Find the Roots (Zeros) f(x)=x^3-4x^2-20x+48

$f (x) = x^{3} - 4 x^{2} - 20 x + 48$

Set $x^{3} - 4 x^{2} - 20 x + 48$ equal to $0$ .

$x^{3} - 4 x^{2} - 20 x + 48 = 0$

Solve for $x$ .

Factor the left side of the equation.

Factor $x^{3} - 4 x^{2} - 20 x + 48$ 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 $2$ 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.

$2^{3} - 4 \cdot 2^{2} - 20 \cdot 2 + 48$

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

$8 - 4 \cdot 2^{2} - 20 \cdot 2 + 48$

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

$8 - 4 \cdot 4 - 20 \cdot 2 + 48$

Multiply $- 4$ by $4$ .

$8 - 16 - 20 \cdot 2 + 48$

Subtract $16$ from $8$ .

$- 8 - 20 \cdot 2 + 48$

Multiply $- 20$ by $2$ .

$- 8 - 40 + 48$

Subtract $40$ from $- 8$ .

$- 48 + 48$

Add $- 48$ and $48$ .

$0$

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.

$\frac{x^{3} - 4 x^{2} - 20 x + 48}{x - 2}$

Divide $x^{3} - 4 x^{2} - 20 x + 48$ 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 $0$ .

$x$

$2$

$x^{3}$

$4 x^{2}$

$20 x$

$48$

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

					$x^{2}$
	$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$20 x$	+	$48$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$20 x$	+	$48$
			+	$x^{3}$	-	$2 x^{2}$

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$20 x$	+	$48$
			-	$x^{3}$	+	$2 x^{2}$

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$20 x$	+	$48$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$20 x$	+	$48$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	-	$20 x$

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

				$x^{2}$	-	$2 x$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$20 x$	+	$48$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	-	$20 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$2 x$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$20 x$	+	$48$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	-	$20 x$
					-	$2 x^{2}$	+	$4 x$

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

				$x^{2}$	-	$2 x$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$20 x$	+	$48$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	-	$20 x$
					+	$2 x^{2}$	-	$4 x$

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

				$x^{2}$	-	$2 x$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$20 x$	+	$48$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	-	$20 x$
					+	$2 x^{2}$	-	$4 x$
							-	$24 x$

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

				$x^{2}$	-	$2 x$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$20 x$	+	$48$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	-	$20 x$
					+	$2 x^{2}$	-	$4 x$
							-	$24 x$	+	$48$

Divide the highest order term in the dividend $- 24 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$2 x$	-	$24$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$20 x$	+	$48$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	-	$20 x$
					+	$2 x^{2}$	-	$4 x$
							-	$24 x$	+	$48$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$2 x$	-	$24$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$20 x$	+	$48$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	-	$20 x$
					+	$2 x^{2}$	-	$4 x$
							-	$24 x$	+	$48$
							-	$24 x$	+	$48$

The expression needs to be subtracted from the dividend, so change all the signs in $- 24 x + 48$

				$x^{2}$	-	$2 x$	-	$24$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$20 x$	+	$48$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	-	$20 x$
					+	$2 x^{2}$	-	$4 x$
							-	$24 x$	+	$48$
							+	$24 x$	-	$48$

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

				$x^{2}$	-	$2 x$	-	$24$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	-	$20 x$	+	$48$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	-	$20 x$
					+	$2 x^{2}$	-	$4 x$
							-	$24 x$	+	$48$
							+	$24 x$	-	$48$
										$0$

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

$x^{2} - 2 x - 24$

Write $x^{3} - 4 x^{2} - 20 x + 48$ as a set of factors.

$(x - 2) (x^{2} - 2 x - 24) = 0$

Factor $x^{2} - 2 x - 24$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 24$ and whose sum is $- 2$ .

$- 6, 4$

Write the factored form using these integers.

$(x - 2) ((x - 6) (x + 4)) = 0$

Remove unnecessary parentheses.

$(x - 2) (x - 6) (x + 4) = 0$

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

$x - 2 = 0$

$x - 6 = 0$

$x + 4 = 0$

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Set $x - 6$ equal to $0$ and solve for $x$ .

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

Set $x + 4$ equal to $0$ and solve for $x$ .

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

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

The final solution is all the values that make $(x - 2) (x - 6) (x + 4) = 0$ true.

$x = 2, 6, - 4$

Find the Roots (Zeros) f(x)=x^3-4x^2-20x+48

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