Find the Roots (Zeros) 4x^5-8x^4-5x^3+10x^2+x-2

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

Step 1

Set $4 x^{5} - 8 x^{4} - 5 x^{3} + 10 x^{2} + x - 2$ equal to $0$ .

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

Step 2

Solve for $x$ .

Factor the left side of the equation.

Regroup terms.

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

Factor $x^{3}$ out of $4 x^{5} - 8 x^{4} - 5 x^{3}$ .

Factor $x^{3}$ out of $4 x^{5}$ .

$x^{3} (4 x^{2}) - 8 x^{4} - 5 x^{3} + 10 x^{2} + x - 2 = 0$

Factor $x^{3}$ out of $- 8 x^{4}$ .

$x^{3} (4 x^{2}) + x^{3} (- 8 x) - 5 x^{3} + 10 x^{2} + x - 2 = 0$

Factor $x^{3}$ out of $- 5 x^{3}$ .

$x^{3} (4 x^{2}) + x^{3} (- 8 x) + x^{3} \cdot - 5 + 10 x^{2} + x - 2 = 0$

Factor $x^{3}$ out of $x^{3} (4 x^{2}) + x^{3} (- 8 x)$ .

$x^{3} (4 x^{2} - 8 x) + x^{3} \cdot - 5 + 10 x^{2} + x - 2 = 0$

Factor $x^{3}$ out of $x^{3} (4 x^{2} - 8 x) + x^{3} \cdot - 5$ .

$x^{3} (4 x^{2} - 8 x - 5) + 10 x^{2} + x - 2 = 0$

Factor.

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 $a \cdot c = 4 \cdot - 5 = - 20$ and whose sum is $b = - 8$ .

Factor $- 8$ out of $- 8 x$ .

$x^{3} (4 x^{2} - 8 x - 5) + 10 x^{2} + x - 2 = 0$

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

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

$x^{3} (2 x (2 x + 1) - 5 (2 x + 1)) + 10 x^{2} + x - 2 = 0$

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$x^{3} ((2 x + 1) (2 x - 5)) + 10 x^{2} + x - 2 = 0$

Remove unnecessary parentheses.

$x^{3} (2 x + 1) (2 x - 5) + 10 x^{2} + x - 2 = 0$

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 $a \cdot c = 10 \cdot - 2 = - 20$ and whose sum is $b = 1$ .

Multiply by $1$ .

$x^{3} (2 x + 1) (2 x - 5) + 10 x^{2} + 1 x - 2 = 0$

Rewrite $1$ as $- 4$ plus $5$

$x^{3} (2 x + 1) (2 x - 5) + 10 x^{2} + (- 4 + 5) x - 2 = 0$

Apply the distributive property.

$x^{3} (2 x + 1) (2 x - 5) + 10 x^{2} - 4 x + 5 x - 2 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$x^{3} (2 x + 1) (2 x - 5) + 10 x^{2} - 4 x + 5 x - 2 = 0$

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

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

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

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

Factor $2 x + 1$ out of $x^{3} (2 x + 1) (2 x - 5) + (5 x - 2) (2 x + 1)$ .

Factor $2 x + 1$ out of $x^{3} (2 x + 1) (2 x - 5)$ .

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

Factor $2 x + 1$ out of $(5 x - 2) (2 x + 1)$ .

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

Factor $2 x + 1$ out of $(2 x + 1) (x^{3} (2 x - 5)) + (2 x + 1) (5 x - 2)$ .

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

Apply the distributive property.

$(2 x + 1) (x^{3} (2 x) + x^{3} \cdot - 5 + 5 x - 2) = 0$

Rewrite using the commutative property of multiplication.

$(2 x + 1) (2 x^{3} x + x^{3} \cdot - 5 + 5 x - 2) = 0$

Move $- 5$ to the left of $x^{3}$ .

$(2 x + 1) (2 x^{3} x - 5 \cdot x^{3} + 5 x - 2) = 0$

Multiply $x^{3}$ by $x$ by adding the exponents.

Move $x$ .

$(2 x + 1) (2 (x \cdot x^{3}) - 5 \cdot x^{3} + 5 x - 2) = 0$

Multiply $x$ by $x^{3}$ .

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

$(2 x + 1) (2 (x \cdot x^{3}) - 5 \cdot x^{3} + 5 x - 2) = 0$

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

$(2 x + 1) (2 x^{1 + 3} - 5 \cdot x^{3} + 5 x - 2) = 0$

Add $1$ and $3$ .

$(2 x + 1) (2 x^{4} - 5 \cdot x^{3} + 5 x - 2) = 0$

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

Factor.

Rewrite $2 x^{4} - 5 x^{3} + 5 x - 2$ in a factored form.

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

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

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

$\pm 1, \pm 0.5, \pm 2$

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.

$2 {(- 1)}^{4} - 5 {(- 1)}^{3} + 5 \cdot - 1 - 2$

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

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

Multiply $2$ by $1$ .

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

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

$2 - 5 \cdot - 1 + 5 \cdot - 1 - 2$

Multiply $- 5$ by $- 1$ .

$2 + 5 + 5 \cdot - 1 - 2$

Add $2$ and $5$ .

$7 + 5 \cdot - 1 - 2$

Multiply $5$ by $- 1$ .

$7 - 5 - 2$

Subtract $5$ from $7$ .

$2 - 2$

Subtract $2$ from $2$ .

$0$

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.

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

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

$x$

$1$

$2 x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x$

$2$

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

$2 x^{3}$

$x$

$1$

$2 x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x$

$2$

Multiply the new quotient term by the divisor.

$2 x^{3}$

$x$

$1$

$2 x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x$

$2$

$2 x^{4}$

$2 x^{3}$

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{4} + 2 x^{3}$

$2 x^{3}$

$x$

$1$

$2 x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x$

$2$

$2 x^{4}$

$2 x^{3}$

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

$2 x^{3}$

$x$

$1$

$2 x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x$

$2$

$2 x^{4}$

$2 x^{3}$

$7 x^{3}$

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

$2 x^{3}$

$x$

$1$

$2 x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x$

$2$

$2 x^{4}$

$2 x^{3}$

$7 x^{3}$

$0 x^{2}$

Divide the highest order term in the dividend $- 7 x^{3}$ by the highest order term in divisor $x$ .

$2 x^{3}$

$7 x^{2}$

$x$

$1$

$2 x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x$

$2$

$2 x^{4}$

$2 x^{3}$

$7 x^{3}$

$0 x^{2}$

Multiply the new quotient term by the divisor.

$2 x^{3}$

$7 x^{2}$

$x$

$1$

$2 x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x$

$2$

$2 x^{4}$

$2 x^{3}$

$7 x^{3}$

$0 x^{2}$

$7 x^{3}$

$7 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 7 x^{3} - 7 x^{2}$

$2 x^{3}$

$7 x^{2}$

$x$

$1$

$2 x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x$

$2$

$2 x^{4}$

$2 x^{3}$

$7 x^{3}$

$0 x^{2}$

$7 x^{3}$

$7 x^{2}$

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

$2 x^{3}$

$7 x^{2}$

$x$

$1$

$2 x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x$

$2$

$2 x^{4}$

$2 x^{3}$

$7 x^{3}$

$0 x^{2}$

$7 x^{3}$

$7 x^{2}$

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

$2 x^{3}$

$7 x^{2}$

$x$

$1$

$2 x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x$

$2$

$2 x^{4}$

$2 x^{3}$

$7 x^{3}$

$0 x^{2}$

$7 x^{3}$

$7 x^{2}$

$5 x$

Divide the highest order term in the dividend $7 x^{2}$ by the highest order term in divisor $x$ .

$2 x^{3}$

$7 x^{2}$

$7 x$

$x$

$1$

$2 x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x$

$2$

$2 x^{4}$

$2 x^{3}$

$7 x^{3}$

$0 x^{2}$

$7 x^{3}$

$7 x^{2}$

$5 x$

Multiply the new quotient term by the divisor.

$2 x^{3}$

$7 x^{2}$

$7 x$

$x$

$1$

$2 x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x$

$2$

$2 x^{4}$

$2 x^{3}$

$7 x^{3}$

$0 x^{2}$

$7 x^{3}$

$7 x^{2}$

$5 x$

$7 x^{2}$

$7 x$

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{2} + 7 x$

$2 x^{3}$

$7 x^{2}$

$7 x$

$x$

$1$

$2 x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x$

$2$

$2 x^{4}$

$2 x^{3}$

$7 x^{3}$

$0 x^{2}$

$7 x^{3}$

$7 x^{2}$

$5 x$

$7 x^{2}$

$7 x$

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

$2 x^{3}$

$7 x^{2}$

$7 x$

$x$

$1$

$2 x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x$

$2$

$2 x^{4}$

$2 x^{3}$

$7 x^{3}$

$0 x^{2}$

$7 x^{3}$

$7 x^{2}$

$5 x$

$7 x^{2}$

$7 x$

$2 x$

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

$2 x^{3}$

$7 x^{2}$

$7 x$

$x$

$1$

$2 x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x$

$2$

$2 x^{4}$

$2 x^{3}$

$7 x^{3}$

$0 x^{2}$

$7 x^{3}$

$7 x^{2}$

$5 x$

$7 x^{2}$

$7 x$

$2 x$

$2$

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

$2 x^{3}$

$7 x^{2}$

$7 x$

$2$

$x$

$1$

$2 x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x$

$2$

$2 x^{4}$

$2 x^{3}$

$7 x^{3}$

$0 x^{2}$

$7 x^{3}$

$7 x^{2}$

$5 x$

$7 x^{2}$

$7 x$

$2 x$

$2$

Multiply the new quotient term by the divisor.

$2 x^{3}$

$7 x^{2}$

$7 x$

$2$

$x$

$1$

$2 x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x$

$2$

$2 x^{4}$

$2 x^{3}$

$7 x^{3}$

$0 x^{2}$

$7 x^{3}$

$7 x^{2}$

$5 x$

$7 x^{2}$

$7 x$

$2 x$

$2$

$2 x$

$2$

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

$2 x^{3}$

$7 x^{2}$

$7 x$

$2$

$x$

$1$

$2 x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x$

$2$

$2 x^{4}$

$2 x^{3}$

$7 x^{3}$

$0 x^{2}$

$7 x^{3}$

$7 x^{2}$

$5 x$

$7 x^{2}$

$7 x$

$2 x$

$2$

$2 x$

$2$

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

$2 x^{3}$

$7 x^{2}$

$7 x$

$2$

$x$

$1$

$2 x^{4}$

$5 x^{3}$

$0 x^{2}$

$5 x$

$2$

$2 x^{4}$

$2 x^{3}$

$7 x^{3}$

$0 x^{2}$

$7 x^{3}$

$7 x^{2}$

$5 x$

$7 x^{2}$

$7 x$

$2 x$

$2$

$2 x$

$2$

$0$

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

$2 x^{3} - 7 x^{2} + 7 x - 2$

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

$(2 x + 1) ((x + 1) (2 x^{3} - 7 x^{2} + 7 x - 2)) = 0$

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

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

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

$\pm 1, \pm 0.5, \pm 2$

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.

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

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

$2 \cdot 1 - 7 \cdot 1^{2} + 7 \cdot 1 - 2$

Multiply $2$ by $1$ .

$2 - 7 \cdot 1^{2} + 7 \cdot 1 - 2$

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

$2 - 7 \cdot 1 + 7 \cdot 1 - 2$

Multiply $- 7$ by $1$ .

$2 - 7 + 7 \cdot 1 - 2$

Subtract $7$ from $2$ .

$- 5 + 7 \cdot 1 - 2$

Multiply $7$ by $1$ .

$- 5 + 7 - 2$

Add $- 5$ and $7$ .

$2 - 2$

Subtract $2$ from $2$ .

$0$

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.

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

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

$x$

$1$

$2 x^{3}$

$7 x^{2}$

$7 x$

$2$

Divide the highest order term in the dividend $2 x^{3}$ by the highest order term in divisor $x$ .

					$2 x^{2}$
	$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$7 x$	-	$2$

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$7 x$	-	$2$
			+	$2 x^{3}$	-	$2 x^{2}$

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

				$2 x^{2}$
$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$7 x$	-	$2$
			-	$2 x^{3}$	+	$2 x^{2}$

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

				$2 x^{2}$
$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$7 x$	-	$2$
			-	$2 x^{3}$	+	$2 x^{2}$
					-	$5 x^{2}$

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

				$2 x^{2}$
$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$7 x$	-	$2$
			-	$2 x^{3}$	+	$2 x^{2}$
					-	$5 x^{2}$	+	$7 x$

Divide the highest order term in the dividend $- 5 x^{2}$ by the highest order term in divisor $x$ .

				$2 x^{2}$	-	$5 x$
$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$7 x$	-	$2$
			-	$2 x^{3}$	+	$2 x^{2}$
					-	$5 x^{2}$	+	$7 x$

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$5 x$
$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$7 x$	-	$2$
			-	$2 x^{3}$	+	$2 x^{2}$
					-	$5 x^{2}$	+	$7 x$
					-	$5 x^{2}$	+	$5 x$

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

				$2 x^{2}$	-	$5 x$
$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$7 x$	-	$2$
			-	$2 x^{3}$	+	$2 x^{2}$
					-	$5 x^{2}$	+	$7 x$
					+	$5 x^{2}$	-	$5 x$

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

				$2 x^{2}$	-	$5 x$
$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$7 x$	-	$2$
			-	$2 x^{3}$	+	$2 x^{2}$
					-	$5 x^{2}$	+	$7 x$
					+	$5 x^{2}$	-	$5 x$
							+	$2 x$

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

				$2 x^{2}$	-	$5 x$
$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$7 x$	-	$2$
			-	$2 x^{3}$	+	$2 x^{2}$
					-	$5 x^{2}$	+	$7 x$
					+	$5 x^{2}$	-	$5 x$
							+	$2 x$	-	$2$

Divide the highest order term in the dividend $2 x$ by the highest order term in divisor $x$ .

				$2 x^{2}$	-	$5 x$	+	$2$
$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$7 x$	-	$2$
			-	$2 x^{3}$	+	$2 x^{2}$
					-	$5 x^{2}$	+	$7 x$
					+	$5 x^{2}$	-	$5 x$
							+	$2 x$	-	$2$

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$5 x$	+	$2$
$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$7 x$	-	$2$
			-	$2 x^{3}$	+	$2 x^{2}$
					-	$5 x^{2}$	+	$7 x$
					+	$5 x^{2}$	-	$5 x$
							+	$2 x$	-	$2$
							+	$2 x$	-	$2$

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

				$2 x^{2}$	-	$5 x$	+	$2$
$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$7 x$	-	$2$
			-	$2 x^{3}$	+	$2 x^{2}$
					-	$5 x^{2}$	+	$7 x$
					+	$5 x^{2}$	-	$5 x$
							+	$2 x$	-	$2$
							-	$2 x$	+	$2$

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

				$2 x^{2}$	-	$5 x$	+	$2$
$x$	-	$1$		$2 x^{3}$	-	$7 x^{2}$	+	$7 x$	-	$2$
			-	$2 x^{3}$	+	$2 x^{2}$
					-	$5 x^{2}$	+	$7 x$
					+	$5 x^{2}$	-	$5 x$
							+	$2 x$	-	$2$
							-	$2 x$	+	$2$
										$0$

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

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

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

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

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 $a \cdot c = 2 \cdot 2 = 4$ and whose sum is $b = - 5$ .

Factor $- 5$ out of $- 5 x$ .

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

Rewrite $- 5$ as $- 1$ plus $- 4$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

$(2 x + 1) ((x + 1) ((x - 1) (x (2 x - 1) - 2 (2 x - 1)))) = 0$

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$(2 x + 1) ((x + 1) ((x - 1) ((2 x - 1) (x - 2)))) = 0$

Remove unnecessary parentheses.

$(2 x + 1) ((x + 1) ((x - 1) (2 x - 1) (x - 2))) = 0$

Remove unnecessary parentheses.

$(2 x + 1) ((x + 1) (x - 1) (2 x - 1) (x - 2)) = 0$

Remove unnecessary parentheses.

$(2 x + 1) (x + 1) (x - 1) (2 x - 1) (x - 2) = 0$

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

$2 x + 1 = 0$

$x + 1 = 0$

$x - 1 = 0$

$2 x - 1 = 0$

$x - 2 = 0$

Set $2 x + 1$ equal to $0$ and solve for $x$ .

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Solve $2 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Divide each term in $2 x = - 1$ by $2$ and simplify.

Divide each term in $2 x = - 1$ by $2$ .

$\frac{2 x}{2} = \frac{- 1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1}{2}$

Divide $x$ by $1$ .

$x = \frac{- 1}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

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

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

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

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.

$x = 1$

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

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Solve $2 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$2 x = 1$

Divide each term in $2 x = 1$ by $2$ and simplify.

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Divide $x$ by $1$ .

$x = \frac{1}{2}$

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.

$x = 2$

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

$x = - \frac{1}{2}, - 1, 1, \frac{1}{2}, 2$

Step 3

Find the Roots (Zeros) 4x^5-8x^4-5x^3+10x^2+x-2

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