Find the Roots (Zeros) x^4-7x^3+14x^2-3x-9

$x^{4} - 7 x^{3} + 14 x^{2} - 3 x - 9$

Step 1

Set $x^{4} - 7 x^{3} + 14 x^{2} - 3 x - 9$ equal to $0$ .

$x^{4} - 7 x^{3} + 14 x^{2} - 3 x - 9 = 0$

Step 2

Solve for $x$ .

Factor the left side of the equation.

Factor $x^{4} - 7 x^{3} + 14 x^{2} - 3 x - 9$ 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.

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

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

$\pm 1, \pm 9, \pm 3$

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.

$3^{4} - 7 \cdot 3^{3} + 14 \cdot 3^{2} - 3 \cdot 3 - 9$

Raise $3$ to the power of $4$ .

$81 - 7 \cdot 3^{3} + 14 \cdot 3^{2} - 3 \cdot 3 - 9$

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

$81 - 7 \cdot 27 + 14 \cdot 3^{2} - 3 \cdot 3 - 9$

Multiply $- 7$ by $27$ .

$81 - 189 + 14 \cdot 3^{2} - 3 \cdot 3 - 9$

Subtract $189$ from $81$ .

$- 108 + 14 \cdot 3^{2} - 3 \cdot 3 - 9$

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

$- 108 + 14 \cdot 9 - 3 \cdot 3 - 9$

Multiply $14$ by $9$ .

$- 108 + 126 - 3 \cdot 3 - 9$

Add $- 108$ and $126$ .

$18 - 3 \cdot 3 - 9$

Multiply $- 3$ by $3$ .

$18 - 9 - 9$

Subtract $9$ from $18$ .

$9 - 9$

Subtract $9$ from $9$ .

$0$

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.

$\frac{x^{4} - 7 x^{3} + 14 x^{2} - 3 x - 9}{x - 3}$

Divide $x^{4} - 7 x^{3} + 14 x^{2} - 3 x - 9$ 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 $0$ .

$x$

$3$

$x^{4}$

$7 x^{3}$

$14 x^{2}$

$3 x$

$9$

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x$ .

					$x^{3}$
	$x$	-	$3$		$x^{4}$	-	$7 x^{3}$	+	$14 x^{2}$	-	$3 x$	-	$9$

Multiply the new quotient term by the divisor.

				$x^{3}$
$x$	-	$3$		$x^{4}$	-	$7 x^{3}$	+	$14 x^{2}$	-	$3 x$	-	$9$
			+	$x^{4}$	-	$3 x^{3}$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} - 3 x^{3}$

				$x^{3}$
$x$	-	$3$		$x^{4}$	-	$7 x^{3}$	+	$14 x^{2}$	-	$3 x$	-	$9$
			-	$x^{4}$	+	$3 x^{3}$

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

				$x^{3}$
$x$	-	$3$		$x^{4}$	-	$7 x^{3}$	+	$14 x^{2}$	-	$3 x$	-	$9$
			-	$x^{4}$	+	$3 x^{3}$
					-	$4 x^{3}$

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

$x^{3}$

$x$

$3$

$x^{4}$

$7 x^{3}$

$14 x^{2}$

$3 x$

$9$

$x^{4}$

$3 x^{3}$

$4 x^{3}$

$14 x^{2}$

Divide the highest order term in the dividend $- 4 x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$4 x^{2}$

$x$

$3$

$x^{4}$

$7 x^{3}$

$14 x^{2}$

$3 x$

$9$

$x^{4}$

$3 x^{3}$

$4 x^{3}$

$14 x^{2}$

Multiply the new quotient term by the divisor.

$x^{3}$

$4 x^{2}$

$x$

$3$

$x^{4}$

$7 x^{3}$

$14 x^{2}$

$3 x$

$9$

$x^{4}$

$3 x^{3}$

$4 x^{3}$

$14 x^{2}$

$4 x^{3}$

$12 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{3} + 12 x^{2}$

$x^{3}$

$4 x^{2}$

$x$

$3$

$x^{4}$

$7 x^{3}$

$14 x^{2}$

$3 x$

$9$

$x^{4}$

$3 x^{3}$

$4 x^{3}$

$14 x^{2}$

$4 x^{3}$

$12 x^{2}$

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

$x^{3}$

$4 x^{2}$

$x$

$3$

$x^{4}$

$7 x^{3}$

$14 x^{2}$

$3 x$

$9$

$x^{4}$

$3 x^{3}$

$4 x^{3}$

$14 x^{2}$

$4 x^{3}$

$12 x^{2}$

$2 x^{2}$

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

$x^{3}$

$4 x^{2}$

$x$

$3$

$x^{4}$

$7 x^{3}$

$14 x^{2}$

$3 x$

$9$

$x^{4}$

$3 x^{3}$

$4 x^{3}$

$14 x^{2}$

$4 x^{3}$

$12 x^{2}$

$2 x^{2}$

$3 x$

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

$x^{3}$

$4 x^{2}$

$2 x$

$x$

$3$

$x^{4}$

$7 x^{3}$

$14 x^{2}$

$3 x$

$9$

$x^{4}$

$3 x^{3}$

$4 x^{3}$

$14 x^{2}$

$4 x^{3}$

$12 x^{2}$

$2 x^{2}$

$3 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$4 x^{2}$

$2 x$

$x$

$3$

$x^{4}$

$7 x^{3}$

$14 x^{2}$

$3 x$

$9$

$x^{4}$

$3 x^{3}$

$4 x^{3}$

$14 x^{2}$

$4 x^{3}$

$12 x^{2}$

$2 x^{2}$

$3 x$

$2 x^{2}$

$6 x$

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{2} - 6 x$

$x^{3}$

$4 x^{2}$

$2 x$

$x$

$3$

$x^{4}$

$7 x^{3}$

$14 x^{2}$

$3 x$

$9$

$x^{4}$

$3 x^{3}$

$4 x^{3}$

$14 x^{2}$

$4 x^{3}$

$12 x^{2}$

$2 x^{2}$

$3 x$

$2 x^{2}$

$6 x$

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

$x^{3}$

$4 x^{2}$

$2 x$

$x$

$3$

$x^{4}$

$7 x^{3}$

$14 x^{2}$

$3 x$

$9$

$x^{4}$

$3 x^{3}$

$4 x^{3}$

$14 x^{2}$

$4 x^{3}$

$12 x^{2}$

$2 x^{2}$

$3 x$

$2 x^{2}$

$6 x$

$3 x$

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

$x^{3}$

$4 x^{2}$

$2 x$

$x$

$3$

$x^{4}$

$7 x^{3}$

$14 x^{2}$

$3 x$

$9$

$x^{4}$

$3 x^{3}$

$4 x^{3}$

$14 x^{2}$

$4 x^{3}$

$12 x^{2}$

$2 x^{2}$

$3 x$

$2 x^{2}$

$6 x$

$3 x$

$9$

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

$x^{3}$

$4 x^{2}$

$2 x$

$3$

$x$

$3$

$x^{4}$

$7 x^{3}$

$14 x^{2}$

$3 x$

$9$

$x^{4}$

$3 x^{3}$

$4 x^{3}$

$14 x^{2}$

$4 x^{3}$

$12 x^{2}$

$2 x^{2}$

$3 x$

$2 x^{2}$

$6 x$

$3 x$

$9$

Multiply the new quotient term by the divisor.

$x^{3}$

$4 x^{2}$

$2 x$

$3$

$x$

$3$

$x^{4}$

$7 x^{3}$

$14 x^{2}$

$3 x$

$9$

$x^{4}$

$3 x^{3}$

$4 x^{3}$

$14 x^{2}$

$4 x^{3}$

$12 x^{2}$

$2 x^{2}$

$3 x$

$2 x^{2}$

$6 x$

$3 x$

$9$

$3 x$

$9$

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

$x^{3}$

$4 x^{2}$

$2 x$

$3$

$x$

$3$

$x^{4}$

$7 x^{3}$

$14 x^{2}$

$3 x$

$9$

$x^{4}$

$3 x^{3}$

$4 x^{3}$

$14 x^{2}$

$4 x^{3}$

$12 x^{2}$

$2 x^{2}$

$3 x$

$2 x^{2}$

$6 x$

$3 x$

$9$

$3 x$

$9$

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

$x^{3}$

$4 x^{2}$

$2 x$

$3$

$x$

$3$

$x^{4}$

$7 x^{3}$

$14 x^{2}$

$3 x$

$9$

$x^{4}$

$3 x^{3}$

$4 x^{3}$

$14 x^{2}$

$4 x^{3}$

$12 x^{2}$

$2 x^{2}$

$3 x$

$2 x^{2}$

$6 x$

$3 x$

$9$

$3 x$

$9$

$0$

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

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

Write $x^{4} - 7 x^{3} + 14 x^{2} - 3 x - 9$ as a set of factors.

$(x - 3) (x^{3} - 4 x^{2} + 2 x + 3) = 0$

Factor $x^{3} - 4 x^{2} + 2 x + 3$ 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.

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

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

$\pm 1, \pm 3$

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.

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

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

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

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

$27 - 4 \cdot 9 + 2 \cdot 3 + 3$

Multiply $- 4$ by $9$ .

$27 - 36 + 2 \cdot 3 + 3$

Subtract $36$ from $27$ .

$- 9 + 2 \cdot 3 + 3$

Multiply $2$ by $3$ .

$- 9 + 6 + 3$

Add $- 9$ and $6$ .

$- 3 + 3$

Add $- 3$ and $3$ .

$0$

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.

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

Divide $x^{3} - 4 x^{2} + 2 x + 3$ 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 $0$ .

$x$

$3$

$x^{3}$

$4 x^{2}$

$2 x$

$3$

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

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

Multiply the new quotient term by the divisor.

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

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

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

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

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

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

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

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

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

Multiply the new quotient term by the divisor.

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

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

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

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

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

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

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

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

				$x^{2}$	-	$x$	-	$1$
$x$	-	$3$		$x^{3}$	-	$4 x^{2}$	+	$2 x$	+	$3$
			-	$x^{3}$	+	$3 x^{2}$
					-	$x^{2}$	+	$2 x$
					+	$x^{2}$	-	$3 x$
							-	$x$	+	$3$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$	-	$1$
$x$	-	$3$		$x^{3}$	-	$4 x^{2}$	+	$2 x$	+	$3$
			-	$x^{3}$	+	$3 x^{2}$
					-	$x^{2}$	+	$2 x$
					+	$x^{2}$	-	$3 x$
							-	$x$	+	$3$
							-	$x$	+	$3$

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

				$x^{2}$	-	$x$	-	$1$
$x$	-	$3$		$x^{3}$	-	$4 x^{2}$	+	$2 x$	+	$3$
			-	$x^{3}$	+	$3 x^{2}$
					-	$x^{2}$	+	$2 x$
					+	$x^{2}$	-	$3 x$
							-	$x$	+	$3$
							+	$x$	-	$3$

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

				$x^{2}$	-	$x$	-	$1$
$x$	-	$3$		$x^{3}$	-	$4 x^{2}$	+	$2 x$	+	$3$
			-	$x^{3}$	+	$3 x^{2}$
					-	$x^{2}$	+	$2 x$
					+	$x^{2}$	-	$3 x$
							-	$x$	+	$3$
							+	$x$	-	$3$
										$0$

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

$x^{2} - x - 1$

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

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

Remove unnecessary parentheses.

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

Combine like factors.

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

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

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

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

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

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

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

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

Use the quadratic formula to find the solutions.

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

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

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

Simplify.

Simplify the numerator.

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

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

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

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 1$ .

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

Add $1$ and $4$ .

$x = \frac{1 \pm \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{1 \pm \sqrt{5}}{2}$

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{5}}{2}, \frac{1 - \sqrt{5}}{2}$

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

$x = 3, \frac{1 + \sqrt{5}}{2}, \frac{1 - \sqrt{5}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 3, \frac{1 + \sqrt{5}}{2}, \frac{1 - \sqrt{5}}{2}$

Decimal Form:

$x = 3, 1.61803398 \dots, - 0.61803398 \dots$

Step 4

Find the Roots (Zeros) x^4-7x^3+14x^2-3x-9

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