Find the Roots (Zeros) f(x)=3x^4+11x^3+11x^2+x-2

$f (x) = 3 x^{4} + 11 x^{3} + 11 x^{2} + x - 2$

Step 1

Set $3 x^{4} + 11 x^{3} + 11 x^{2} + x - 2$ equal to $0$ .

$3 x^{4} + 11 x^{3} + 11 x^{2} + x - 2 = 0$

Step 2

Solve for $x$ .

Factor the left side of the equation.

Regroup terms.

$11 x^{3} + 11 x^{2} + 3 x^{4} + x - 2 = 0$

Factor $11 x^{2}$ out of $11 x^{3} + 11 x^{2}$ .

Factor $11 x^{2}$ out of $11 x^{3}$ .

$11 x^{2} (x) + 11 x^{2} + 3 x^{4} + x - 2 = 0$

Factor $11 x^{2}$ out of $11 x^{2}$ .

$11 x^{2} (x) + 11 x^{2} (1) + 3 x^{4} + x - 2 = 0$

Factor $11 x^{2}$ out of $11 x^{2} (x) + 11 x^{2} (1)$ .

$11 x^{2} (x + 1) + 3 x^{4} + x - 2 = 0$

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

$p = \pm 1, \pm 2 q = \pm 1, \pm 3$

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

$\pm 1, \pm 0.\overline{3}, \pm 2, \pm 0.\overline{6}$

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.

$3 {(- 1)}^{4} - 1 - 2$

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

$3 \cdot 1 - 1 - 2$

Multiply $3$ by $1$ .

$3 - 1 - 2$

Subtract $1$ from $3$ .

$2 - 2$

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.

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

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

$x$

$1$

$3 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$2$

Divide the highest order term in the dividend $3 x^{4}$ by the highest order term in divisor $x$ .

$3 x^{3}$

$x$

$1$

$3 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$2$

Multiply the new quotient term by the divisor.

$3 x^{3}$

$x$

$1$

$3 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$2$

$3 x^{4}$

$3 x^{3}$

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

$3 x^{3}$

$x$

$1$

$3 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$2$

$3 x^{4}$

$3 x^{3}$

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

$3 x^{3}$

$x$

$1$

$3 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$2$

$3 x^{4}$

$3 x^{3}$

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

$3 x^{3}$

$x$

$1$

$3 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$2$

$3 x^{4}$

$3 x^{3}$

$0 x^{2}$

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

$3 x^{3}$

$3 x^{2}$

$x$

$1$

$3 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$2$

$3 x^{4}$

$3 x^{3}$

$0 x^{2}$

Multiply the new quotient term by the divisor.

$3 x^{3}$

$3 x^{2}$

$x$

$1$

$3 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$2$

$3 x^{4}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$3 x^{2}$

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

$3 x^{3}$

$3 x^{2}$

$x$

$1$

$3 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$2$

$3 x^{4}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$3 x^{2}$

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

$3 x^{3}$

$3 x^{2}$

$x$

$1$

$3 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$2$

$3 x^{4}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$3 x^{2}$

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

$3 x^{3}$

$3 x^{2}$

$x$

$1$

$3 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$2$

$3 x^{4}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$3 x^{2}$

$x$

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

$3 x^{3}$

$3 x^{2}$

$3 x$

$x$

$1$

$3 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$2$

$3 x^{4}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$3 x^{2}$

$x$

Multiply the new quotient term by the divisor.

$3 x^{3}$

$3 x^{2}$

$3 x$

$x$

$1$

$3 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$2$

$3 x^{4}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$3 x^{2}$

$x$

$3 x^{2}$

$3 x$

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

$3 x^{3}$

$3 x^{2}$

$3 x$

$x$

$1$

$3 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$2$

$3 x^{4}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$3 x^{2}$

$x$

$3 x^{2}$

$3 x$

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

$3 x^{3}$

$3 x^{2}$

$3 x$

$x$

$1$

$3 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$2$

$3 x^{4}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$3 x^{2}$

$x$

$3 x^{2}$

$3 x$

$2 x$

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

$3 x^{3}$

$3 x^{2}$

$3 x$

$x$

$1$

$3 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$2$

$3 x^{4}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$3 x^{2}$

$x$

$3 x^{2}$

$3 x$

$2 x$

$2$

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

$3 x^{3}$

$3 x^{2}$

$3 x$

$2$

$x$

$1$

$3 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$2$

$3 x^{4}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$3 x^{2}$

$x$

$3 x^{2}$

$3 x$

$2 x$

$2$

Multiply the new quotient term by the divisor.

$3 x^{3}$

$3 x^{2}$

$3 x$

$2$

$x$

$1$

$3 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$2$

$3 x^{4}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$3 x^{2}$

$x$

$3 x^{2}$

$3 x$

$2 x$

$2$

$2 x$

$2$

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

$3 x^{3}$

$3 x^{2}$

$3 x$

$2$

$x$

$1$

$3 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$2$

$3 x^{4}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$3 x^{2}$

$x$

$3 x^{2}$

$3 x$

$2 x$

$2$

$2 x$

$2$

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

$3 x^{3}$

$3 x^{2}$

$3 x$

$2$

$x$

$1$

$3 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$2$

$3 x^{4}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$3 x^{2}$

$x$

$3 x^{2}$

$3 x$

$2 x$

$2$

$2 x$

$2$

$0$

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

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

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

$11 x^{2} (x + 1) + (x + 1) (3 x^{3} - 3 x^{2} + 3 x - 2) = 0$

Factor $x + 1$ out of $11 x^{2} (x + 1) + (x + 1) (3 x^{3} - 3 x^{2} + 3 x - 2)$ .

Factor $x + 1$ out of $11 x^{2} (x + 1)$ .

$(x + 1) (11 x^{2}) + (x + 1) (3 x^{3} - 3 x^{2} + 3 x - 2) = 0$

Factor $x + 1$ out of $(x + 1) (11 x^{2}) + (x + 1) (3 x^{3} - 3 x^{2} + 3 x - 2)$ .

$(x + 1) (11 x^{2} + 3 x^{3} - 3 x^{2} + 3 x - 2) = 0$

Subtract $3 x^{2}$ from $11 x^{2}$ .

$(x + 1) (3 x^{3} + 8 x^{2} + 3 x - 2) = 0$

Factor.

Rewrite $3 x^{3} + 8 x^{2} + 3 x - 2$ in a factored form.

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

$p = \pm 1, \pm 2 q = \pm 1, \pm 3$

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

$\pm 1, \pm 0.\overline{3}, \pm 2, \pm 0.\overline{6}$

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.

$3 {(- 1)}^{3} + 8 {(- 1)}^{2} + 3 \cdot - 1 - 2$

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

$3 \cdot - 1 + 8 {(- 1)}^{2} + 3 \cdot - 1 - 2$

Multiply $3$ by $- 1$ .

$- 3 + 8 {(- 1)}^{2} + 3 \cdot - 1 - 2$

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

$- 3 + 8 \cdot 1 + 3 \cdot - 1 - 2$

Multiply $8$ by $1$ .

$- 3 + 8 + 3 \cdot - 1 - 2$

Add $- 3$ and $8$ .

$5 + 3 \cdot - 1 - 2$

Multiply $3$ by $- 1$ .

$5 - 3 - 2$

Subtract $3$ from $5$ .

$2 - 2$

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.

$\frac{3 x^{3} + 8 x^{2} + 3 x - 2}{x + 1}$

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

$x$

$1$

$3 x^{3}$

$8 x^{2}$

$3 x$

$2$

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

					$3 x^{2}$
	$x$	+	$1$		$3 x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$2$

Multiply the new quotient term by the divisor.

				$3 x^{2}$
$x$	+	$1$		$3 x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$2$
			+	$3 x^{3}$	+	$3 x^{2}$

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

				$3 x^{2}$
$x$	+	$1$		$3 x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$2$
			-	$3 x^{3}$	-	$3 x^{2}$

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

				$3 x^{2}$
$x$	+	$1$		$3 x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$2$
			-	$3 x^{3}$	-	$3 x^{2}$
					+	$5 x^{2}$

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

				$3 x^{2}$
$x$	+	$1$		$3 x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$2$
			-	$3 x^{3}$	-	$3 x^{2}$
					+	$5 x^{2}$	+	$3 x$

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

				$3 x^{2}$	+	$5 x$
$x$	+	$1$		$3 x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$2$
			-	$3 x^{3}$	-	$3 x^{2}$
					+	$5 x^{2}$	+	$3 x$

Multiply the new quotient term by the divisor.

				$3 x^{2}$	+	$5 x$
$x$	+	$1$		$3 x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$2$
			-	$3 x^{3}$	-	$3 x^{2}$
					+	$5 x^{2}$	+	$3 x$
					+	$5 x^{2}$	+	$5 x$

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

				$3 x^{2}$	+	$5 x$
$x$	+	$1$		$3 x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$2$
			-	$3 x^{3}$	-	$3 x^{2}$
					+	$5 x^{2}$	+	$3 x$
					-	$5 x^{2}$	-	$5 x$

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

				$3 x^{2}$	+	$5 x$
$x$	+	$1$		$3 x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$2$
			-	$3 x^{3}$	-	$3 x^{2}$
					+	$5 x^{2}$	+	$3 x$
					-	$5 x^{2}$	-	$5 x$
							-	$2 x$

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

				$3 x^{2}$	+	$5 x$
$x$	+	$1$		$3 x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$2$
			-	$3 x^{3}$	-	$3 x^{2}$
					+	$5 x^{2}$	+	$3 x$
					-	$5 x^{2}$	-	$5 x$
							-	$2 x$	-	$2$

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

				$3 x^{2}$	+	$5 x$	-	$2$
$x$	+	$1$		$3 x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$2$
			-	$3 x^{3}$	-	$3 x^{2}$
					+	$5 x^{2}$	+	$3 x$
					-	$5 x^{2}$	-	$5 x$
							-	$2 x$	-	$2$

Multiply the new quotient term by the divisor.

				$3 x^{2}$	+	$5 x$	-	$2$
$x$	+	$1$		$3 x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$2$
			-	$3 x^{3}$	-	$3 x^{2}$
					+	$5 x^{2}$	+	$3 x$
					-	$5 x^{2}$	-	$5 x$
							-	$2 x$	-	$2$
							-	$2 x$	-	$2$

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

				$3 x^{2}$	+	$5 x$	-	$2$
$x$	+	$1$		$3 x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$2$
			-	$3 x^{3}$	-	$3 x^{2}$
					+	$5 x^{2}$	+	$3 x$
					-	$5 x^{2}$	-	$5 x$
							-	$2 x$	-	$2$
							+	$2 x$	+	$2$

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

				$3 x^{2}$	+	$5 x$	-	$2$
$x$	+	$1$		$3 x^{3}$	+	$8 x^{2}$	+	$3 x$	-	$2$
			-	$3 x^{3}$	-	$3 x^{2}$
					+	$5 x^{2}$	+	$3 x$
					-	$5 x^{2}$	-	$5 x$
							-	$2 x$	-	$2$
							+	$2 x$	+	$2$
										$0$

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

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

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

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

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 $a \cdot c = 3 \cdot - 2 = - 6$ and whose sum is $b = 5$ .

Factor $5$ out of $5 x$ .

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

Rewrite $5$ as $- 1$ plus $6$

$(x + 1) ((x + 1) (3 x^{2} + (- 1 + 6) x - 2)) = 0$

Apply the distributive property.

$(x + 1) ((x + 1) (3 x^{2} - 1 x + 6 x - 2)) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x + 1) ((x + 1) ((3 x^{2} - 1 x) + 6 x - 2)) = 0$

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

$(x + 1) ((x + 1) (x (3 x - 1) + 2 (3 x - 1))) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$(x + 1) ((x + 1) ((3 x - 1) (x + 2))) = 0$

Remove unnecessary parentheses.

$(x + 1) ((x + 1) (3 x - 1) (x + 2)) = 0$

Remove unnecessary parentheses.

$(x + 1) (x + 1) (3 x - 1) (x + 2) = 0$

Combine exponents.

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

$(x + 1) (x + 1) (3 x - 1) (x + 2) = 0$

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

$(x + 1) (x + 1) (3 x - 1) (x + 2) = 0$

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

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

Add $1$ and $1$ .

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

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

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

$3 x - 1 = 0$

$x + 2 = 0$

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 $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Set $3 x - 1$ equal to $0$ and solve for $x$ .

Set $3 x - 1$ equal to $0$ .

$3 x - 1 = 0$

Solve $3 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$3 x = 1$

Divide each term in $3 x = 1$ by $3$ and simplify.

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Divide $x$ by $1$ .

$x = \frac{1}{3}$

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.

$x = - 2$

The final solution is all the values that make ${(x + 1)}^{2} (3 x - 1) (x + 2) = 0$ true.

$x = - 1, \frac{1}{3}, - 2$

Step 3

Find the Roots (Zeros) f(x)=3x^4+11x^3+11x^2+x-2

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