1.5

Raise $- 1$ to the power of $2$ .

$9 - 10 + 19 \cdot 1 + 20 (- 1) + 2$

Step 4.1.6

Multiply $19$ by $1$ .

$9 - 10 + 19 + 20 (- 1) + 2$

Step 4.1.7

Multiply $20$ by $- 1$ .

$9 - 10 + 19 - 20 + 2$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Subtract $10$ from $9$ .

$- 1 + 19 - 20 + 2$

Step 4.2.2

Add $- 1$ and $19$ .

$18 - 20 + 2$

Step 4.2.3

Subtract $20$ from $18$ .

$- 2 + 2$

Step 4.2.4

Add $- 2$ and $2$ .

$0$

Step 5

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{9 x^{4} + 10 x^{3} + 19 x^{2} + 20 x + 2}{x + 1}$

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.

$- 1$	$9$	$10$	$19$	$20$	$2$

Step 6.2

The first number in the dividend $(9)$ is put into the first position of the result area (below the horizontal line).

$- 1$	$9$	$10$	$19$	$20$	$2$

	$9$

Step 6.3

Multiply the newest entry in the result $(9)$ by the divisor $(- 1)$ and place the result of $(- 9)$ under the next term in the dividend $(10)$ .

$- 1$	$9$	$10$	$19$	$20$	$2$
		$- 9$
	$9$

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.

$- 1$	$9$	$10$	$19$	$20$	$2$
		$- 9$
	$9$	$1$

Step 6.5

Multiply the newest entry in the result $(1)$ by the divisor $(- 1)$ and place the result of $(- 1)$ under the next term in the dividend $(19)$ .

$- 1$	$9$	$10$	$19$	$20$	$2$
		$- 9$	$- 1$
	$9$	$1$

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.

$- 1$	$9$	$10$	$19$	$20$	$2$
		$- 9$	$- 1$
	$9$	$1$	$18$

Step 6.7

Multiply the newest entry in the result $(18)$ by the divisor $(- 1)$ and place the result of $(- 18)$ under the next term in the dividend $(20)$ .

$- 1$	$9$	$10$	$19$	$20$	$2$
		$- 9$	$- 1$	$- 18$
	$9$	$1$	$18$

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.

$- 1$	$9$	$10$	$19$	$20$	$2$
		$- 9$	$- 1$	$- 18$
	$9$	$1$	$18$	$2$

Step 6.9

Multiply the newest entry in the result $(2)$ by the divisor $(- 1)$ and place the result of $(- 2)$ under the next term in the dividend $(2)$ .

$- 1$	$9$	$10$	$19$	$20$	$2$
		$- 9$	$- 1$	$- 18$	$- 2$
	$9$	$1$	$18$	$2$

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.

$- 1$	$9$	$10$	$19$	$20$	$2$
		$- 9$	$- 1$	$- 18$	$- 2$
	$9$	$1$	$18$	$2$	$0$

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.

$9 x^{3} + 1 x^{2} + (18) x + 2$

Step 6.12

Simplify the quotient polynomial.

$9 x^{3} + x^{2} + 18 x + 2$

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.

$(x + 1) (9 x^{3} + x^{2}) + 18 x + 2$

Step 7.2

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

$(x + 1) + x^{2} (9 x + 1) + 2 (9 x + 1)$

$(x + 1) x^{2} (9 x + 1) + 2 (9 x + 1)$

Step 8

Factor the polynomial by factoring out the greatest common factor, $9 x + 1$ .

$(x + 1) (9 x + 1) (x^{2} + 2)$

Step 9

Factor the left side of the equation.

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Step 9.1

Regroup terms.

$10 x^{3} + 20 x + 9 x^{4} + 19 x^{2} + 2 = 0$

Step 9.2

Factor $10 x$ out of $10 x^{3} + 20 x$ .

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Step 9.2.1

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

$10 x (x^{2}) + 20 x + 9 x^{4} + 19 x^{2} + 2 = 0$

Step 9.2.2

Factor $10 x$ out of $20 x$ .

$10 x (x^{2}) + 10 x (2) + 9 x^{4} + 19 x^{2} + 2 = 0$

Step 9.2.3

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

$10 x (x^{2} + 2) + 9 x^{4} + 19 x^{2} + 2 = 0$

Step 9.3

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

$10 x (x^{2} + 2) + 9 {(x^{2})}^{2} + 19 x^{2} + 2 = 0$

Step 9.4

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

$10 x (x^{2} + 2) + 9 u^{2} + 19 u + 2 = 0$

Step 9.5

Factor by grouping.

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Step 9.5.1

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 $a \cdot c = 9 \cdot 2 = 18$ and whose sum is $b = 19$ .

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Step 9.5.1.1

Factor $19$ out of $19 u$ .

$10 x (x^{2} + 2) + 9 u^{2} + 19 (u) + 2 = 0$

Step 9.5.1.2

Rewrite $19$ as $1$ plus $18$

$10 x (x^{2} + 2) + 9 u^{2} + (1 + 18) u + 2 = 0$

Step 9.5.1.3

Apply the distributive property.

$10 x (x^{2} + 2) + 9 u^{2} + 1 u + 18 u + 2 = 0$

Step 9.5.1.4

Multiply $u$ by $1$ .

$10 x (x^{2} + 2) + 9 u^{2} + u + 18 u + 2 = 0$

Step 9.5.2

Factor out the greatest common factor from each group.

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Step 9.5.2.1

Group the first two terms and the last two terms.

$10 x (x^{2} + 2) + 9 u^{2} + u + 18 u + 2 = 0$

Step 9.5.2.2

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

$10 x (x^{2} + 2) + u (9 u + 1) + 2 (9 u + 1) = 0$

Step 9.5.3

Factor the polynomial by factoring out the greatest common factor, $9 u + 1$ .

$10 x (x^{2} + 2) + (9 u + 1) (u + 2) = 0$

Step 9.6

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

$10 x (x^{2} + 2) + (9 x^{2} + 1) (x^{2} + 2) = 0$

Step 9.7

Factor $x^{2} + 2$ out of $10 x (x^{2} + 2) + (9 x^{2} + 1) (x^{2} + 2)$ .

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Step 9.7.1

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

$(x^{2} + 2) (10 x) + (9 x^{2} + 1) (x^{2} + 2) = 0$

Step 9.7.2

Factor $x^{2} + 2$ out of $(9 x^{2} + 1) (x^{2} + 2)$ .

$(x^{2} + 2) (10 x) + (x^{2} + 2) (9 x^{2} + 1) = 0$

Step 9.7.3

Factor $x^{2} + 2$ out of $(x^{2} + 2) (10 x) + (x^{2} + 2) (9 x^{2} + 1)$ .

$(x^{2} + 2) (10 x + 9 x^{2} + 1) = 0$

Step 9.8

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

$(x^{2} + 2) (10 u + 9 u^{2} + 1) = 0$

Step 9.9

Factor by grouping.

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Step 9.9.1

Reorder terms.

$(x^{2} + 2) (9 u^{2} + 10 u + 1) = 0$

Step 9.9.2

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 $a \cdot c = 9 \cdot 1 = 9$ and whose sum is $b = 10$ .

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Step 9.9.2.1

Factor $10$ out of $10 u$ .

$(x^{2} + 2) (9 u^{2} + 10 (u) + 1) = 0$

Step 9.9.2.2

Rewrite $10$ as $1$ plus $9$

$(x^{2} + 2) (9 u^{2} + (1 + 9) u + 1) = 0$

Step 9.9.2.3

Apply the distributive property.

$(x^{2} + 2) (9 u^{2} + 1 u + 9 u + 1) = 0$

Step 9.9.2.4

Multiply $u$ by $1$ .

$(x^{2} + 2) (9 u^{2} + u + 9 u + 1) = 0$

Step 9.9.3

Factor out the greatest common factor from each group.

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Step 9.9.3.1

Group the first two terms and the last two terms.

$(x^{2} + 2) ((9 u^{2} + u) + 9 u + 1) = 0$

Step 9.9.3.2

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

$(x^{2} + 2) (u (9 u + 1) + 1 (9 u + 1)) = 0$

Step 9.9.4

Factor the polynomial by factoring out the greatest common factor, $9 u + 1$ .

$(x^{2} + 2) ((9 u + 1) (u + 1)) = 0$

Step 9.10

Factor.

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Step 9.10.1

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

$(x^{2} + 2) ((9 x + 1) (x + 1)) = 0$

Step 9.10.2

Remove unnecessary parentheses.

$(x^{2} + 2) (9 x + 1) (x + 1) = 0$

Step 10

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

$x^{2} + 2 = 0$

$9 x + 1 = 0$

$x + 1 = 0$

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 $9 x + 1$ equal to $0$ and solve for $x$ .

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Step 12.1

Set $9 x + 1$ equal to $0$ .

$9 x + 1 = 0$

Step 12.2

Solve $9 x + 1 = 0$ for $x$ .

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Step 12.2.1

Subtract $1$ from both sides of the equation.

$9 x = - 1$

Step 12.2.2

Divide each term in $9 x = - 1$ by $9$ and simplify.

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Step 12.2.2.1

Divide each term in $9 x = - 1$ by $9$ .

$\frac{9 x}{9} = \frac{- 1}{9}$

Step 12.2.2.2

Simplify the left side.

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Step 12.2.2.2.1

Cancel the common factor of $9$ .

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Step 12.2.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{- 1}{9}$

Step 12.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{9}$

Step 12.2.2.3

Simplify the right side.

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Step 12.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{9}$

Step 13

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

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Step 13.1

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

$x + 1 = 0$

Step 13.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 14

The final solution is all the values that make $(x^{2} + 2) (9 x + 1) (x + 1) = 0$ true.

$x = i \sqrt{2}, - i \sqrt{2}, - \frac{1}{9}, - 1$

Step 15