Find the Roots/Zeros Using the Rational Roots Test 2x^3-3x^2+1

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

Step 1

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.

$p = \pm 1$

$q = \pm 1, \pm 2$

Step 2

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

$\pm 1, \pm \frac{1}{2}$

Step 3

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

$2 {(1)}^{3} - 3 {(1)}^{2} + 1$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = 1$ is a root of the polynomial.

Simplify each term.

One to any power is one.

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

Multiply $2$ by $1$ .

$2 - 3 {(1)}^{2} + 1$

One to any power is one.

$2 - 3 \cdot 1 + 1$

Multiply $- 3$ by $1$ .

$2 - 3 + 1$

Simplify by adding and subtracting.

Subtract $3$ from $2$ .

$- 1 + 1$

Add $- 1$ and $1$ .

$0$

Step 5

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.

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

Step 6

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$1$	$2$	$- 3$	$0$	$1$
	\xa0			\xa0

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

$1$	$2$	$- 3$	$0$	$1$
	\xa0			\xa0
	$2$

Multiply the newest entry in the result $(2)$ by the divisor $(1)$ and place the result of $(2)$ under the next term in the dividend $(- 3)$ .

$1$	$2$	$- 3$	$0$	$1$
	\xa0	$2$		\xa0
	$2$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$1$	$2$	$- 3$	$0$	$1$
	\xa0	$2$		\xa0
	$2$	$- 1$

Multiply the newest entry in the result $(- 1)$ by the divisor $(1)$ and place the result of $(- 1)$ under the next term in the dividend $(0)$ .

$1$	$2$	$- 3$	$0$	$1$
	\xa0	$2$	$- 1$	\xa0
	$2$	$- 1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$1$	$2$	$- 3$	$0$	$1$
	\xa0	$2$	$- 1$	\xa0
	$2$	$- 1$	$- 1$

Multiply the newest entry in the result $(- 1)$ by the divisor $(1)$ and place the result of $(- 1)$ under the next term in the dividend $(1)$ .

$1$	$2$	$- 3$	$0$	$1$
	\xa0	$2$	$- 1$	$- 1$
	$2$	$- 1$	$- 1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$1$	$2$	$- 3$	$0$	$1$
	\xa0	$2$	$- 1$	$- 1$
	$2$	$- 1$	$- 1$	$0$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

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

Simplify the quotient polynomial.

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

Step 7

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

Factor $- 1$ out of $- x$ .

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

Rewrite $- 1$ as $1$ plus $- 2$

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

Apply the distributive property.

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

Multiply $x$ by $1$ .

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

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

Step 8

Factor the left side of the equation.

Factor $2 x^{3} - 3 x^{2} + 1$ 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.

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

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

$\pm 1, \pm 0.5$

Substitute $- 0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.5$ is a root of the polynomial.

Substitute $- 0.5$ into the polynomial.

$2 {(- 0.5)}^{3} - 3 {(- 0.5)}^{2} + 1$

Raise $- 0.5$ to the power of $3$ .

$2 \cdot - 0.125 - 3 {(- 0.5)}^{2} + 1$

Multiply $2$ by $- 0.125$ .

$- 0.25 - 3 {(- 0.5)}^{2} + 1$

Raise $- 0.5$ to the power of $2$ .

$- 0.25 - 3 \cdot 0.25 + 1$

Multiply $- 3$ by $0.25$ .

$- 0.25 - 0.75 + 1$

Subtract $0.75$ from $- 0.25$ .

$- 1 + 1$

Add $- 1$ and $1$ .

$0$

Since $- 0.5$ is a known root, divide the polynomial by $2 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

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

Divide $2 x^{3} - 3 x^{2} + 1$ by $2 x + 1$ .

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

$2 x$

$1$

$2 x^{3}$

$3 x^{2}$

$0 x$

$1$

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

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

Multiply the new quotient term by the divisor.

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

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

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

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

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

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

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

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

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

Multiply the new quotient term by the divisor.

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

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

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

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

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

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

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

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

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

Multiply the new quotient term by the divisor.

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

The expression needs to be subtracted from the dividend, so change all the signs in $2 x + 1$

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

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

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

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

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

Write $2 x^{3} - 3 x^{2} + 1$ as a set of factors.

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

Factor using the perfect square rule.

Rewrite $1$ as $1^{2}$ .

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

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

$2 x = 2 \cdot x \cdot 1$

Rewrite the polynomial.

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

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Step 9

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

$2 x + 1 = 0$

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

Step 10

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.

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

Step 11

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

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

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

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Step 12

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

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

Step 13

Find the Roots/Zeros Using the Rational Roots Test 2x^3-3x^2+1

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