Find the Roots (Zeros) 3x^4+x^3-17x^2+19x-6

$3 x^{4} + x^{3} - 17 x^{2} + 19 x - 6$

Step 1

Set $3 x^{4} + x^{3} - 17 x^{2} + 19 x - 6$ equal to $0$ .

$3 x^{4} + x^{3} - 17 x^{2} + 19 x - 6 = 0$

Step 2

Solve for $x$ .

Factor the left side of the equation.

Factor $3 x^{4} + x^{3} - 17 x^{2} + 19 x - 6$ 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.

$p = \pm 1, \pm 6, \pm 2, \pm 3 q = \pm 1, \pm 3$

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

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

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.

$3 \cdot 1^{4} + 1^{3} - 17 \cdot 1^{2} + 19 \cdot 1 - 6$

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

$3 \cdot 1 + 1^{3} - 17 \cdot 1^{2} + 19 \cdot 1 - 6$

Multiply $3$ by $1$ .

$3 + 1^{3} - 17 \cdot 1^{2} + 19 \cdot 1 - 6$

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

$3 + 1 - 17 \cdot 1^{2} + 19 \cdot 1 - 6$

Add $3$ and $1$ .

$4 - 17 \cdot 1^{2} + 19 \cdot 1 - 6$

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

$4 - 17 \cdot 1 + 19 \cdot 1 - 6$

Multiply $- 17$ by $1$ .

$4 - 17 + 19 \cdot 1 - 6$

Subtract $17$ from $4$ .

$- 13 + 19 \cdot 1 - 6$

Multiply $19$ by $1$ .

$- 13 + 19 - 6$

Add $- 13$ and $19$ .

$6 - 6$

Subtract $6$ from $6$ .

$0$

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.

$\frac{3 x^{4} + x^{3} - 17 x^{2} + 19 x - 6}{x - 1}$

Divide $3 x^{4} + x^{3} - 17 x^{2} + 19 x - 6$ 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 $0$ .

$x$

$1$

$3 x^{4}$

$x^{3}$

$17 x^{2}$

$19 x$

$6$

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

$3 x^{3}$

$x$

$1$

$3 x^{4}$

$x^{3}$

$17 x^{2}$

$19 x$

$6$

Multiply the new quotient term by the divisor.

$3 x^{3}$

$x$

$1$

$3 x^{4}$

$x^{3}$

$17 x^{2}$

$19 x$

$6$

$3 x^{4}$

$3 x^{3}$

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

$3 x^{3}$

$x$

$1$

$3 x^{4}$

$x^{3}$

$17 x^{2}$

$19 x$

$6$

$3 x^{4}$

$3 x^{3}$

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

$3 x^{3}$

$x$

$1$

$3 x^{4}$

$x^{3}$

$17 x^{2}$

$19 x$

$6$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

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

$3 x^{3}$

$x$

$1$

$3 x^{4}$

$x^{3}$

$17 x^{2}$

$19 x$

$6$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$17 x^{2}$

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

$3 x^{3}$

$4 x^{2}$

$x$

$1$

$3 x^{4}$

$x^{3}$

$17 x^{2}$

$19 x$

$6$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$17 x^{2}$

Multiply the new quotient term by the divisor.

$3 x^{3}$

$4 x^{2}$

$x$

$1$

$3 x^{4}$

$x^{3}$

$17 x^{2}$

$19 x$

$6$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$17 x^{2}$

$4 x^{3}$

$4 x^{2}$

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

$3 x^{3}$

$4 x^{2}$

$x$

$1$

$3 x^{4}$

$x^{3}$

$17 x^{2}$

$19 x$

$6$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$17 x^{2}$

$4 x^{3}$

$4 x^{2}$

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

$3 x^{3}$

$4 x^{2}$

$x$

$1$

$3 x^{4}$

$x^{3}$

$17 x^{2}$

$19 x$

$6$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$17 x^{2}$

$4 x^{3}$

$4 x^{2}$

$13 x^{2}$

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

$3 x^{3}$

$4 x^{2}$

$x$

$1$

$3 x^{4}$

$x^{3}$

$17 x^{2}$

$19 x$

$6$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$17 x^{2}$

$4 x^{3}$

$4 x^{2}$

$13 x^{2}$

$19 x$

Divide the highest order term in the dividend $- 13 x^{2}$ by the highest order term in divisor $x$ .

$3 x^{3}$

$4 x^{2}$

$13 x$

$x$

$1$

$3 x^{4}$

$x^{3}$

$17 x^{2}$

$19 x$

$6$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$17 x^{2}$

$4 x^{3}$

$4 x^{2}$

$13 x^{2}$

$19 x$

Multiply the new quotient term by the divisor.

$3 x^{3}$

$4 x^{2}$

$13 x$

$x$

$1$

$3 x^{4}$

$x^{3}$

$17 x^{2}$

$19 x$

$6$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$17 x^{2}$

$4 x^{3}$

$4 x^{2}$

$13 x^{2}$

$19 x$

$13 x^{2}$

$13 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 13 x^{2} + 13 x$

$3 x^{3}$

$4 x^{2}$

$13 x$

$x$

$1$

$3 x^{4}$

$x^{3}$

$17 x^{2}$

$19 x$

$6$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$17 x^{2}$

$4 x^{3}$

$4 x^{2}$

$13 x^{2}$

$19 x$

$13 x^{2}$

$13 x$

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

$3 x^{3}$

$4 x^{2}$

$13 x$

$x$

$1$

$3 x^{4}$

$x^{3}$

$17 x^{2}$

$19 x$

$6$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$17 x^{2}$

$4 x^{3}$

$4 x^{2}$

$13 x^{2}$

$19 x$

$13 x^{2}$

$13 x$

$6 x$

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

$3 x^{3}$

$4 x^{2}$

$13 x$

$x$

$1$

$3 x^{4}$

$x^{3}$

$17 x^{2}$

$19 x$

$6$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$17 x^{2}$

$4 x^{3}$

$4 x^{2}$

$13 x^{2}$

$19 x$

$13 x^{2}$

$13 x$

$6 x$

$6$

Divide the highest order term in the dividend $6 x$ by the highest order term in divisor $x$ .

$3 x^{3}$

$4 x^{2}$

$13 x$

$6$

$x$

$1$

$3 x^{4}$

$x^{3}$

$17 x^{2}$

$19 x$

$6$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$17 x^{2}$

$4 x^{3}$

$4 x^{2}$

$13 x^{2}$

$19 x$

$13 x^{2}$

$13 x$

$6 x$

$6$

Multiply the new quotient term by the divisor.

$3 x^{3}$

$4 x^{2}$

$13 x$

$6$

$x$

$1$

$3 x^{4}$

$x^{3}$

$17 x^{2}$

$19 x$

$6$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$17 x^{2}$

$4 x^{3}$

$4 x^{2}$

$13 x^{2}$

$19 x$

$13 x^{2}$

$13 x$

$6 x$

$6$

$6 x$

$6$

The expression needs to be subtracted from the dividend, so change all the signs in $6 x - 6$

$3 x^{3}$

$4 x^{2}$

$13 x$

$6$

$x$

$1$

$3 x^{4}$

$x^{3}$

$17 x^{2}$

$19 x$

$6$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$17 x^{2}$

$4 x^{3}$

$4 x^{2}$

$13 x^{2}$

$19 x$

$13 x^{2}$

$13 x$

$6 x$

$6$

$6 x$

$6$

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

$3 x^{3}$

$4 x^{2}$

$13 x$

$6$

$x$

$1$

$3 x^{4}$

$x^{3}$

$17 x^{2}$

$19 x$

$6$

$3 x^{4}$

$3 x^{3}$

$4 x^{3}$

$17 x^{2}$

$4 x^{3}$

$4 x^{2}$

$13 x^{2}$

$19 x$

$13 x^{2}$

$13 x$

$6 x$

$6$

$6 x$

$6$

$0$

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

$3 x^{3} + 4 x^{2} - 13 x + 6$

Write $3 x^{4} + x^{3} - 17 x^{2} + 19 x - 6$ as a set of factors.

$(x - 1) (3 x^{3} + 4 x^{2} - 13 x + 6) = 0$

Factor $3 x^{3} + 4 x^{2} - 13 x + 6$ 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.

$p = \pm 1, \pm 6, \pm 2, \pm 3 q = \pm 1, \pm 3$

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

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

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.

$3 \cdot 1^{3} + 4 \cdot 1^{2} - 13 \cdot 1 + 6$

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

$3 \cdot 1 + 4 \cdot 1^{2} - 13 \cdot 1 + 6$

Multiply $3$ by $1$ .

$3 + 4 \cdot 1^{2} - 13 \cdot 1 + 6$

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

$3 + 4 \cdot 1 - 13 \cdot 1 + 6$

Multiply $4$ by $1$ .

$3 + 4 - 13 \cdot 1 + 6$

Add $3$ and $4$ .

$7 - 13 \cdot 1 + 6$

Multiply $- 13$ by $1$ .

$7 - 13 + 6$

Subtract $13$ from $7$ .

$- 6 + 6$

Add $- 6$ and $6$ .

$0$

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.

$\frac{3 x^{3} + 4 x^{2} - 13 x + 6}{x - 1}$

Divide $3 x^{3} + 4 x^{2} - 13 x + 6$ 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 $0$ .

$x$

$1$

$3 x^{3}$

$4 x^{2}$

$13 x$

$6$

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

					$3 x^{2}$
	$x$	-	$1$		$3 x^{3}$	+	$4 x^{2}$	-	$13 x$	+	$6$

Multiply the new quotient term by the divisor.

				$3 x^{2}$
$x$	-	$1$		$3 x^{3}$	+	$4 x^{2}$	-	$13 x$	+	$6$
			+	$3 x^{3}$	-	$3 x^{2}$

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

				$3 x^{2}$
$x$	-	$1$		$3 x^{3}$	+	$4 x^{2}$	-	$13 x$	+	$6$
			-	$3 x^{3}$	+	$3 x^{2}$

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

				$3 x^{2}$
$x$	-	$1$		$3 x^{3}$	+	$4 x^{2}$	-	$13 x$	+	$6$
			-	$3 x^{3}$	+	$3 x^{2}$
					+	$7 x^{2}$

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

				$3 x^{2}$
$x$	-	$1$		$3 x^{3}$	+	$4 x^{2}$	-	$13 x$	+	$6$
			-	$3 x^{3}$	+	$3 x^{2}$
					+	$7 x^{2}$	-	$13 x$

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

				$3 x^{2}$	+	$7 x$
$x$	-	$1$		$3 x^{3}$	+	$4 x^{2}$	-	$13 x$	+	$6$
			-	$3 x^{3}$	+	$3 x^{2}$
					+	$7 x^{2}$	-	$13 x$

Multiply the new quotient term by the divisor.

				$3 x^{2}$	+	$7 x$
$x$	-	$1$		$3 x^{3}$	+	$4 x^{2}$	-	$13 x$	+	$6$
			-	$3 x^{3}$	+	$3 x^{2}$
					+	$7 x^{2}$	-	$13 x$
					+	$7 x^{2}$	-	$7 x$

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

				$3 x^{2}$	+	$7 x$
$x$	-	$1$		$3 x^{3}$	+	$4 x^{2}$	-	$13 x$	+	$6$
			-	$3 x^{3}$	+	$3 x^{2}$
					+	$7 x^{2}$	-	$13 x$
					-	$7 x^{2}$	+	$7 x$

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

				$3 x^{2}$	+	$7 x$
$x$	-	$1$		$3 x^{3}$	+	$4 x^{2}$	-	$13 x$	+	$6$
			-	$3 x^{3}$	+	$3 x^{2}$
					+	$7 x^{2}$	-	$13 x$
					-	$7 x^{2}$	+	$7 x$
							-	$6 x$

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

				$3 x^{2}$	+	$7 x$
$x$	-	$1$		$3 x^{3}$	+	$4 x^{2}$	-	$13 x$	+	$6$
			-	$3 x^{3}$	+	$3 x^{2}$
					+	$7 x^{2}$	-	$13 x$
					-	$7 x^{2}$	+	$7 x$
							-	$6 x$	+	$6$

Divide the highest order term in the dividend $- 6 x$ by the highest order term in divisor $x$ .

				$3 x^{2}$	+	$7 x$	-	$6$
$x$	-	$1$		$3 x^{3}$	+	$4 x^{2}$	-	$13 x$	+	$6$
			-	$3 x^{3}$	+	$3 x^{2}$
					+	$7 x^{2}$	-	$13 x$
					-	$7 x^{2}$	+	$7 x$
							-	$6 x$	+	$6$

Multiply the new quotient term by the divisor.

				$3 x^{2}$	+	$7 x$	-	$6$
$x$	-	$1$		$3 x^{3}$	+	$4 x^{2}$	-	$13 x$	+	$6$
			-	$3 x^{3}$	+	$3 x^{2}$
					+	$7 x^{2}$	-	$13 x$
					-	$7 x^{2}$	+	$7 x$
							-	$6 x$	+	$6$
							-	$6 x$	+	$6$

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x + 6$

				$3 x^{2}$	+	$7 x$	-	$6$
$x$	-	$1$		$3 x^{3}$	+	$4 x^{2}$	-	$13 x$	+	$6$
			-	$3 x^{3}$	+	$3 x^{2}$
					+	$7 x^{2}$	-	$13 x$
					-	$7 x^{2}$	+	$7 x$
							-	$6 x$	+	$6$
							+	$6 x$	-	$6$

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

				$3 x^{2}$	+	$7 x$	-	$6$
$x$	-	$1$		$3 x^{3}$	+	$4 x^{2}$	-	$13 x$	+	$6$
			-	$3 x^{3}$	+	$3 x^{2}$
					+	$7 x^{2}$	-	$13 x$
					-	$7 x^{2}$	+	$7 x$
							-	$6 x$	+	$6$
							+	$6 x$	-	$6$
										$0$

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

$3 x^{2} + 7 x - 6$

Write $3 x^{3} + 4 x^{2} - 13 x + 6$ as a set of factors.

$(x - 1) ((x - 1) (3 x^{2} + 7 x - 6)) = 0$

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 $a \cdot c = 3 \cdot - 6 = - 18$ and whose sum is $b = 7$ .

Factor $7$ out of $7 x$ .

$(x - 1) ((x - 1) (3 x^{2} + 7 (x) - 6)) = 0$

Rewrite $7$ as $- 2$ plus $9$

$(x - 1) ((x - 1) (3 x^{2} + (- 2 + 9) x - 6)) = 0$

Apply the distributive property.

$(x - 1) ((x - 1) (3 x^{2} - 2 x + 9 x - 6)) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x - 1) ((x - 1) ((3 x^{2} - 2 x) + 9 x - 6)) = 0$

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

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

Factor the polynomial by factoring out the greatest common factor, $3 x - 2$ .

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

Remove unnecessary parentheses.

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

Remove unnecessary parentheses.

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

Combine like factors.

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

$3 x - 2 = 0$

$x + 3 = 0$

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 $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Set $3 x - 2$ equal to $0$ and solve for $x$ .

Set $3 x - 2$ equal to $0$ .

$3 x - 2 = 0$

Solve $3 x - 2 = 0$ for $x$ .

Add $2$ to both sides of the equation.

$3 x = 2$

Divide each term in $3 x = 2$ by $3$ and simplify.

Divide each term in $3 x = 2$ by $3$ .

$\frac{3 x}{3} = \frac{2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{2}{3}$

Divide $x$ by $1$ .

$x = \frac{2}{3}$

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

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

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

Step 3

Find the Roots (Zeros) 3x^4+x^3-17x^2+19x-6

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