Find the Roots (Zeros) x^3-6x^2+4x+15

$x^{3} - 6 x^{2} + 4 x + 15$

Step 1

Set $x^{3} - 6 x^{2} + 4 x + 15$ equal to $0$ .

$x^{3} - 6 x^{2} + 4 x + 15 = 0$

Step 2

Solve for $x$ .

Factor $x^{3} - 6 x^{2} + 4 x + 15$ using the rational roots test.

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.

$p = \pm 1, \pm 15, \pm 3, \pm 5 q = \pm 1$

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

$\pm 1, \pm 15, \pm 3, \pm 5$

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

Substitute $3$ into the polynomial.

$3^{3} - 6 \cdot 3^{2} + 4 \cdot 3 + 15$

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

$27 - 6 \cdot 3^{2} + 4 \cdot 3 + 15$

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

$27 - 6 \cdot 9 + 4 \cdot 3 + 15$

Multiply $- 6$ by $9$ .

$27 - 54 + 4 \cdot 3 + 15$

Subtract $54$ from $27$ .

$- 27 + 4 \cdot 3 + 15$

Multiply $4$ by $3$ .

$- 27 + 12 + 15$

Add $- 27$ and $12$ .

$- 15 + 15$

Add $- 15$ and $15$ .

$0$

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.

$\frac{x^{3} - 6 x^{2} + 4 x + 15}{x - 3}$

Divide $x^{3} - 6 x^{2} + 4 x + 15$ by $x - 3$ .

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

$x$

$3$

$x^{3}$

$6 x^{2}$

$4 x$

$15$

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

					$x^{2}$
	$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$4 x$	+	$15$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$4 x$	+	$15$
			+	$x^{3}$	-	$3 x^{2}$

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$4 x$	+	$15$
			-	$x^{3}$	+	$3 x^{2}$

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$4 x$	+	$15$
			-	$x^{3}$	+	$3 x^{2}$
					-	$3 x^{2}$

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$4 x$	+	$15$
			-	$x^{3}$	+	$3 x^{2}$
					-	$3 x^{2}$	+	$4 x$

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

				$x^{2}$	-	$3 x$
$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$4 x$	+	$15$
			-	$x^{3}$	+	$3 x^{2}$
					-	$3 x^{2}$	+	$4 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$3 x$
$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$4 x$	+	$15$
			-	$x^{3}$	+	$3 x^{2}$
					-	$3 x^{2}$	+	$4 x$
					-	$3 x^{2}$	+	$9 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x^{2} + 9 x$

				$x^{2}$	-	$3 x$
$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$4 x$	+	$15$
			-	$x^{3}$	+	$3 x^{2}$
					-	$3 x^{2}$	+	$4 x$
					+	$3 x^{2}$	-	$9 x$

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

				$x^{2}$	-	$3 x$
$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$4 x$	+	$15$
			-	$x^{3}$	+	$3 x^{2}$
					-	$3 x^{2}$	+	$4 x$
					+	$3 x^{2}$	-	$9 x$
							-	$5 x$

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

				$x^{2}$	-	$3 x$
$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$4 x$	+	$15$
			-	$x^{3}$	+	$3 x^{2}$
					-	$3 x^{2}$	+	$4 x$
					+	$3 x^{2}$	-	$9 x$
							-	$5 x$	+	$15$

Divide the highest order term in the dividend $- 5 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$3 x$	-	$5$
$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$4 x$	+	$15$
			-	$x^{3}$	+	$3 x^{2}$
					-	$3 x^{2}$	+	$4 x$
					+	$3 x^{2}$	-	$9 x$
							-	$5 x$	+	$15$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$3 x$	-	$5$
$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$4 x$	+	$15$
			-	$x^{3}$	+	$3 x^{2}$
					-	$3 x^{2}$	+	$4 x$
					+	$3 x^{2}$	-	$9 x$
							-	$5 x$	+	$15$
							-	$5 x$	+	$15$

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x + 15$

				$x^{2}$	-	$3 x$	-	$5$
$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$4 x$	+	$15$
			-	$x^{3}$	+	$3 x^{2}$
					-	$3 x^{2}$	+	$4 x$
					+	$3 x^{2}$	-	$9 x$
							-	$5 x$	+	$15$
							+	$5 x$	-	$15$

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

				$x^{2}$	-	$3 x$	-	$5$
$x$	-	$3$		$x^{3}$	-	$6 x^{2}$	+	$4 x$	+	$15$
			-	$x^{3}$	+	$3 x^{2}$
					-	$3 x^{2}$	+	$4 x$
					+	$3 x^{2}$	-	$9 x$
							-	$5 x$	+	$15$
							+	$5 x$	-	$15$
										$0$

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

$x^{2} - 3 x - 5$

Write $x^{3} - 6 x^{2} + 4 x + 15$ as a set of factors.

$(x - 3) (x^{2} - 3 x - 5) = 0$

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

$x - 3 = 0$

$x^{2} - 3 x - 5 = 0$

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

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Set $x^{2} - 3 x - 5$ equal to $0$ and solve for $x$ .

Set $x^{2} - 3 x - 5$ equal to $0$ .

$x^{2} - 3 x - 5 = 0$

Solve $x^{2} - 3 x - 5 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 3$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (1 \cdot - 5)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 5$ .

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 5}}{2 \cdot 1}$

Multiply $- 4$ by $- 5$ .

$x = \frac{3 \pm \sqrt{9 + 20}}{2 \cdot 1}$

Add $9$ and $20$ .

$x = \frac{3 \pm \sqrt{29}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{3 \pm \sqrt{29}}{2}$

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{29}}{2}, \frac{3 - \sqrt{29}}{2}$

The final solution is all the values that make $(x - 3) (x^{2} - 3 x - 5) = 0$ true.

$x = 3, \frac{3 + \sqrt{29}}{2}, \frac{3 - \sqrt{29}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 3, \frac{3 + \sqrt{29}}{2}, \frac{3 - \sqrt{29}}{2}$

Decimal Form:

$x = 3, 4.19258240 \dots, - 1.19258240 \dots$

Step 4

Find the Roots (Zeros) x^3-6x^2+4x+15

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