Find the Roots (Zeros) x^4+2x^3-23x^2-24x+144=0

$x^{4} + 2 x^{3} - 23 x^{2} - 24 x + 144 = 0$

Step 1

Factor the left side of the equation.

Factor $x^{4} + 2 x^{3} - 23 x^{2} - 24 x + 144$ 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.

$p = \pm 1, \pm 144, \pm 2, \pm 72, \pm 3, \pm 48, \pm 4, \pm 36, \pm 6, \pm 24, \pm 8, \pm 18, \pm 9, \pm 16, \pm 12 q = \pm 1$

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

$\pm 1, \pm 144, \pm 2, \pm 72, \pm 3, \pm 48, \pm 4, \pm 36, \pm 6, \pm 24, \pm 8, \pm 18, \pm 9, \pm 16, \pm 12$

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.

$3^{4} + 2 \cdot 3^{3} - 23 \cdot 3^{2} - 24 \cdot 3 + 144$

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

$81 + 2 \cdot 3^{3} - 23 \cdot 3^{2} - 24 \cdot 3 + 144$

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

$81 + 2 \cdot 27 - 23 \cdot 3^{2} - 24 \cdot 3 + 144$

Multiply $2$ by $27$ .

$81 + 54 - 23 \cdot 3^{2} - 24 \cdot 3 + 144$

Add $81$ and $54$ .

$135 - 23 \cdot 3^{2} - 24 \cdot 3 + 144$

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

$135 - 23 \cdot 9 - 24 \cdot 3 + 144$

Multiply $- 23$ by $9$ .

$135 - 207 - 24 \cdot 3 + 144$

Subtract $207$ from $135$ .

$- 72 - 24 \cdot 3 + 144$

Multiply $- 24$ by $3$ .

$- 72 - 72 + 144$

Subtract $72$ from $- 72$ .

$- 144 + 144$

Add $- 144$ and $144$ .

$0$

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.

$\frac{x^{4} + 2 x^{3} - 23 x^{2} - 24 x + 144}{x - 3}$

Divide $x^{4} + 2 x^{3} - 23 x^{2} - 24 x + 144$ 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 $0$ .

$x$

$3$

$x^{4}$

$2 x^{3}$

$23 x^{2}$

$24 x$

$144$

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

$x^{3}$

$x$

$3$

$x^{4}$

$2 x^{3}$

$23 x^{2}$

$24 x$

$144$

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$3$

$x^{4}$

$2 x^{3}$

$23 x^{2}$

$24 x$

$144$

$x^{4}$

$3 x^{3}$

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

$x^{3}$

$x$

$3$

$x^{4}$

$2 x^{3}$

$23 x^{2}$

$24 x$

$144$

$x^{4}$

$3 x^{3}$

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

$x^{3}$

$x$

$3$

$x^{4}$

$2 x^{3}$

$23 x^{2}$

$24 x$

$144$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

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

$x^{3}$

$x$

$3$

$x^{4}$

$2 x^{3}$

$23 x^{2}$

$24 x$

$144$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$23 x^{2}$

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

$x^{3}$

$5 x^{2}$

$x$

$3$

$x^{4}$

$2 x^{3}$

$23 x^{2}$

$24 x$

$144$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$23 x^{2}$

Multiply the new quotient term by the divisor.

$x^{3}$

$5 x^{2}$

$x$

$3$

$x^{4}$

$2 x^{3}$

$23 x^{2}$

$24 x$

$144$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$23 x^{2}$

$5 x^{3}$

$15 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{3} - 15 x^{2}$

$x^{3}$

$5 x^{2}$

$x$

$3$

$x^{4}$

$2 x^{3}$

$23 x^{2}$

$24 x$

$144$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$23 x^{2}$

$5 x^{3}$

$15 x^{2}$

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

$x^{3}$

$5 x^{2}$

$x$

$3$

$x^{4}$

$2 x^{3}$

$23 x^{2}$

$24 x$

$144$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$23 x^{2}$

$5 x^{3}$

$15 x^{2}$

$8 x^{2}$

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

$x^{3}$

$5 x^{2}$

$x$

$3$

$x^{4}$

$2 x^{3}$

$23 x^{2}$

$24 x$

$144$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$23 x^{2}$

$5 x^{3}$

$15 x^{2}$

$8 x^{2}$

$24 x$

Divide the highest order term in the dividend $- 8 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$5 x^{2}$

$8 x$

$x$

$3$

$x^{4}$

$2 x^{3}$

$23 x^{2}$

$24 x$

$144$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$23 x^{2}$

$5 x^{3}$

$15 x^{2}$

$8 x^{2}$

$24 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$5 x^{2}$

$8 x$

$x$

$3$

$x^{4}$

$2 x^{3}$

$23 x^{2}$

$24 x$

$144$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$23 x^{2}$

$5 x^{3}$

$15 x^{2}$

$8 x^{2}$

$24 x$

$8 x^{2}$

$24 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 x^{2} + 24 x$

$x^{3}$

$5 x^{2}$

$8 x$

$x$

$3$

$x^{4}$

$2 x^{3}$

$23 x^{2}$

$24 x$

$144$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$23 x^{2}$

$5 x^{3}$

$15 x^{2}$

$8 x^{2}$

$24 x$

$8 x^{2}$

$24 x$

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

$x^{3}$

$5 x^{2}$

$8 x$

$x$

$3$

$x^{4}$

$2 x^{3}$

$23 x^{2}$

$24 x$

$144$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$23 x^{2}$

$5 x^{3}$

$15 x^{2}$

$8 x^{2}$

$24 x$

$8 x^{2}$

$24 x$

$48 x$

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

$x^{3}$

$5 x^{2}$

$8 x$

$x$

$3$

$x^{4}$

$2 x^{3}$

$23 x^{2}$

$24 x$

$144$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$23 x^{2}$

$5 x^{3}$

$15 x^{2}$

$8 x^{2}$

$24 x$

$8 x^{2}$

$24 x$

$48 x$

$144$

Divide the highest order term in the dividend $- 48 x$ by the highest order term in divisor $x$ .

$x^{3}$

$5 x^{2}$

$8 x$

$48$

$x$

$3$

$x^{4}$

$2 x^{3}$

$23 x^{2}$

$24 x$

$144$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$23 x^{2}$

$5 x^{3}$

$15 x^{2}$

$8 x^{2}$

$24 x$

$8 x^{2}$

$24 x$

$48 x$

$144$

Multiply the new quotient term by the divisor.

$x^{3}$

$5 x^{2}$

$8 x$

$48$

$x$

$3$

$x^{4}$

$2 x^{3}$

$23 x^{2}$

$24 x$

$144$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$23 x^{2}$

$5 x^{3}$

$15 x^{2}$

$8 x^{2}$

$24 x$

$8 x^{2}$

$24 x$

$48 x$

$144$

$48 x$

$144$

The expression needs to be subtracted from the dividend, so change all the signs in $- 48 x + 144$

$x^{3}$

$5 x^{2}$

$8 x$

$48$

$x$

$3$

$x^{4}$

$2 x^{3}$

$23 x^{2}$

$24 x$

$144$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$23 x^{2}$

$5 x^{3}$

$15 x^{2}$

$8 x^{2}$

$24 x$

$8 x^{2}$

$24 x$

$48 x$

$144$

$48 x$

$144$

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

$x^{3}$

$5 x^{2}$

$8 x$

$48$

$x$

$3$

$x^{4}$

$2 x^{3}$

$23 x^{2}$

$24 x$

$144$

$x^{4}$

$3 x^{3}$

$5 x^{3}$

$23 x^{2}$

$5 x^{3}$

$15 x^{2}$

$8 x^{2}$

$24 x$

$8 x^{2}$

$24 x$

$48 x$

$144$

$48 x$

$144$

$0$

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

$x^{3} + 5 x^{2} - 8 x - 48$

Write $x^{4} + 2 x^{3} - 23 x^{2} - 24 x + 144$ as a set of factors.

$(x - 3) (x^{3} + 5 x^{2} - 8 x - 48) = 0$

Factor $x^{3} + 5 x^{2} - 8 x - 48$ 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.

$p = \pm 1, \pm 48, \pm 2, \pm 24, \pm 3, \pm 16, \pm 4, \pm 12, \pm 6, \pm 8 q = \pm 1$

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

$\pm 1, \pm 48, \pm 2, \pm 24, \pm 3, \pm 16, \pm 4, \pm 12, \pm 6, \pm 8$

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.

$3^{3} + 5 \cdot 3^{2} - 8 \cdot 3 - 48$

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

$27 + 5 \cdot 3^{2} - 8 \cdot 3 - 48$

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

$27 + 5 \cdot 9 - 8 \cdot 3 - 48$

Multiply $5$ by $9$ .

$27 + 45 - 8 \cdot 3 - 48$

Add $27$ and $45$ .

$72 - 8 \cdot 3 - 48$

Multiply $- 8$ by $3$ .

$72 - 24 - 48$

Subtract $24$ from $72$ .

$48 - 48$

Subtract $48$ from $48$ .

$0$

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.

$\frac{x^{3} + 5 x^{2} - 8 x - 48}{x - 3}$

Divide $x^{3} + 5 x^{2} - 8 x - 48$ 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 $0$ .

$x$

$3$

$x^{3}$

$5 x^{2}$

$8 x$

$48$

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

					$x^{2}$
	$x$	-	$3$		$x^{3}$	+	$5 x^{2}$	-	$8 x$	-	$48$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$3$		$x^{3}$	+	$5 x^{2}$	-	$8 x$	-	$48$
			+	$x^{3}$	-	$3 x^{2}$

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	+	$5 x^{2}$	-	$8 x$	-	$48$
			-	$x^{3}$	+	$3 x^{2}$

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	+	$5 x^{2}$	-	$8 x$	-	$48$
			-	$x^{3}$	+	$3 x^{2}$
					+	$8 x^{2}$

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	+	$5 x^{2}$	-	$8 x$	-	$48$
			-	$x^{3}$	+	$3 x^{2}$
					+	$8 x^{2}$	-	$8 x$

Divide the highest order term in the dividend $8 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$8 x$
$x$	-	$3$		$x^{3}$	+	$5 x^{2}$	-	$8 x$	-	$48$
			-	$x^{3}$	+	$3 x^{2}$
					+	$8 x^{2}$	-	$8 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$8 x$
$x$	-	$3$		$x^{3}$	+	$5 x^{2}$	-	$8 x$	-	$48$
			-	$x^{3}$	+	$3 x^{2}$
					+	$8 x^{2}$	-	$8 x$
					+	$8 x^{2}$	-	$24 x$

The expression needs to be subtracted from the dividend, so change all the signs in $8 x^{2} - 24 x$

				$x^{2}$	+	$8 x$
$x$	-	$3$		$x^{3}$	+	$5 x^{2}$	-	$8 x$	-	$48$
			-	$x^{3}$	+	$3 x^{2}$
					+	$8 x^{2}$	-	$8 x$
					-	$8 x^{2}$	+	$24 x$

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

				$x^{2}$	+	$8 x$
$x$	-	$3$		$x^{3}$	+	$5 x^{2}$	-	$8 x$	-	$48$
			-	$x^{3}$	+	$3 x^{2}$
					+	$8 x^{2}$	-	$8 x$
					-	$8 x^{2}$	+	$24 x$
							+	$16 x$

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

				$x^{2}$	+	$8 x$
$x$	-	$3$		$x^{3}$	+	$5 x^{2}$	-	$8 x$	-	$48$
			-	$x^{3}$	+	$3 x^{2}$
					+	$8 x^{2}$	-	$8 x$
					-	$8 x^{2}$	+	$24 x$
							+	$16 x$	-	$48$

Divide the highest order term in the dividend $16 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$8 x$	+	$16$
$x$	-	$3$		$x^{3}$	+	$5 x^{2}$	-	$8 x$	-	$48$
			-	$x^{3}$	+	$3 x^{2}$
					+	$8 x^{2}$	-	$8 x$
					-	$8 x^{2}$	+	$24 x$
							+	$16 x$	-	$48$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$8 x$	+	$16$
$x$	-	$3$		$x^{3}$	+	$5 x^{2}$	-	$8 x$	-	$48$
			-	$x^{3}$	+	$3 x^{2}$
					+	$8 x^{2}$	-	$8 x$
					-	$8 x^{2}$	+	$24 x$
							+	$16 x$	-	$48$
							+	$16 x$	-	$48$

The expression needs to be subtracted from the dividend, so change all the signs in $16 x - 48$

				$x^{2}$	+	$8 x$	+	$16$
$x$	-	$3$		$x^{3}$	+	$5 x^{2}$	-	$8 x$	-	$48$
			-	$x^{3}$	+	$3 x^{2}$
					+	$8 x^{2}$	-	$8 x$
					-	$8 x^{2}$	+	$24 x$
							+	$16 x$	-	$48$
							-	$16 x$	+	$48$

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

				$x^{2}$	+	$8 x$	+	$16$
$x$	-	$3$		$x^{3}$	+	$5 x^{2}$	-	$8 x$	-	$48$
			-	$x^{3}$	+	$3 x^{2}$
					+	$8 x^{2}$	-	$8 x$
					-	$8 x^{2}$	+	$24 x$
							+	$16 x$	-	$48$
							-	$16 x$	+	$48$
										$0$

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

$x^{2} + 8 x + 16$

Write $x^{3} + 5 x^{2} - 8 x - 48$ as a set of factors.

$(x - 3) ((x - 3) (x^{2} + 8 x + 16)) = 0$

Factor using the perfect square rule.

Rewrite $16$ as $4^{2}$ .

$(x - 3) ((x - 3) (x^{2} + 8 x + 4^{2})) = 0$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 x = 2 \cdot x \cdot 4$

Rewrite the polynomial.

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

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 4$ .

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

Remove unnecessary parentheses.

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

Combine like factors.

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

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

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

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

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

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

Add $1$ and $1$ .

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

Step 2

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

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

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

Step 3

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.

$x = 3$

Step 4

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

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

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

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

Set the $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 5

The final solution is all the values that make ${(x - 3)}^{2} {(x + 4)}^{2} = 0$ true.

$x = 3, - 4$

Step 6

Find the Roots (Zeros) x^4+2x^3-23x^2-24x+144=0

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