Precalculus Examples

$3 x^{4} + 8 x^{3} + 6 x^{2} + 3 x - 2$

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.

$p = \pm 1, \pm 2$

$q = \pm 1, \pm 3$

Step 2

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

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

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.

$3 {(- 2)}^{4} + 8 {(- 2)}^{3} + 6 {(- 2)}^{2} + 3 (- 2) - 2$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 2$ is a root of the polynomial.

Tap for more steps...

Step 4.1

Simplify each term.

Tap for more steps...

Step 4.1.1

Raise $- 2$ to the power of $4$ .

$3 \cdot 16 + 8 {(- 2)}^{3} + 6 {(- 2)}^{2} + 3 (- 2) - 2$

Step 4.1.2

Multiply $3$ by $16$ .

$48 + 8 {(- 2)}^{3} + 6 {(- 2)}^{2} + 3 (- 2) - 2$

Step 4.1.3

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

$48 + 8 \cdot - 8 + 6 {(- 2)}^{2} + 3 (- 2) - 2$

Step 4.1.4

Multiply $8$ by $- 8$ .

$48 - 64 + 6 {(- 2)}^{2} + 3 (- 2) - 2$

Step 4.1.5

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

$48 - 64 + 6 \cdot 4 + 3 (- 2) - 2$

Step 4.1.6

Multiply $6$ by $4$ .

$48 - 64 + 24 + 3 (- 2) - 2$

Step 4.1.7

Multiply $3$ by $- 2$ .

$48 - 64 + 24 - 6 - 2$

Step 4.2

Simplify by adding and subtracting.

Tap for more steps...

Step 4.2.1

Subtract $64$ from $48$ .

$- 16 + 24 - 6 - 2$

Step 4.2.2

Add $- 16$ and $24$ .

$8 - 6 - 2$

Step 4.2.3

Subtract $6$ from $8$ .

$2 - 2$

Step 4.2.4

Subtract $2$ from $2$ .

$0$

Step 5

Since $- 2$ is a known root, divide the polynomial by $x + 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{3 x^{4} + 8 x^{3} + 6 x^{2} + 3 x - 2}{x + 2}$

Step 6

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

Tap for more steps...

Step 6.1

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

$- 2$	$3$	$8$	$6$	$3$	$- 2$

Step 6.2

The first number in the dividend $(3)$ is put into the first position of the result area (below the horizontal line).

$- 2$	$3$	$8$	$6$	$3$	$- 2$

	$3$

Step 6.3

Multiply the newest entry in the result $(3)$ by the divisor $(- 2)$ and place the result of $(- 6)$ under the next term in the dividend $(8)$ .

$- 2$	$3$	$8$	$6$	$3$	$- 2$
		$- 6$
	$3$

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.

$- 2$	$3$	$8$	$6$	$3$	$- 2$
		$- 6$
	$3$	$2$

Step 6.5

Multiply the newest entry in the result $(2)$ by the divisor $(- 2)$ and place the result of $(- 4)$ under the next term in the dividend $(6)$ .

$- 2$	$3$	$8$	$6$	$3$	$- 2$
		$- 6$	$- 4$
	$3$	$2$

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.

$- 2$	$3$	$8$	$6$	$3$	$- 2$
		$- 6$	$- 4$
	$3$	$2$	$2$

Step 6.7

Multiply the newest entry in the result $(2)$ by the divisor $(- 2)$ and place the result of $(- 4)$ under the next term in the dividend $(3)$ .

$- 2$	$3$	$8$	$6$	$3$	$- 2$
		$- 6$	$- 4$	$- 4$
	$3$	$2$	$2$

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.

$- 2$	$3$	$8$	$6$	$3$	$- 2$
		$- 6$	$- 4$	$- 4$
	$3$	$2$	$2$	$- 1$

Step 6.9

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

$- 2$	$3$	$8$	$6$	$3$	$- 2$
		$- 6$	$- 4$	$- 4$	$2$
	$3$	$2$	$2$	$- 1$

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.

$- 2$	$3$	$8$	$6$	$3$	$- 2$
		$- 6$	$- 4$	$- 4$	$2$
	$3$	$2$	$2$	$- 1$	$0$

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.

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

Step 6.12

Simplify the quotient polynomial.

$3 x^{3} + 2 x^{2} + 2 x - 1$

Step 7

Solve the equation to find any remaining roots.

Tap for more steps...

Step 7.1

Factor $3 x^{3} + 2 x^{2} + 2 x - 1$ using the rational roots test.

Tap for more steps...

Step 7.1.1

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.

$p = \pm 1$

$q = \pm 1, \pm 3$

Step 7.1.2

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

$\pm 1, \pm 0.\overline{3}$

Step 7.1.3

Substitute $0.\overline{3}$ and simplify the expression. In this case, the expression is equal to $0$ so $0.\overline{3}$ is a root of the polynomial.

Tap for more steps...

Step 7.1.3.1

Substitute $0.\overline{3}$ into the polynomial.

$3 \cdot {0.\overline{3}}^{3} + 2 \cdot {0.\overline{3}}^{2} + 2 \cdot 0.\overline{3} - 1$

Step 7.1.3.2

Raise $0.\overline{3}$ to the power of $3$ .

$3 \cdot 0.\overline{037} + 2 \cdot {0.\overline{3}}^{2} + 2 \cdot 0.\overline{3} - 1$

Step 7.1.3.3

Multiply $3$ by $0.\overline{037}$ .

$0.\overline{1} + 2 \cdot {0.\overline{3}}^{2} + 2 \cdot 0.\overline{3} - 1$

Step 7.1.3.4

Raise $0.\overline{3}$ to the power of $2$ .

$0.\overline{1} + 2 \cdot 0.\overline{1} + 2 \cdot 0.\overline{3} - 1$

Step 7.1.3.5

Multiply $2$ by $0.\overline{1}$ .

$0.\overline{1} + 0.\overline{2} + 2 \cdot 0.\overline{3} - 1$

Step 7.1.3.6

Add $0.\overline{1}$ and $0.\overline{2}$ .

$0.\overline{3} + 2 \cdot 0.\overline{3} - 1$

Step 7.1.3.7

Multiply $2$ by $0.\overline{3}$ .

$0.\overline{3} + 0.\overline{6} - 1$

Step 7.1.3.8

Add $0.\overline{3}$ and $0.\overline{6}$ .

$1 - 1$

Step 7.1.3.9

Subtract $1$ from $1$ .

$0$

Step 7.1.4

Since $0.\overline{3}$ is a known root, divide the polynomial by $3 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{3 x^{3} + 2 x^{2} + 2 x - 1}{3 x - 1}$

Step 7.1.5

Divide $3 x^{3} + 2 x^{2} + 2 x - 1$ by $3 x - 1$ .

Tap for more steps...

Step 7.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$3 x$

$1$

$3 x^{3}$

$2 x^{2}$

$2 x$

$1$

Step 7.1.5.2

Divide the highest order term in the dividend $3 x^{3}$ by the highest order term in divisor $3 x$ .

					$x^{2}$
	$3 x$	-	$1$		$3 x^{3}$	+	$2 x^{2}$	+	$2 x$	-	$1$

Step 7.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$3 x$	-	$1$		$3 x^{3}$	+	$2 x^{2}$	+	$2 x$	-	$1$
			+	$3 x^{3}$	-	$x^{2}$

Step 7.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{3} - x^{2}$

				$x^{2}$
$3 x$	-	$1$		$3 x^{3}$	+	$2 x^{2}$	+	$2 x$	-	$1$
			-	$3 x^{3}$	+	$x^{2}$

Step 7.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$
$3 x$	-	$1$		$3 x^{3}$	+	$2 x^{2}$	+	$2 x$	-	$1$
			-	$3 x^{3}$	+	$x^{2}$
					+	$3 x^{2}$

Step 7.1.5.6

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$
$3 x$	-	$1$		$3 x^{3}$	+	$2 x^{2}$	+	$2 x$	-	$1$
			-	$3 x^{3}$	+	$x^{2}$
					+	$3 x^{2}$	+	$2 x$

Step 7.1.5.7

Divide the highest order term in the dividend $3 x^{2}$ by the highest order term in divisor $3 x$ .

				$x^{2}$	+	$x$
$3 x$	-	$1$		$3 x^{3}$	+	$2 x^{2}$	+	$2 x$	-	$1$
			-	$3 x^{3}$	+	$x^{2}$
					+	$3 x^{2}$	+	$2 x$

Step 7.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$x$
$3 x$	-	$1$		$3 x^{3}$	+	$2 x^{2}$	+	$2 x$	-	$1$
			-	$3 x^{3}$	+	$x^{2}$
					+	$3 x^{2}$	+	$2 x$
					+	$3 x^{2}$	-	$x$

Step 7.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{2} - x$

				$x^{2}$	+	$x$
$3 x$	-	$1$		$3 x^{3}$	+	$2 x^{2}$	+	$2 x$	-	$1$
			-	$3 x^{3}$	+	$x^{2}$
					+	$3 x^{2}$	+	$2 x$
					-	$3 x^{2}$	+	$x$

Step 7.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	+	$x$
$3 x$	-	$1$		$3 x^{3}$	+	$2 x^{2}$	+	$2 x$	-	$1$
			-	$3 x^{3}$	+	$x^{2}$
					+	$3 x^{2}$	+	$2 x$
					-	$3 x^{2}$	+	$x$
							+	$3 x$

Step 7.1.5.11

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$	+	$x$
$3 x$	-	$1$		$3 x^{3}$	+	$2 x^{2}$	+	$2 x$	-	$1$
			-	$3 x^{3}$	+	$x^{2}$
					+	$3 x^{2}$	+	$2 x$
					-	$3 x^{2}$	+	$x$
							+	$3 x$	-	$1$

Step 7.1.5.12

Divide the highest order term in the dividend $3 x$ by the highest order term in divisor $3 x$ .

				$x^{2}$	+	$x$	+	$1$
$3 x$	-	$1$		$3 x^{3}$	+	$2 x^{2}$	+	$2 x$	-	$1$
			-	$3 x^{3}$	+	$x^{2}$
					+	$3 x^{2}$	+	$2 x$
					-	$3 x^{2}$	+	$x$
							+	$3 x$	-	$1$

Step 7.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$x$	+	$1$
$3 x$	-	$1$		$3 x^{3}$	+	$2 x^{2}$	+	$2 x$	-	$1$
			-	$3 x^{3}$	+	$x^{2}$
					+	$3 x^{2}$	+	$2 x$
					-	$3 x^{2}$	+	$x$
							+	$3 x$	-	$1$
							+	$3 x$	-	$1$

Step 7.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $3 x - 1$

				$x^{2}$	+	$x$	+	$1$
$3 x$	-	$1$		$3 x^{3}$	+	$2 x^{2}$	+	$2 x$	-	$1$
			-	$3 x^{3}$	+	$x^{2}$
					+	$3 x^{2}$	+	$2 x$
					-	$3 x^{2}$	+	$x$
							+	$3 x$	-	$1$
							-	$3 x$	+	$1$

Step 7.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	+	$x$	+	$1$
$3 x$	-	$1$		$3 x^{3}$	+	$2 x^{2}$	+	$2 x$	-	$1$
			-	$3 x^{3}$	+	$x^{2}$
					+	$3 x^{2}$	+	$2 x$
					-	$3 x^{2}$	+	$x$
							+	$3 x$	-	$1$
							-	$3 x$	+	$1$
										$0$

Step 7.1.5.16

Since the remander is $0$ , the final answer is the quotient.

$x^{2} + x + 1$

Step 7.1.6

Write $3 x^{3} + 2 x^{2} + 2 x - 1$ as a set of factors.

$(3 x - 1) (x^{2} + x + 1) = 0$

Step 7.2

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

$3 x - 1 = 0$

$x^{2} + x + 1 = 0$

Step 7.3

Set $3 x - 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 7.3.1

Set $3 x - 1$ equal to $0$ .

$3 x - 1 = 0$

Step 7.3.2

Solve $3 x - 1 = 0$ for $x$ .

Tap for more steps...

Step 7.3.2.1

Add $1$ to both sides of the equation.

$3 x = 1$

Step 7.3.2.2

Divide each term in $3 x = 1$ by $3$ and simplify.

Tap for more steps...

Step 7.3.2.2.1

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

Step 7.3.2.2.2

Simplify the left side.

Tap for more steps...

Step 7.3.2.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 7.3.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Step 7.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{3}$

Step 7.4

Set $x^{2} + x + 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 7.4.1

Set $x^{2} + x + 1$ equal to $0$ .

$x^{2} + x + 1 = 0$

Step 7.4.2

Solve $x^{2} + x + 1 = 0$ for $x$ .

Tap for more steps...

Step 7.4.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 7.4.2.2

Substitute the values $a = 1$ , $b = 1$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 7.4.2.3

Simplify.

Tap for more steps...

Step 7.4.2.3.1

Simplify the numerator.

Tap for more steps...

Step 7.4.2.3.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 7.4.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

Tap for more steps...

Step 7.4.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 7.4.2.3.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 7.4.2.3.1.3

Subtract $4$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 7.4.2.3.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 7.4.2.3.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 7.4.2.3.1.6

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

$x = \frac{- 1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 7.4.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{3}}{2}$

Step 7.4.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Tap for more steps...

Step 7.4.2.4.1

Simplify the numerator.

Tap for more steps...

Step 7.4.2.4.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 7.4.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

Tap for more steps...

Step 7.4.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 7.4.2.4.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 7.4.2.4.1.3

Subtract $4$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 7.4.2.4.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 7.4.2.4.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 7.4.2.4.1.6

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

$x = \frac{- 1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 7.4.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{3}}{2}$

Step 7.4.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 1 + i \sqrt{3}}{2}$

Step 7.4.2.4.4

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

$x = \frac{- 1 \cdot 1 + i \sqrt{3}}{2}$

Step 7.4.2.4.5

Factor $- 1$ out of $i \sqrt{3}$ .

$x = \frac{- 1 \cdot 1 - (- i \sqrt{3})}{2}$

Step 7.4.2.4.6

Factor $- 1$ out of $- 1 (1) - (- i \sqrt{3})$ .

$x = \frac{- 1 (1 - i \sqrt{3})}{2}$

Step 7.4.2.4.7

Move the negative in front of the fraction.

$x = - \frac{1 - i \sqrt{3}}{2}$

Step 7.4.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Tap for more steps...

Step 7.4.2.5.1

Simplify the numerator.

Tap for more steps...

Step 7.4.2.5.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 7.4.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

Tap for more steps...

Step 7.4.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 7.4.2.5.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 7.4.2.5.1.3

Subtract $4$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 7.4.2.5.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 7.4.2.5.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 7.4.2.5.1.6

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

$x = \frac{- 1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 7.4.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm i \sqrt{3}}{2}$

Step 7.4.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 1 - i \sqrt{3}}{2}$

Step 7.4.2.5.4

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

$x = \frac{- 1 \cdot 1 - i \sqrt{3}}{2}$

Step 7.4.2.5.5

Factor $- 1$ out of $- i \sqrt{3}$ .

$x = \frac{- 1 \cdot 1 - (i \sqrt{3})}{2}$

Step 7.4.2.5.6

Factor $- 1$ out of $- 1 (1) - (i \sqrt{3})$ .

$x = \frac{- 1 (1 + i \sqrt{3})}{2}$

Step 7.4.2.5.7

Move the negative in front of the fraction.

$x = - \frac{1 + i \sqrt{3}}{2}$

Step 7.4.2.6

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$

Step 7.5

The final solution is all the values that make $(3 x - 1) (x^{2} + x + 1) = 0$ true.

$x = \frac{1}{3}, - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$

Step 8

The polynomial can be written as a set of linear factors.

$(x + 2) (x - \frac{1}{3}) (x + \frac{1 - i \sqrt{3}}{2}) (x + \frac{1 + i \sqrt{3}}{2})$

Step 9

These are the roots (zeros) of the polynomial $3 x^{4} + 8 x^{3} + 6 x^{2} + 3 x - 2$ .

$x = - 2, \frac{1}{3}, - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$

Step 10