Precalculus Examples

$y = 7 x^{3} - 15 x^{2} + 23 x - 3$

Step 1

Set $7 x^{3} - 15 x^{2} + 23 x - 3$ equal to $0$ .

$7 x^{3} - 15 x^{2} + 23 x - 3 = 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

Factor $7 x^{3} - 15 x^{2} + 23 x - 3$ using the rational roots test.

Tap for more steps...

Step 2.1.1

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.

$p = \pm 1, \pm 3$

$q = \pm 1, \pm 7$

Step 2.1.2

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

$\pm 1, \pm 0.\overline{142857}, \pm 3, \pm 0.\overline{428571}$

Step 2.1.3

Substitute $0.\overline{142857}$ and simplify the expression. In this case, the expression is equal to $0$ so $0.\overline{142857}$ is a root of the polynomial.

Tap for more steps...

Step 2.1.3.1

Substitute $0.\overline{142857}$ into the polynomial.

$7 \cdot {0.\overline{142857}}^{3} - 15 \cdot {0.\overline{142857}}^{2} + 23 \cdot 0.\overline{142857} - 3$

Step 2.1.3.2

Raise $0.\overline{142857}$ to the power of $3$ .

$7 \cdot 0.00291545 - 15 \cdot {0.\overline{142857}}^{2} + 23 \cdot 0.\overline{142857} - 3$

Step 2.1.3.3

Multiply $7$ by $0.00291545$ .

$0.02040816 - 15 \cdot {0.\overline{142857}}^{2} + 23 \cdot 0.\overline{142857} - 3$

Step 2.1.3.4

Raise $0.\overline{142857}$ to the power of $2$ .

$0.02040816 - 15 \cdot 0.02040816 + 23 \cdot 0.\overline{142857} - 3$

Step 2.1.3.5

Multiply $- 15$ by $0.02040816$ .

$0.02040816 - 0.30612244 + 23 \cdot 0.\overline{142857} - 3$

Step 2.1.3.6

Subtract $0.30612244$ from $0.02040816$ .

$- 0.28571428 + 23 \cdot 0.\overline{142857} - 3$

Step 2.1.3.7

Multiply $23$ by $0.\overline{142857}$ .

$- 0.28571428 + 3.28571428 - 3$

Step 2.1.3.8

Add $- 0.28571428$ and $3.28571428$ .

$3 - 3$

Step 2.1.3.9

Subtract $3$ from $3$ .

$0$

Step 2.1.4

Since $0.\overline{142857}$ is a known root, divide the polynomial by $7 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{7 x^{3} - 15 x^{2} + 23 x - 3}{7 x - 1}$

Step 2.1.5

Divide $7 x^{3} - 15 x^{2} + 23 x - 3$ by $7 x - 1$ .

Tap for more steps...

Step 2.1.5.1

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

$7 x$

$1$

$7 x^{3}$

$15 x^{2}$

$23 x$

$3$

Step 2.1.5.2

Divide the highest order term in the dividend $7 x^{3}$ by the highest order term in divisor $7 x$ .

					$x^{2}$
	$7 x$	-	$1$		$7 x^{3}$	-	$15 x^{2}$	+	$23 x$	-	$3$

Step 2.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$7 x$	-	$1$		$7 x^{3}$	-	$15 x^{2}$	+	$23 x$	-	$3$
			+	$7 x^{3}$	-	$x^{2}$

Step 2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{3} - x^{2}$

				$x^{2}$
$7 x$	-	$1$		$7 x^{3}$	-	$15 x^{2}$	+	$23 x$	-	$3$
			-	$7 x^{3}$	+	$x^{2}$

Step 2.1.5.5

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

				$x^{2}$
$7 x$	-	$1$		$7 x^{3}$	-	$15 x^{2}$	+	$23 x$	-	$3$
			-	$7 x^{3}$	+	$x^{2}$
					-	$14 x^{2}$

Step 2.1.5.6

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

				$x^{2}$
$7 x$	-	$1$		$7 x^{3}$	-	$15 x^{2}$	+	$23 x$	-	$3$
			-	$7 x^{3}$	+	$x^{2}$
					-	$14 x^{2}$	+	$23 x$

Step 2.1.5.7

Divide the highest order term in the dividend $- 14 x^{2}$ by the highest order term in divisor $7 x$ .

				$x^{2}$	-	$2 x$
$7 x$	-	$1$		$7 x^{3}$	-	$15 x^{2}$	+	$23 x$	-	$3$
			-	$7 x^{3}$	+	$x^{2}$
					-	$14 x^{2}$	+	$23 x$

Step 2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$2 x$
$7 x$	-	$1$		$7 x^{3}$	-	$15 x^{2}$	+	$23 x$	-	$3$
			-	$7 x^{3}$	+	$x^{2}$
					-	$14 x^{2}$	+	$23 x$
					-	$14 x^{2}$	+	$2 x$

Step 2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 14 x^{2} + 2 x$

				$x^{2}$	-	$2 x$
$7 x$	-	$1$		$7 x^{3}$	-	$15 x^{2}$	+	$23 x$	-	$3$
			-	$7 x^{3}$	+	$x^{2}$
					-	$14 x^{2}$	+	$23 x$
					+	$14 x^{2}$	-	$2 x$

Step 2.1.5.10

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

				$x^{2}$	-	$2 x$
$7 x$	-	$1$		$7 x^{3}$	-	$15 x^{2}$	+	$23 x$	-	$3$
			-	$7 x^{3}$	+	$x^{2}$
					-	$14 x^{2}$	+	$23 x$
					+	$14 x^{2}$	-	$2 x$
							+	$21 x$

Step 2.1.5.11

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

				$x^{2}$	-	$2 x$
$7 x$	-	$1$		$7 x^{3}$	-	$15 x^{2}$	+	$23 x$	-	$3$
			-	$7 x^{3}$	+	$x^{2}$
					-	$14 x^{2}$	+	$23 x$
					+	$14 x^{2}$	-	$2 x$
							+	$21 x$	-	$3$

Step 2.1.5.12

Divide the highest order term in the dividend $21 x$ by the highest order term in divisor $7 x$ .

				$x^{2}$	-	$2 x$	+	$3$
$7 x$	-	$1$		$7 x^{3}$	-	$15 x^{2}$	+	$23 x$	-	$3$
			-	$7 x^{3}$	+	$x^{2}$
					-	$14 x^{2}$	+	$23 x$
					+	$14 x^{2}$	-	$2 x$
							+	$21 x$	-	$3$

Step 2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$2 x$	+	$3$
$7 x$	-	$1$		$7 x^{3}$	-	$15 x^{2}$	+	$23 x$	-	$3$
			-	$7 x^{3}$	+	$x^{2}$
					-	$14 x^{2}$	+	$23 x$
					+	$14 x^{2}$	-	$2 x$
							+	$21 x$	-	$3$
							+	$21 x$	-	$3$

Step 2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $21 x - 3$

				$x^{2}$	-	$2 x$	+	$3$
$7 x$	-	$1$		$7 x^{3}$	-	$15 x^{2}$	+	$23 x$	-	$3$
			-	$7 x^{3}$	+	$x^{2}$
					-	$14 x^{2}$	+	$23 x$
					+	$14 x^{2}$	-	$2 x$
							+	$21 x$	-	$3$
							-	$21 x$	+	$3$

Step 2.1.5.15

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

				$x^{2}$	-	$2 x$	+	$3$
$7 x$	-	$1$		$7 x^{3}$	-	$15 x^{2}$	+	$23 x$	-	$3$
			-	$7 x^{3}$	+	$x^{2}$
					-	$14 x^{2}$	+	$23 x$
					+	$14 x^{2}$	-	$2 x$
							+	$21 x$	-	$3$
							-	$21 x$	+	$3$
										$0$

Step 2.1.5.16

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

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

Step 2.1.6

Write $7 x^{3} - 15 x^{2} + 23 x - 3$ as a set of factors.

$(7 x - 1) (x^{2} - 2 x + 3) = 0$

Step 2.2

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

$7 x - 1 = 0$

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

Step 2.3

Set $7 x - 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.3.1

Set $7 x - 1$ equal to $0$ .

$7 x - 1 = 0$

Step 2.3.2

Solve $7 x - 1 = 0$ for $x$ .

Tap for more steps...

Step 2.3.2.1

Add $1$ to both sides of the equation.

$7 x = 1$

Step 2.3.2.2

Divide each term in $7 x = 1$ by $7$ and simplify.

Tap for more steps...

Step 2.3.2.2.1

Divide each term in $7 x = 1$ by $7$ .

$\frac{7 x}{7} = \frac{1}{7}$

Step 2.3.2.2.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.2.2.1

Cancel the common factor of $7$ .

Tap for more steps...

Step 2.3.2.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{1}{7}$

Step 2.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{7}$

Step 2.4

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

Tap for more steps...

Step 2.4.1

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

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

Step 2.4.2

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

Tap for more steps...

Step 2.4.2.1

Use the quadratic formula to find the solutions.

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

Step 2.4.2.2

Substitute the values $a = 1$ , $b = - 2$ , and $c = 3$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot 3)}}{2 \cdot 1}$

Step 2.4.2.3

Simplify.

Tap for more steps...

Step 2.4.2.3.1

Simplify the numerator.

Tap for more steps...

Step 2.4.2.3.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

Step 2.4.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

Tap for more steps...

Step 2.4.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 3}}{2 \cdot 1}$

Step 2.4.2.3.1.2.2

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12}}{2 \cdot 1}$

Step 2.4.2.3.1.3

Subtract $12$ from $4$ .

$x = \frac{2 \pm \sqrt{- 8}}{2 \cdot 1}$

Step 2.4.2.3.1.4

Rewrite $- 8$ as $- 1 (8)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 1}$

Step 2.4.2.3.1.5

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 1}$

Step 2.4.2.3.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{8}}{2 \cdot 1}$

Step 2.4.2.3.1.7

Rewrite $8$ as $2^{2} \cdot 2$ .

Tap for more steps...

Step 2.4.2.3.1.7.1

Factor $4$ out of $8$ .

$x = \frac{2 \pm i \cdot \sqrt{4 (2)}}{2 \cdot 1}$

Step 2.4.2.3.1.7.2

Rewrite $4$ as $2^{2}$ .

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

Step 2.4.2.3.1.8

Pull terms out from under the radical.

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

Step 2.4.2.3.1.9

Move $2$ to the left of $i$ .

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

Step 2.4.2.3.2

Multiply $2$ by $1$ .

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

Step 2.4.2.3.3

Simplify $\frac{2 \pm 2 i \sqrt{2}}{2}$ .

$x = 1 \pm i \sqrt{2}$

Step 2.4.2.4

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

Tap for more steps...

Step 2.4.2.4.1

Simplify the numerator.

Tap for more steps...

Step 2.4.2.4.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

Step 2.4.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

Tap for more steps...

Step 2.4.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 3}}{2 \cdot 1}$

Step 2.4.2.4.1.2.2

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12}}{2 \cdot 1}$

Step 2.4.2.4.1.3

Subtract $12$ from $4$ .

$x = \frac{2 \pm \sqrt{- 8}}{2 \cdot 1}$

Step 2.4.2.4.1.4

Rewrite $- 8$ as $- 1 (8)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 1}$

Step 2.4.2.4.1.5

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 1}$

Step 2.4.2.4.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{8}}{2 \cdot 1}$

Step 2.4.2.4.1.7

Rewrite $8$ as $2^{2} \cdot 2$ .

Tap for more steps...

Step 2.4.2.4.1.7.1

Factor $4$ out of $8$ .

$x = \frac{2 \pm i \cdot \sqrt{4 (2)}}{2 \cdot 1}$

Step 2.4.2.4.1.7.2

Rewrite $4$ as $2^{2}$ .

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

Step 2.4.2.4.1.8

Pull terms out from under the radical.

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

Step 2.4.2.4.1.9

Move $2$ to the left of $i$ .

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

Step 2.4.2.4.2

Multiply $2$ by $1$ .

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

Step 2.4.2.4.3

Simplify $\frac{2 \pm 2 i \sqrt{2}}{2}$ .

$x = 1 \pm i \sqrt{2}$

Step 2.4.2.4.4

Change the $\pm$ to $+$ .

$x = 1 + i \sqrt{2}$

Step 2.4.2.5

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

Tap for more steps...

Step 2.4.2.5.1

Simplify the numerator.

Tap for more steps...

Step 2.4.2.5.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

Step 2.4.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

Tap for more steps...

Step 2.4.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 3}}{2 \cdot 1}$

Step 2.4.2.5.1.2.2

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12}}{2 \cdot 1}$

Step 2.4.2.5.1.3

Subtract $12$ from $4$ .

$x = \frac{2 \pm \sqrt{- 8}}{2 \cdot 1}$

Step 2.4.2.5.1.4

Rewrite $- 8$ as $- 1 (8)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 1}$

Step 2.4.2.5.1.5

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 1}$

Step 2.4.2.5.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{8}}{2 \cdot 1}$

Step 2.4.2.5.1.7

Rewrite $8$ as $2^{2} \cdot 2$ .

Tap for more steps...

Step 2.4.2.5.1.7.1

Factor $4$ out of $8$ .

$x = \frac{2 \pm i \cdot \sqrt{4 (2)}}{2 \cdot 1}$

Step 2.4.2.5.1.7.2

Rewrite $4$ as $2^{2}$ .

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

Step 2.4.2.5.1.8

Pull terms out from under the radical.

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

Step 2.4.2.5.1.9

Move $2$ to the left of $i$ .

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

Step 2.4.2.5.2

Multiply $2$ by $1$ .

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

Step 2.4.2.5.3

Simplify $\frac{2 \pm 2 i \sqrt{2}}{2}$ .

$x = 1 \pm i \sqrt{2}$

Step 2.4.2.5.4

Change the $\pm$ to $-$ .

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

Step 2.4.2.6

The final answer is the combination of both solutions.

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

Step 2.5

The final solution is all the values that make $(7 x - 1) (x^{2} - 2 x + 3) = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = \frac{1}{7}$ (Multiplicity of $1$ )

$x = 1 + i \sqrt{2}$ (Multiplicity of $1$ )

$x = 1 - i \sqrt{2}$ (Multiplicity of $1$ )

$x = \frac{1}{7}$ (Multiplicity of $1$ )

$x = 1 + i \sqrt{2}$ (Multiplicity of $1$ )

$x = 1 - i \sqrt{2}$ (Multiplicity of $1$ )

Step 3