Find the Roots (Zeros) f(x)=x^4-2x^3-5x^2+8x+4

$f (x) = x^{4} - 2 x^{3} - 5 x^{2} + 8 x + 4$

Step 1

Set $x^{4} - 2 x^{3} - 5 x^{2} + 8 x + 4$ equal to $0$ .

$x^{4} - 2 x^{3} - 5 x^{2} + 8 x + 4 = 0$

Step 2

Solve for $x$ .

Factor the left side of the equation.

Regroup terms.

$x^{4} - 2 x^{3} - 5 x^{2} + 8 x + 4 = 0$

Factor $x^{3}$ out of $x^{4} - 2 x^{3}$ .

Factor $x^{3}$ out of $x^{4}$ .

$x^{3} x - 2 x^{3} - 5 x^{2} + 8 x + 4 = 0$

Factor $x^{3}$ out of $- 2 x^{3}$ .

$x^{3} x + x^{3} \cdot - 2 - 5 x^{2} + 8 x + 4 = 0$

Factor $x^{3}$ out of $x^{3} x + x^{3} \cdot - 2$ .

$x^{3} (x - 2) - 5 x^{2} + 8 x + 4 = 0$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 5 \cdot 4 = - 20$ and whose sum is $b = 8$ .

Factor $8$ out of $8 x$ .

$x^{3} (x - 2) - 5 x^{2} + 8 (x) + 4 = 0$

Rewrite $8$ as $- 2$ plus $10$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

Factor out the greatest common factor (GCF) from each group.

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

Factor the polynomial by factoring out the greatest common factor, $- 5 x - 2$ .

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

Factor $x - 2$ out of $x^{3} (x - 2) + (- 5 x - 2) (x - 2)$ .

Factor $x - 2$ out of $x^{3} (x - 2)$ .

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

Factor $x - 2$ out of $(- 5 x - 2) (x - 2)$ .

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

Factor $x - 2$ out of $(x - 2) x^{3} + (x - 2) (- 5 x - 2)$ .

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

Factor.

Factor $x^{3} - 5 x - 2$ 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.

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

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

$\pm 1, \pm 2$

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.

${(- 2)}^{3} - 5 \cdot - 2 - 2$

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

$- 8 - 5 \cdot - 2 - 2$

Multiply $- 5$ by $- 2$ .

$- 8 + 10 - 2$

Add $- 8$ and $10$ .

$2 - 2$

Subtract $2$ from $2$ .

$0$

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.

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

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

$x$

$2$

$x^{3}$

$0 x^{2}$

$5 x$

$2$

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

					$x^{2}$
	$x$	+	$2$		$x^{3}$	+	$0 x^{2}$	-	$5 x$	-	$2$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$2$		$x^{3}$	+	$0 x^{2}$	-	$5 x$	-	$2$
			+	$x^{3}$	+	$2 x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	+	$0 x^{2}$	-	$5 x$	-	$2$
			-	$x^{3}$	-	$2 x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	+	$0 x^{2}$	-	$5 x$	-	$2$
			-	$x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	+	$0 x^{2}$	-	$5 x$	-	$2$
			-	$x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$5 x$

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

				$x^{2}$	-	$2 x$
$x$	+	$2$		$x^{3}$	+	$0 x^{2}$	-	$5 x$	-	$2$
			-	$x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$5 x$

Multiply the new quotient term by the divisor.

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

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

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

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

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

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

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

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

				$x^{2}$	-	$2 x$	-	$1$
$x$	+	$2$		$x^{3}$	+	$0 x^{2}$	-	$5 x$	-	$2$
			-	$x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$5 x$
					+	$2 x^{2}$	+	$4 x$
							-	$x$	-	$2$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$2 x$	-	$1$
$x$	+	$2$		$x^{3}$	+	$0 x^{2}$	-	$5 x$	-	$2$
			-	$x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$5 x$
					+	$2 x^{2}$	+	$4 x$
							-	$x$	-	$2$
							-	$x$	-	$2$

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

				$x^{2}$	-	$2 x$	-	$1$
$x$	+	$2$		$x^{3}$	+	$0 x^{2}$	-	$5 x$	-	$2$
			-	$x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$5 x$
					+	$2 x^{2}$	+	$4 x$
							-	$x$	-	$2$
							+	$x$	+	$2$

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

				$x^{2}$	-	$2 x$	-	$1$
$x$	+	$2$		$x^{3}$	+	$0 x^{2}$	-	$5 x$	-	$2$
			-	$x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$5 x$
					+	$2 x^{2}$	+	$4 x$
							-	$x$	-	$2$
							+	$x$	+	$2$
										$0$

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

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

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

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

Remove unnecessary parentheses.

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

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

$x - 2 = 0$

$x + 2 = 0$

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

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

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

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

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

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

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

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

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 2$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

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

Simplify.

Simplify the numerator.

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

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

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

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 1$ .

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

Add $4$ and $4$ .

$x = \frac{2 \pm \sqrt{8}}{2 \cdot 1}$

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $1$ .

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

Simplify $\frac{2 \pm 2 \sqrt{2}}{2}$ .

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

The final answer is the combination of both solutions.

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

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

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 2, - 2, 2.41421356 \dots, - 0.41421356 \dots$

Step 4

Find the Roots (Zeros) f(x)=x^4-2x^3-5x^2+8x+4

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