Precalculus Examples

$9 x^{4} + 46 x^{3} + 68 x^{2} + 322 x + 35$

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.

$p = \pm 1, \pm 5, \pm 7, \pm 35$

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

Step 2

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

$\pm 1, \pm \frac{1}{3}, \pm \frac{1}{9}, \pm 5, \pm \frac{5}{3}, \pm \frac{5}{9}, \pm 7, \pm \frac{7}{3}, \pm \frac{7}{9}, \pm 35, \pm \frac{35}{3}, \pm \frac{35}{9}$

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.

$9 {(- 5)}^{4} + 46 {(- 5)}^{3} + 68 {(- 5)}^{2} + 322 (- 5) + 35$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 5$ 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 $- 5$ to the power of $4$ .

$9 \cdot 625 + 46 {(- 5)}^{3} + 68 {(- 5)}^{2} + 322 (- 5) + 35$

Step 4.1.2

Multiply $9$ by $625$ .

$5625 + 46 {(- 5)}^{3} + 68 {(- 5)}^{2} + 322 (- 5) + 35$

Step 4.1.3

Raise $- 5$ to the power of $3$ .

$5625 + 46 \cdot - 125 + 68 {(- 5)}^{2} + 322 (- 5) + 35$

Step 4.1.4

Multiply $46$ by $- 125$ .

$5625 - 5750 + 68 {(- 5)}^{2} + 322 (- 5) + 35$

Step 4.1.5

Raise $- 5$ to the power of $2$ .

$5625 - 5750 + 68 \cdot 25 + 322 (- 5) + 35$

Step 4.1.6

Multiply $68$ by $25$ .

$5625 - 5750 + 1700 + 322 (- 5) + 35$

Step 4.1.7

Multiply $322$ by $- 5$ .

$5625 - 5750 + 1700 - 1610 + 35$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Subtract $5750$ from $5625$ .

$- 125 + 1700 - 1610 + 35$

Step 4.2.2

Add $- 125$ and $1700$ .

$1575 - 1610 + 35$

Step 4.2.3

Subtract $1610$ from $1575$ .

$- 35 + 35$

Step 4.2.4

Add $- 35$ and $35$ .

$0$

Step 5

Since $- 5$ is a known root, divide the polynomial by $x + 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{9 x^{4} + 46 x^{3} + 68 x^{2} + 322 x + 35}{x + 5}$

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.

$- 5$	$9$	$46$	$68$	$322$	$35$

Step 6.2

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

$- 5$	$9$	$46$	$68$	$322$	$35$

	$9$

Step 6.3

Multiply the newest entry in the result $(9)$ by the divisor $(- 5)$ and place the result of $(- 45)$ under the next term in the dividend $(46)$ .

$- 5$	$9$	$46$	$68$	$322$	$35$
		$- 45$
	$9$

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.

$- 5$	$9$	$46$	$68$	$322$	$35$
		$- 45$
	$9$	$1$

Step 6.5

Multiply the newest entry in the result $(1)$ by the divisor $(- 5)$ and place the result of $(- 5)$ under the next term in the dividend $(68)$ .

$- 5$	$9$	$46$	$68$	$322$	$35$
		$- 45$	$- 5$
	$9$	$1$

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.

$- 5$	$9$	$46$	$68$	$322$	$35$
		$- 45$	$- 5$
	$9$	$1$	$63$

Step 6.7

Multiply the newest entry in the result $(63)$ by the divisor $(- 5)$ and place the result of $(- 315)$ under the next term in the dividend $(322)$ .

$- 5$	$9$	$46$	$68$	$322$	$35$
		$- 45$	$- 5$	$- 315$
	$9$	$1$	$63$

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.

$- 5$	$9$	$46$	$68$	$322$	$35$
		$- 45$	$- 5$	$- 315$
	$9$	$1$	$63$	$7$

Step 6.9

Multiply the newest entry in the result $(7)$ by the divisor $(- 5)$ and place the result of $(- 35)$ under the next term in the dividend $(35)$ .

$- 5$	$9$	$46$	$68$	$322$	$35$
		$- 45$	$- 5$	$- 315$	$- 35$
	$9$	$1$	$63$	$7$

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.

$- 5$	$9$	$46$	$68$	$322$	$35$
		$- 45$	$- 5$	$- 315$	$- 35$
	$9$	$1$	$63$	$7$	$0$

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.

$9 x^{3} + 1 x^{2} + (63) x + 7$

Step 6.12

Simplify the quotient polynomial.

$9 x^{3} + x^{2} + 63 x + 7$

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.

$(x + 5) (9 x^{3} + x^{2}) + 63 x + 7$

Step 7.2

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

$(x + 5) + x^{2} (9 x + 1) + 7 (9 x + 1)$

$(x + 5) x^{2} (9 x + 1) + 7 (9 x + 1)$

Step 8

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

$(x + 5) (9 x + 1) (x^{2} + 7)$

Step 9

Factor the left side of the equation.

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Step 9.1

Regroup terms.

$46 x^{3} + 322 x + 9 x^{4} + 68 x^{2} + 35 = 0$

Step 9.2

Factor $46 x$ out of $46 x^{3} + 322 x$ .

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Step 9.2.1

Factor $46 x$ out of $46 x^{3}$ .

$46 x (x^{2}) + 322 x + 9 x^{4} + 68 x^{2} + 35 = 0$

Step 9.2.2

Factor $46 x$ out of $322 x$ .

$46 x (x^{2}) + 46 x (7) + 9 x^{4} + 68 x^{2} + 35 = 0$

Step 9.2.3

Factor $46 x$ out of $46 x (x^{2}) + 46 x (7)$ .

$46 x (x^{2} + 7) + 9 x^{4} + 68 x^{2} + 35 = 0$

Step 9.3

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

$46 x (x^{2} + 7) + 9 {(x^{2})}^{2} + 68 x^{2} + 35 = 0$

Step 9.4

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

$46 x (x^{2} + 7) + 9 u^{2} + 68 u + 35 = 0$

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 $a \cdot c = 9 \cdot 35 = 315$ and whose sum is $b = 68$ .

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Step 9.5.1.1

Factor $68$ out of $68 u$ .

$46 x (x^{2} + 7) + 9 u^{2} + 68 (u) + 35 = 0$

Step 9.5.1.2

Rewrite $68$ as $5$ plus $63$

$46 x (x^{2} + 7) + 9 u^{2} + (5 + 63) u + 35 = 0$

Step 9.5.1.3

Apply the distributive property.

$46 x (x^{2} + 7) + 9 u^{2} + 5 u + 63 u + 35 = 0$

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.

$46 x (x^{2} + 7) + 9 u^{2} + 5 u + 63 u + 35 = 0$

Step 9.5.2.2

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

$46 x (x^{2} + 7) + u (9 u + 5) + 7 (9 u + 5) = 0$

Step 9.5.3

Factor the polynomial by factoring out the greatest common factor, $9 u + 5$ .

$46 x (x^{2} + 7) + (9 u + 5) (u + 7) = 0$

Step 9.6

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

$46 x (x^{2} + 7) + (9 x^{2} + 5) (x^{2} + 7) = 0$

Step 9.7

Factor $x^{2} + 7$ out of $46 x (x^{2} + 7) + (9 x^{2} + 5) (x^{2} + 7)$ .

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Step 9.7.1

Factor $x^{2} + 7$ out of $46 x (x^{2} + 7)$ .

$(x^{2} + 7) (46 x) + (9 x^{2} + 5) (x^{2} + 7) = 0$

Step 9.7.2

Factor $x^{2} + 7$ out of $(9 x^{2} + 5) (x^{2} + 7)$ .

$(x^{2} + 7) (46 x) + (x^{2} + 7) (9 x^{2} + 5) = 0$

Step 9.7.3

Factor $x^{2} + 7$ out of $(x^{2} + 7) (46 x) + (x^{2} + 7) (9 x^{2} + 5)$ .

$(x^{2} + 7) (46 x + 9 x^{2} + 5) = 0$

Step 9.8

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

$(x^{2} + 7) (46 u + 9 u^{2} + 5) = 0$

Step 9.9

Factor by grouping.

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Step 9.9.1

Reorder terms.

$(x^{2} + 7) (9 u^{2} + 46 u + 5) = 0$

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 $a \cdot c = 9 \cdot 5 = 45$ and whose sum is $b = 46$ .

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Step 9.9.2.1

Factor $46$ out of $46 u$ .

$(x^{2} + 7) (9 u^{2} + 46 (u) + 5) = 0$

Step 9.9.2.2

Rewrite $46$ as $1$ plus $45$

$(x^{2} + 7) (9 u^{2} + (1 + 45) u + 5) = 0$

Step 9.9.2.3

Apply the distributive property.

$(x^{2} + 7) (9 u^{2} + 1 u + 45 u + 5) = 0$

Step 9.9.2.4

Multiply $u$ by $1$ .

$(x^{2} + 7) (9 u^{2} + u + 45 u + 5) = 0$

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.

$(x^{2} + 7) ((9 u^{2} + u) + 45 u + 5) = 0$

Step 9.9.3.2

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

$(x^{2} + 7) (u (9 u + 1) + 5 (9 u + 1)) = 0$

Step 9.9.4

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

$(x^{2} + 7) ((9 u + 1) (u + 5)) = 0$

Step 9.10

Factor.

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Step 9.10.1

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

$(x^{2} + 7) ((9 x + 1) (x + 5)) = 0$

Step 9.10.2

Remove unnecessary parentheses.

$(x^{2} + 7) (9 x + 1) (x + 5) = 0$

Step 10

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

$x^{2} + 7 = 0$

$9 x + 1 = 0$

$x + 5 = 0$

Step 11

Set $x^{2} + 7$ equal to $0$ and solve for $x$ .

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Step 11.1

Set $x^{2} + 7$ equal to $0$ .

$x^{2} + 7 = 0$

Step 11.2

Solve $x^{2} + 7 = 0$ for $x$ .

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Step 11.2.1

Subtract $7$ from both sides of the equation.

$x^{2} = - 7$

Step 11.2.2

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

$x = \pm \sqrt{- 7}$

Step 11.2.3

Simplify $\pm \sqrt{- 7}$ .

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Step 11.2.3.1

Rewrite $- 7$ as $- 1 (7)$ .

$x = \pm \sqrt{- 1 (7)}$

Step 11.2.3.2

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{7}$

Step 11.2.3.3

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

$x = \pm i \sqrt{7}$

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.

$x = i \sqrt{7}$

Step 11.2.4.2

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

$x = - i \sqrt{7}$

Step 11.2.4.3

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

$x = i \sqrt{7}, - i \sqrt{7}$

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 $x + 5$ equal to $0$ and solve for $x$ .

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Step 13.1

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 13.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 14

The final solution is all the values that make $(x^{2} + 7) (9 x + 1) (x + 5) = 0$ true.

$x = i \sqrt{7}, - i \sqrt{7}, - \frac{1}{9}, - 5$

Step 15