Find the Roots (Zeros) f(x)=x^4-4x^3+23x^2+62x+34

$f (x) = x^{4} - 4 x^{3} + 23 x^{2} + 62 x + 34$

Step 1

Set $x^{4} - 4 x^{3} + 23 x^{2} + 62 x + 34$ equal to $0$ .

$x^{4} - 4 x^{3} + 23 x^{2} + 62 x + 34 = 0$

Step 2

Solve for $x$ .

Factor the left side of the equation.

Factor $x^{4} - 4 x^{3} + 23 x^{2} + 62 x + 34$ 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.

$p = \pm 1, \pm 34, \pm 2, \pm 17 q = \pm 1$

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

$\pm 1, \pm 34, \pm 2, \pm 17$

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

Substitute $- 1$ into the polynomial.

${(- 1)}^{4} - 4 {(- 1)}^{3} + 23 {(- 1)}^{2} + 62 \cdot - 1 + 34$

Raise $- 1$ to the power of $4$ .

$1 - 4 {(- 1)}^{3} + 23 {(- 1)}^{2} + 62 \cdot - 1 + 34$

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

$1 - 4 \cdot - 1 + 23 {(- 1)}^{2} + 62 \cdot - 1 + 34$

Multiply $- 4$ by $- 1$ .

$1 + 4 + 23 {(- 1)}^{2} + 62 \cdot - 1 + 34$

Add $1$ and $4$ .

$5 + 23 {(- 1)}^{2} + 62 \cdot - 1 + 34$

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

$5 + 23 \cdot 1 + 62 \cdot - 1 + 34$

Multiply $23$ by $1$ .

$5 + 23 + 62 \cdot - 1 + 34$

Add $5$ and $23$ .

$28 + 62 \cdot - 1 + 34$

Multiply $62$ by $- 1$ .

$28 - 62 + 34$

Subtract $62$ from $28$ .

$- 34 + 34$

Add $- 34$ and $34$ .

$0$

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} - 4 x^{3} + 23 x^{2} + 62 x + 34}{x + 1}$

Divide $x^{4} - 4 x^{3} + 23 x^{2} + 62 x + 34$ by $x + 1$ .

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

$x$

$1$

$x^{4}$

$4 x^{3}$

$23 x^{2}$

$62 x$

$34$

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

$x^{3}$

$x$

$1$

$x^{4}$

$4 x^{3}$

$23 x^{2}$

$62 x$

$34$

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$1$

$x^{4}$

$4 x^{3}$

$23 x^{2}$

$62 x$

$34$

$x^{4}$

$x^{3}$

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

$x^{3}$

$x$

$1$

$x^{4}$

$4 x^{3}$

$23 x^{2}$

$62 x$

$34$

$x^{4}$

$x^{3}$

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

$x^{3}$

$x$

$1$

$x^{4}$

$4 x^{3}$

$23 x^{2}$

$62 x$

$34$

$x^{4}$

$x^{3}$

$5 x^{3}$

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

$x^{3}$

$x$

$1$

$x^{4}$

$4 x^{3}$

$23 x^{2}$

$62 x$

$34$

$x^{4}$

$x^{3}$

$5 x^{3}$

$23 x^{2}$

Divide the highest order term in the dividend $- 5 x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$5 x^{2}$

$x$

$1$

$x^{4}$

$4 x^{3}$

$23 x^{2}$

$62 x$

$34$

$x^{4}$

$x^{3}$

$5 x^{3}$

$23 x^{2}$

Multiply the new quotient term by the divisor.

$x^{3}$

$5 x^{2}$

$x$

$1$

$x^{4}$

$4 x^{3}$

$23 x^{2}$

$62 x$

$34$

$x^{4}$

$x^{3}$

$5 x^{3}$

$23 x^{2}$

$5 x^{3}$

$5 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x^{3} - 5 x^{2}$

$x^{3}$

$5 x^{2}$

$x$

$1$

$x^{4}$

$4 x^{3}$

$23 x^{2}$

$62 x$

$34$

$x^{4}$

$x^{3}$

$5 x^{3}$

$23 x^{2}$

$5 x^{3}$

$5 x^{2}$

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

$x^{3}$

$5 x^{2}$

$x$

$1$

$x^{4}$

$4 x^{3}$

$23 x^{2}$

$62 x$

$34$

$x^{4}$

$x^{3}$

$5 x^{3}$

$23 x^{2}$

$5 x^{3}$

$5 x^{2}$

$28 x^{2}$

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

$x^{3}$

$5 x^{2}$

$x$

$1$

$x^{4}$

$4 x^{3}$

$23 x^{2}$

$62 x$

$34$

$x^{4}$

$x^{3}$

$5 x^{3}$

$23 x^{2}$

$5 x^{3}$

$5 x^{2}$

$28 x^{2}$

$62 x$

Divide the highest order term in the dividend $28 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$5 x^{2}$

$28 x$

$x$

$1$

$x^{4}$

$4 x^{3}$

$23 x^{2}$

$62 x$

$34$

$x^{4}$

$x^{3}$

$5 x^{3}$

$23 x^{2}$

$5 x^{3}$

$5 x^{2}$

$28 x^{2}$

$62 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$5 x^{2}$

$28 x$

$x$

$1$

$x^{4}$

$4 x^{3}$

$23 x^{2}$

$62 x$

$34$

$x^{4}$

$x^{3}$

$5 x^{3}$

$23 x^{2}$

$5 x^{3}$

$5 x^{2}$

$28 x^{2}$

$62 x$

$28 x^{2}$

$28 x$

The expression needs to be subtracted from the dividend, so change all the signs in $28 x^{2} + 28 x$

$x^{3}$

$5 x^{2}$

$28 x$

$x$

$1$

$x^{4}$

$4 x^{3}$

$23 x^{2}$

$62 x$

$34$

$x^{4}$

$x^{3}$

$5 x^{3}$

$23 x^{2}$

$5 x^{3}$

$5 x^{2}$

$28 x^{2}$

$62 x$

$28 x^{2}$

$28 x$

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

$x^{3}$

$5 x^{2}$

$28 x$

$x$

$1$

$x^{4}$

$4 x^{3}$

$23 x^{2}$

$62 x$

$34$

$x^{4}$

$x^{3}$

$5 x^{3}$

$23 x^{2}$

$5 x^{3}$

$5 x^{2}$

$28 x^{2}$

$62 x$

$28 x^{2}$

$28 x$

$34 x$

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

$x^{3}$

$5 x^{2}$

$28 x$

$x$

$1$

$x^{4}$

$4 x^{3}$

$23 x^{2}$

$62 x$

$34$

$x^{4}$

$x^{3}$

$5 x^{3}$

$23 x^{2}$

$5 x^{3}$

$5 x^{2}$

$28 x^{2}$

$62 x$

$28 x^{2}$

$28 x$

$34 x$

$34$

Divide the highest order term in the dividend $34 x$ by the highest order term in divisor $x$ .

$x^{3}$

$5 x^{2}$

$28 x$

$34$

$x$

$1$

$x^{4}$

$4 x^{3}$

$23 x^{2}$

$62 x$

$34$

$x^{4}$

$x^{3}$

$5 x^{3}$

$23 x^{2}$

$5 x^{3}$

$5 x^{2}$

$28 x^{2}$

$62 x$

$28 x^{2}$

$28 x$

$34 x$

$34$

Multiply the new quotient term by the divisor.

$x^{3}$

$5 x^{2}$

$28 x$

$34$

$x$

$1$

$x^{4}$

$4 x^{3}$

$23 x^{2}$

$62 x$

$34$

$x^{4}$

$x^{3}$

$5 x^{3}$

$23 x^{2}$

$5 x^{3}$

$5 x^{2}$

$28 x^{2}$

$62 x$

$28 x^{2}$

$28 x$

$34 x$

$34$

$34 x$

$34$

The expression needs to be subtracted from the dividend, so change all the signs in $34 x + 34$

$x^{3}$

$5 x^{2}$

$28 x$

$34$

$x$

$1$

$x^{4}$

$4 x^{3}$

$23 x^{2}$

$62 x$

$34$

$x^{4}$

$x^{3}$

$5 x^{3}$

$23 x^{2}$

$5 x^{3}$

$5 x^{2}$

$28 x^{2}$

$62 x$

$28 x^{2}$

$28 x$

$34 x$

$34$

$34 x$

$34$

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

$x^{3}$

$5 x^{2}$

$28 x$

$34$

$x$

$1$

$x^{4}$

$4 x^{3}$

$23 x^{2}$

$62 x$

$34$

$x^{4}$

$x^{3}$

$5 x^{3}$

$23 x^{2}$

$5 x^{3}$

$5 x^{2}$

$28 x^{2}$

$62 x$

$28 x^{2}$

$28 x$

$34 x$

$34$

$34 x$

$34$

$0$

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

$x^{3} - 5 x^{2} + 28 x + 34$

Write $x^{4} - 4 x^{3} + 23 x^{2} + 62 x + 34$ as a set of factors.

$(x + 1) (x^{3} - 5 x^{2} + 28 x + 34) = 0$

Factor $x^{3} - 5 x^{2} + 28 x + 34$ 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.

$p = \pm 1, \pm 34, \pm 2, \pm 17 q = \pm 1$

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

$\pm 1, \pm 34, \pm 2, \pm 17$

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

Substitute $- 1$ into the polynomial.

${(- 1)}^{3} - 5 {(- 1)}^{2} + 28 \cdot - 1 + 34$

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

$- 1 - 5 {(- 1)}^{2} + 28 \cdot - 1 + 34$

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

$- 1 - 5 \cdot 1 + 28 \cdot - 1 + 34$

Multiply $- 5$ by $1$ .

$- 1 - 5 + 28 \cdot - 1 + 34$

Subtract $5$ from $- 1$ .

$- 6 + 28 \cdot - 1 + 34$

Multiply $28$ by $- 1$ .

$- 6 - 28 + 34$

Subtract $28$ from $- 6$ .

$- 34 + 34$

Add $- 34$ and $34$ .

$0$

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 5 x^{2} + 28 x + 34}{x + 1}$

Divide $x^{3} - 5 x^{2} + 28 x + 34$ by $x + 1$ .

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

$x$

$1$

$x^{3}$

$5 x^{2}$

$28 x$

$34$

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

					$x^{2}$
	$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$28 x$	+	$34$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$28 x$	+	$34$
			+	$x^{3}$	+	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$28 x$	+	$34$
			-	$x^{3}$	-	$x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$28 x$	+	$34$
			-	$x^{3}$	-	$x^{2}$
					-	$6 x^{2}$

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$28 x$	+	$34$
			-	$x^{3}$	-	$x^{2}$
					-	$6 x^{2}$	+	$28 x$

Divide the highest order term in the dividend $- 6 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$6 x$
$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$28 x$	+	$34$
			-	$x^{3}$	-	$x^{2}$
					-	$6 x^{2}$	+	$28 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$6 x$
$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$28 x$	+	$34$
			-	$x^{3}$	-	$x^{2}$
					-	$6 x^{2}$	+	$28 x$
					-	$6 x^{2}$	-	$6 x$

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

				$x^{2}$	-	$6 x$
$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$28 x$	+	$34$
			-	$x^{3}$	-	$x^{2}$
					-	$6 x^{2}$	+	$28 x$
					+	$6 x^{2}$	+	$6 x$

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

				$x^{2}$	-	$6 x$
$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$28 x$	+	$34$
			-	$x^{3}$	-	$x^{2}$
					-	$6 x^{2}$	+	$28 x$
					+	$6 x^{2}$	+	$6 x$
							+	$34 x$

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

				$x^{2}$	-	$6 x$
$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$28 x$	+	$34$
			-	$x^{3}$	-	$x^{2}$
					-	$6 x^{2}$	+	$28 x$
					+	$6 x^{2}$	+	$6 x$
							+	$34 x$	+	$34$

Divide the highest order term in the dividend $34 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$6 x$	+	$34$
$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$28 x$	+	$34$
			-	$x^{3}$	-	$x^{2}$
					-	$6 x^{2}$	+	$28 x$
					+	$6 x^{2}$	+	$6 x$
							+	$34 x$	+	$34$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$6 x$	+	$34$
$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$28 x$	+	$34$
			-	$x^{3}$	-	$x^{2}$
					-	$6 x^{2}$	+	$28 x$
					+	$6 x^{2}$	+	$6 x$
							+	$34 x$	+	$34$
							+	$34 x$	+	$34$

The expression needs to be subtracted from the dividend, so change all the signs in $34 x + 34$

				$x^{2}$	-	$6 x$	+	$34$
$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$28 x$	+	$34$
			-	$x^{3}$	-	$x^{2}$
					-	$6 x^{2}$	+	$28 x$
					+	$6 x^{2}$	+	$6 x$
							+	$34 x$	+	$34$
							-	$34 x$	-	$34$

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

				$x^{2}$	-	$6 x$	+	$34$
$x$	+	$1$		$x^{3}$	-	$5 x^{2}$	+	$28 x$	+	$34$
			-	$x^{3}$	-	$x^{2}$
					-	$6 x^{2}$	+	$28 x$
					+	$6 x^{2}$	+	$6 x$
							+	$34 x$	+	$34$
							-	$34 x$	-	$34$
										$0$

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

$x^{2} - 6 x + 34$

Write $x^{3} - 5 x^{2} + 28 x + 34$ as a set of factors.

$(x + 1) ((x + 1) (x^{2} - 6 x + 34)) = 0$

Remove unnecessary parentheses.

$(x + 1) (x + 1) (x^{2} - 6 x + 34) = 0$

Combine like factors.

Raise $x + 1$ to the power of $1$ .

$(x + 1) (x + 1) (x^{2} - 6 x + 34) = 0$

Raise $x + 1$ to the power of $1$ .

$(x + 1) (x + 1) (x^{2} - 6 x + 34) = 0$

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

${(x + 1)}^{1 + 1} (x^{2} - 6 x + 34) = 0$

Add $1$ and $1$ .

${(x + 1)}^{2} (x^{2} - 6 x + 34) = 0$

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

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

$x^{2} - 6 x + 34 = 0$

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

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

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

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

Set the $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Set $x^{2} - 6 x + 34$ equal to $0$ and solve for $x$ .

Set $x^{2} - 6 x + 34$ equal to $0$ .

$x^{2} - 6 x + 34 = 0$

Solve $x^{2} - 6 x + 34 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 6$ , and $c = 34$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot 34)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 34}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 34$ .

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 34}}{2 \cdot 1}$

Multiply $- 4$ by $34$ .

$x = \frac{6 \pm \sqrt{36 - 136}}{2 \cdot 1}$

Subtract $136$ from $36$ .

$x = \frac{6 \pm \sqrt{- 100}}{2 \cdot 1}$

Rewrite $- 100$ as $- 1 (100)$ .

$x = \frac{6 \pm \sqrt{- 1 \cdot 100}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (100)}$ as $\sqrt{- 1} \cdot \sqrt{100}$ .

$x = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{100}}{2 \cdot 1}$

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

$x = \frac{6 \pm i \cdot \sqrt{100}}{2 \cdot 1}$

Rewrite $100$ as $10^{2}$ .

$x = \frac{6 \pm i \cdot \sqrt{10^{2}}}{2 \cdot 1}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{6 \pm i \cdot 10}{2 \cdot 1}$

Move $10$ to the left of $i$ .

$x = \frac{6 \pm 10 i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{6 \pm 10 i}{2}$

Simplify $\frac{6 \pm 10 i}{2}$ .

$x = 3 \pm 5 i$

The final answer is the combination of both solutions.

$x = 3 + 5 i, 3 - 5 i$

The final solution is all the values that make ${(x + 1)}^{2} (x^{2} - 6 x + 34) = 0$ true.

$x = - 1, 3 + 5 i, 3 - 5 i$

Step 3

Find the Roots (Zeros) f(x)=x^4-4x^3+23x^2+62x+34

Related Questions

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