Find the Roots (Zeros) x^4+5x^3+7x^2-3x-10=0

$x^{4} + 5 x^{3} + 7 x^{2} - 3 x - 10 = 0$

Step 1

Factor the left side of the equation.

Factor $x^{4} + 5 x^{3} + 7 x^{2} - 3 x - 10$ 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.

$p = \pm 1, \pm 10, \pm 2, \pm 5 q = \pm 1$

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

$\pm 1, \pm 10, \pm 2, \pm 5$

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.

$1^{4} + 5 \cdot 1^{3} + 7 \cdot 1^{2} - 3 \cdot 1 - 10$

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

$1 + 5 \cdot 1^{3} + 7 \cdot 1^{2} - 3 \cdot 1 - 10$

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

$1 + 5 \cdot 1 + 7 \cdot 1^{2} - 3 \cdot 1 - 10$

Multiply $5$ by $1$ .

$1 + 5 + 7 \cdot 1^{2} - 3 \cdot 1 - 10$

Add $1$ and $5$ .

$6 + 7 \cdot 1^{2} - 3 \cdot 1 - 10$

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

$6 + 7 \cdot 1 - 3 \cdot 1 - 10$

Multiply $7$ by $1$ .

$6 + 7 - 3 \cdot 1 - 10$

Add $6$ and $7$ .

$13 - 3 \cdot 1 - 10$

Multiply $- 3$ by $1$ .

$13 - 3 - 10$

Subtract $3$ from $13$ .

$10 - 10$

Subtract $10$ from $10$ .

$0$

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.

$\frac{x^{4} + 5 x^{3} + 7 x^{2} - 3 x - 10}{x - 1}$

Divide $x^{4} + 5 x^{3} + 7 x^{2} - 3 x - 10$ 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 $0$ .

$x$

$1$

$x^{4}$

$5 x^{3}$

$7 x^{2}$

$3 x$

$10$

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

$x^{3}$

$x$

$1$

$x^{4}$

$5 x^{3}$

$7 x^{2}$

$3 x$

$10$

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$1$

$x^{4}$

$5 x^{3}$

$7 x^{2}$

$3 x$

$10$

$x^{4}$

$x^{3}$

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

$x^{3}$

$x$

$1$

$x^{4}$

$5 x^{3}$

$7 x^{2}$

$3 x$

$10$

$x^{4}$

$x^{3}$

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

$x^{3}$

$x$

$1$

$x^{4}$

$5 x^{3}$

$7 x^{2}$

$3 x$

$10$

$x^{4}$

$x^{3}$

$6 x^{3}$

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

$x^{3}$

$x$

$1$

$x^{4}$

$5 x^{3}$

$7 x^{2}$

$3 x$

$10$

$x^{4}$

$x^{3}$

$6 x^{3}$

$7 x^{2}$

Divide the highest order term in the dividend $6 x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$6 x^{2}$

$x$

$1$

$x^{4}$

$5 x^{3}$

$7 x^{2}$

$3 x$

$10$

$x^{4}$

$x^{3}$

$6 x^{3}$

$7 x^{2}$

Multiply the new quotient term by the divisor.

$x^{3}$

$6 x^{2}$

$x$

$1$

$x^{4}$

$5 x^{3}$

$7 x^{2}$

$3 x$

$10$

$x^{4}$

$x^{3}$

$6 x^{3}$

$7 x^{2}$

$6 x^{3}$

$6 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{3} - 6 x^{2}$

$x^{3}$

$6 x^{2}$

$x$

$1$

$x^{4}$

$5 x^{3}$

$7 x^{2}$

$3 x$

$10$

$x^{4}$

$x^{3}$

$6 x^{3}$

$7 x^{2}$

$6 x^{3}$

$6 x^{2}$

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

$x^{3}$

$6 x^{2}$

$x$

$1$

$x^{4}$

$5 x^{3}$

$7 x^{2}$

$3 x$

$10$

$x^{4}$

$x^{3}$

$6 x^{3}$

$7 x^{2}$

$6 x^{3}$

$6 x^{2}$

$13 x^{2}$

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

$x^{3}$

$6 x^{2}$

$x$

$1$

$x^{4}$

$5 x^{3}$

$7 x^{2}$

$3 x$

$10$

$x^{4}$

$x^{3}$

$6 x^{3}$

$7 x^{2}$

$6 x^{3}$

$6 x^{2}$

$13 x^{2}$

$3 x$

Divide the highest order term in the dividend $13 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$6 x^{2}$

$13 x$

$x$

$1$

$x^{4}$

$5 x^{3}$

$7 x^{2}$

$3 x$

$10$

$x^{4}$

$x^{3}$

$6 x^{3}$

$7 x^{2}$

$6 x^{3}$

$6 x^{2}$

$13 x^{2}$

$3 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$6 x^{2}$

$13 x$

$x$

$1$

$x^{4}$

$5 x^{3}$

$7 x^{2}$

$3 x$

$10$

$x^{4}$

$x^{3}$

$6 x^{3}$

$7 x^{2}$

$6 x^{3}$

$6 x^{2}$

$13 x^{2}$

$3 x$

$13 x^{2}$

$13 x$

The expression needs to be subtracted from the dividend, so change all the signs in $13 x^{2} - 13 x$

$x^{3}$

$6 x^{2}$

$13 x$

$x$

$1$

$x^{4}$

$5 x^{3}$

$7 x^{2}$

$3 x$

$10$

$x^{4}$

$x^{3}$

$6 x^{3}$

$7 x^{2}$

$6 x^{3}$

$6 x^{2}$

$13 x^{2}$

$3 x$

$13 x^{2}$

$13 x$

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

$x^{3}$

$6 x^{2}$

$13 x$

$x$

$1$

$x^{4}$

$5 x^{3}$

$7 x^{2}$

$3 x$

$10$

$x^{4}$

$x^{3}$

$6 x^{3}$

$7 x^{2}$

$6 x^{3}$

$6 x^{2}$

$13 x^{2}$

$3 x$

$13 x^{2}$

$13 x$

$10 x$

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

$x^{3}$

$6 x^{2}$

$13 x$

$x$

$1$

$x^{4}$

$5 x^{3}$

$7 x^{2}$

$3 x$

$10$

$x^{4}$

$x^{3}$

$6 x^{3}$

$7 x^{2}$

$6 x^{3}$

$6 x^{2}$

$13 x^{2}$

$3 x$

$13 x^{2}$

$13 x$

$10 x$

$10$

Divide the highest order term in the dividend $10 x$ by the highest order term in divisor $x$ .

$x^{3}$

$6 x^{2}$

$13 x$

$10$

$x$

$1$

$x^{4}$

$5 x^{3}$

$7 x^{2}$

$3 x$

$10$

$x^{4}$

$x^{3}$

$6 x^{3}$

$7 x^{2}$

$6 x^{3}$

$6 x^{2}$

$13 x^{2}$

$3 x$

$13 x^{2}$

$13 x$

$10 x$

$10$

Multiply the new quotient term by the divisor.

$x^{3}$

$6 x^{2}$

$13 x$

$10$

$x$

$1$

$x^{4}$

$5 x^{3}$

$7 x^{2}$

$3 x$

$10$

$x^{4}$

$x^{3}$

$6 x^{3}$

$7 x^{2}$

$6 x^{3}$

$6 x^{2}$

$13 x^{2}$

$3 x$

$13 x^{2}$

$13 x$

$10 x$

$10$

$10 x$

$10$

The expression needs to be subtracted from the dividend, so change all the signs in $10 x - 10$

$x^{3}$

$6 x^{2}$

$13 x$

$10$

$x$

$1$

$x^{4}$

$5 x^{3}$

$7 x^{2}$

$3 x$

$10$

$x^{4}$

$x^{3}$

$6 x^{3}$

$7 x^{2}$

$6 x^{3}$

$6 x^{2}$

$13 x^{2}$

$3 x$

$13 x^{2}$

$13 x$

$10 x$

$10$

$10 x$

$10$

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

$x^{3}$

$6 x^{2}$

$13 x$

$10$

$x$

$1$

$x^{4}$

$5 x^{3}$

$7 x^{2}$

$3 x$

$10$

$x^{4}$

$x^{3}$

$6 x^{3}$

$7 x^{2}$

$6 x^{3}$

$6 x^{2}$

$13 x^{2}$

$3 x$

$13 x^{2}$

$13 x$

$10 x$

$10$

$10 x$

$10$

$0$

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

$x^{3} + 6 x^{2} + 13 x + 10$

Write $x^{4} + 5 x^{3} + 7 x^{2} - 3 x - 10$ as a set of factors.

$(x - 1) (x^{3} + 6 x^{2} + 13 x + 10) = 0$

Factor $x^{3} + 6 x^{2} + 13 x + 10$ 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.

$p = \pm 1, \pm 10, \pm 2, \pm 5 q = \pm 1$

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

$\pm 1, \pm 10, \pm 2, \pm 5$

Substitute $- 2$ and simplify the expression. In this case, the expression is equal to $0$ so $- 2$ is a root of the polynomial.

Substitute $- 2$ into the polynomial.

${(- 2)}^{3} + 6 {(- 2)}^{2} + 13 \cdot - 2 + 10$

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

$- 8 + 6 {(- 2)}^{2} + 13 \cdot - 2 + 10$

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

$- 8 + 6 \cdot 4 + 13 \cdot - 2 + 10$

Multiply $6$ by $4$ .

$- 8 + 24 + 13 \cdot - 2 + 10$

Add $- 8$ and $24$ .

$16 + 13 \cdot - 2 + 10$

Multiply $13$ by $- 2$ .

$16 - 26 + 10$

Subtract $26$ from $16$ .

$- 10 + 10$

Add $- 10$ and $10$ .

$0$

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.

$\frac{x^{3} + 6 x^{2} + 13 x + 10}{x + 2}$

Divide $x^{3} + 6 x^{2} + 13 x + 10$ by $x + 2$ .

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

$x$

$2$

$x^{3}$

$6 x^{2}$

$13 x$

$10$

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

					$x^{2}$
	$x$	+	$2$		$x^{3}$	+	$6 x^{2}$	+	$13 x$	+	$10$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$2$		$x^{3}$	+	$6 x^{2}$	+	$13 x$	+	$10$
			+	$x^{3}$	+	$2 x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	+	$6 x^{2}$	+	$13 x$	+	$10$
			-	$x^{3}$	-	$2 x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	+	$6 x^{2}$	+	$13 x$	+	$10$
			-	$x^{3}$	-	$2 x^{2}$
					+	$4 x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	+	$6 x^{2}$	+	$13 x$	+	$10$
			-	$x^{3}$	-	$2 x^{2}$
					+	$4 x^{2}$	+	$13 x$

Divide the highest order term in the dividend $4 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$4 x$
$x$	+	$2$		$x^{3}$	+	$6 x^{2}$	+	$13 x$	+	$10$
			-	$x^{3}$	-	$2 x^{2}$
					+	$4 x^{2}$	+	$13 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$4 x$
$x$	+	$2$		$x^{3}$	+	$6 x^{2}$	+	$13 x$	+	$10$
			-	$x^{3}$	-	$2 x^{2}$
					+	$4 x^{2}$	+	$13 x$
					+	$4 x^{2}$	+	$8 x$

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{2} + 8 x$

				$x^{2}$	+	$4 x$
$x$	+	$2$		$x^{3}$	+	$6 x^{2}$	+	$13 x$	+	$10$
			-	$x^{3}$	-	$2 x^{2}$
					+	$4 x^{2}$	+	$13 x$
					-	$4 x^{2}$	-	$8 x$

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

				$x^{2}$	+	$4 x$
$x$	+	$2$		$x^{3}$	+	$6 x^{2}$	+	$13 x$	+	$10$
			-	$x^{3}$	-	$2 x^{2}$
					+	$4 x^{2}$	+	$13 x$
					-	$4 x^{2}$	-	$8 x$
							+	$5 x$

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

				$x^{2}$	+	$4 x$
$x$	+	$2$		$x^{3}$	+	$6 x^{2}$	+	$13 x$	+	$10$
			-	$x^{3}$	-	$2 x^{2}$
					+	$4 x^{2}$	+	$13 x$
					-	$4 x^{2}$	-	$8 x$
							+	$5 x$	+	$10$

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

				$x^{2}$	+	$4 x$	+	$5$
$x$	+	$2$		$x^{3}$	+	$6 x^{2}$	+	$13 x$	+	$10$
			-	$x^{3}$	-	$2 x^{2}$
					+	$4 x^{2}$	+	$13 x$
					-	$4 x^{2}$	-	$8 x$
							+	$5 x$	+	$10$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$4 x$	+	$5$
$x$	+	$2$		$x^{3}$	+	$6 x^{2}$	+	$13 x$	+	$10$
			-	$x^{3}$	-	$2 x^{2}$
					+	$4 x^{2}$	+	$13 x$
					-	$4 x^{2}$	-	$8 x$
							+	$5 x$	+	$10$
							+	$5 x$	+	$10$

The expression needs to be subtracted from the dividend, so change all the signs in $5 x + 10$

				$x^{2}$	+	$4 x$	+	$5$
$x$	+	$2$		$x^{3}$	+	$6 x^{2}$	+	$13 x$	+	$10$
			-	$x^{3}$	-	$2 x^{2}$
					+	$4 x^{2}$	+	$13 x$
					-	$4 x^{2}$	-	$8 x$
							+	$5 x$	+	$10$
							-	$5 x$	-	$10$

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

				$x^{2}$	+	$4 x$	+	$5$
$x$	+	$2$		$x^{3}$	+	$6 x^{2}$	+	$13 x$	+	$10$
			-	$x^{3}$	-	$2 x^{2}$
					+	$4 x^{2}$	+	$13 x$
					-	$4 x^{2}$	-	$8 x$
							+	$5 x$	+	$10$
							-	$5 x$	-	$10$
										$0$

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

$x^{2} + 4 x + 5$

Write $x^{3} + 6 x^{2} + 13 x + 10$ as a set of factors.

$(x - 1) ((x + 2) (x^{2} + 4 x + 5)) = 0$

Remove unnecessary parentheses.

$(x - 1) (x + 2) (x^{2} + 4 x + 5) = 0$

Step 2

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

$x - 1 = 0$

$x + 2 = 0$

$x^{2} + 4 x + 5 = 0$

Step 3

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Step 4

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5

Set $x^{2} + 4 x + 5$ equal to $0$ and solve for $x$ .

Set $x^{2} + 4 x + 5$ equal to $0$ .

$x^{2} + 4 x + 5 = 0$

Solve $x^{2} + 4 x + 5 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = 4$ , and $c = 5$ into the quadratic formula and solve for $x$ .

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

Simplify.

Simplify the numerator.

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

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

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 5}}{2 \cdot 1}$

Multiply $- 4$ by $5$ .

$x = \frac{- 4 \pm \sqrt{16 - 20}}{2 \cdot 1}$

Subtract $20$ from $16$ .

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

Rewrite $- 4$ as $- 1 (4)$ .

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

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

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

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

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

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

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

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

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

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

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 2 i}{2}$

Simplify $\frac{- 4 \pm 2 i}{2}$ .

$x = - 2 \pm i$

The final answer is the combination of both solutions.

$x = - 2 + i, - 2 - i$

Step 6

The final solution is all the values that make $(x - 1) (x + 2) (x^{2} + 4 x + 5) = 0$ true.

$x = 1, - 2, - 2 + i, - 2 - i$

Step 7

Find the Roots (Zeros) x^4+5x^3+7x^2-3x-10=0

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