Find the Roots (Zeros) x^4-x^3-19x^2+49x-30

$x^{4} - x^{3} - 19 x^{2} + 49 x - 30$

Step 1

Set $x^{4} - x^{3} - 19 x^{2} + 49 x - 30$ equal to $0$ .

$x^{4} - x^{3} - 19 x^{2} + 49 x - 30 = 0$

Step 2

Solve for $x$ .

Factor the left side of the equation.

Factor $x^{4} - x^{3} - 19 x^{2} + 49 x - 30$ 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.

$p = \pm 1, \pm 30, \pm 2, \pm 15, \pm 3, \pm 10, \pm 5, \pm 6 q = \pm 1$

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

$\pm 1, \pm 30, \pm 2, \pm 15, \pm 3, \pm 10, \pm 5, \pm 6$

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.

$1^{4} - 1^{3} - 19 \cdot 1^{2} + 49 \cdot 1 - 30$

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

$1 - 1^{3} - 19 \cdot 1^{2} + 49 \cdot 1 - 30$

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

$1 - 1 \cdot 1 - 19 \cdot 1^{2} + 49 \cdot 1 - 30$

Multiply $- 1$ by $1$ .

$1 - 1 - 19 \cdot 1^{2} + 49 \cdot 1 - 30$

Subtract $1$ from $1$ .

$0 - 19 \cdot 1^{2} + 49 \cdot 1 - 30$

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

$0 - 19 \cdot 1 + 49 \cdot 1 - 30$

Multiply $- 19$ by $1$ .

$0 - 19 + 49 \cdot 1 - 30$

Subtract $19$ from $0$ .

$- 19 + 49 \cdot 1 - 30$

Multiply $49$ by $1$ .

$- 19 + 49 - 30$

Add $- 19$ and $49$ .

$30 - 30$

Subtract $30$ from $30$ .

$0$

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.

$\frac{x^{4} - x^{3} - 19 x^{2} + 49 x - 30}{x - 1}$

Divide $x^{4} - x^{3} - 19 x^{2} + 49 x - 30$ 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 $0$ .

$x$

$1$

$x^{4}$

$x^{3}$

$19 x^{2}$

$49 x$

$30$

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

					$x^{3}$
	$x$	-	$1$		$x^{4}$	-	$x^{3}$	-	$19 x^{2}$	+	$49 x$	-	$30$

Multiply the new quotient term by the divisor.

				$x^{3}$
$x$	-	$1$		$x^{4}$	-	$x^{3}$	-	$19 x^{2}$	+	$49 x$	-	$30$
			+	$x^{4}$	-	$x^{3}$

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

				$x^{3}$
$x$	-	$1$		$x^{4}$	-	$x^{3}$	-	$19 x^{2}$	+	$49 x$	-	$30$
			-	$x^{4}$	+	$x^{3}$

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

				$x^{3}$
$x$	-	$1$		$x^{4}$	-	$x^{3}$	-	$19 x^{2}$	+	$49 x$	-	$30$
			-	$x^{4}$	+	$x^{3}$
						$0$

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

				$x^{3}$
$x$	-	$1$		$x^{4}$	-	$x^{3}$	-	$19 x^{2}$	+	$49 x$	-	$30$
			-	$x^{4}$	+	$x^{3}$
						$0$	-	$19 x^{2}$	+	$49 x$

Divide the highest order term in the dividend $- 19 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$0 x^{2}$

$19 x$

$x$

$1$

$x^{4}$

$x^{3}$

$19 x^{2}$

$49 x$

$30$

$x^{4}$

$x^{3}$

$0$

$19 x^{2}$

$49 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$0 x^{2}$

$19 x$

$x$

$1$

$x^{4}$

$x^{3}$

$19 x^{2}$

$49 x$

$30$

$x^{4}$

$x^{3}$

$0$

$19 x^{2}$

$49 x$

$19 x^{2}$

$19 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 19 x^{2} + 19 x$

$x^{3}$

$0 x^{2}$

$19 x$

$x$

$1$

$x^{4}$

$x^{3}$

$19 x^{2}$

$49 x$

$30$

$x^{4}$

$x^{3}$

$0$

$19 x^{2}$

$49 x$

$19 x^{2}$

$19 x$

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

$x^{3}$

$0 x^{2}$

$19 x$

$x$

$1$

$x^{4}$

$x^{3}$

$19 x^{2}$

$49 x$

$30$

$x^{4}$

$x^{3}$

$0$

$19 x^{2}$

$49 x$

$19 x^{2}$

$19 x$

$30 x$

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

$x^{3}$

$0 x^{2}$

$19 x$

$x$

$1$

$x^{4}$

$x^{3}$

$19 x^{2}$

$49 x$

$30$

$x^{4}$

$x^{3}$

$0$

$19 x^{2}$

$49 x$

$19 x^{2}$

$19 x$

$30 x$

$30$

Divide the highest order term in the dividend $30 x$ by the highest order term in divisor $x$ .

$x^{3}$

$0 x^{2}$

$19 x$

$30$

$x$

$1$

$x^{4}$

$x^{3}$

$19 x^{2}$

$49 x$

$30$

$x^{4}$

$x^{3}$

$0$

$19 x^{2}$

$49 x$

$19 x^{2}$

$19 x$

$30 x$

$30$

Multiply the new quotient term by the divisor.

				$x^{3}$	+	$0 x^{2}$	-	$19 x$	+	$30$
$x$	-	$1$		$x^{4}$	-	$x^{3}$	-	$19 x^{2}$	+	$49 x$	-	$30$
			-	$x^{4}$	+	$x^{3}$
						$0$	-	$19 x^{2}$	+	$49 x$
							+	$19 x^{2}$	-	$19 x$
									+	$30 x$	-	$30$
									+	$30 x$	-	$30$

The expression needs to be subtracted from the dividend, so change all the signs in $30 x - 30$

				$x^{3}$	+	$0 x^{2}$	-	$19 x$	+	$30$
$x$	-	$1$		$x^{4}$	-	$x^{3}$	-	$19 x^{2}$	+	$49 x$	-	$30$
			-	$x^{4}$	+	$x^{3}$
						$0$	-	$19 x^{2}$	+	$49 x$
							+	$19 x^{2}$	-	$19 x$
									+	$30 x$	-	$30$
									-	$30 x$	+	$30$

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

				$x^{3}$	+	$0 x^{2}$	-	$19 x$	+	$30$
$x$	-	$1$		$x^{4}$	-	$x^{3}$	-	$19 x^{2}$	+	$49 x$	-	$30$
			-	$x^{4}$	+	$x^{3}$
						$0$	-	$19 x^{2}$	+	$49 x$
							+	$19 x^{2}$	-	$19 x$
									+	$30 x$	-	$30$
									-	$30 x$	+	$30$
												$0$

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

$x^{3} + 0 x^{2} - 19 x + 30$

Write $x^{4} - x^{3} - 19 x^{2} + 49 x - 30$ as a set of factors.

$(x - 1) (x^{3} + 0 x^{2} - 19 x + 30) = 0$

Multiply $0$ by $x^{2}$ .

$(x - 1) (x^{3} + 0 - 19 x + 30) = 0$

Add $x^{3}$ and $0$ .

$(x - 1) (x^{3} - 19 x + 30) = 0$

Rewrite $x^{3} - 19 x + 30$ in a factored form.

Factor $x^{3} - 19 x + 30$ 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.

$p = \pm 1, \pm 30, \pm 2, \pm 15, \pm 3, \pm 10, \pm 5, \pm 6 q = \pm 1$

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

$\pm 1, \pm 30, \pm 2, \pm 15, \pm 3, \pm 10, \pm 5, \pm 6$

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.

$2^{3} - 19 \cdot 2 + 30$

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

$8 - 19 \cdot 2 + 30$

Multiply $- 19$ by $2$ .

$8 - 38 + 30$

Subtract $38$ from $8$ .

$- 30 + 30$

Add $- 30$ and $30$ .

$0$

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.

$\frac{x^{3} - 19 x + 30}{x - 2}$

Divide $x^{3} - 19 x + 30$ 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 $0$ .

$x$

$2$

$x^{3}$

$0 x^{2}$

$19 x$

$30$

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

					$x^{2}$
	$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$19 x$	+	$30$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$19 x$	+	$30$
			+	$x^{3}$	-	$2 x^{2}$

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$19 x$	+	$30$
			-	$x^{3}$	+	$2 x^{2}$

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$19 x$	+	$30$
			-	$x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$19 x$	+	$30$
			-	$x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	-	$19 x$

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

				$x^{2}$	+	$2 x$
$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$19 x$	+	$30$
			-	$x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	-	$19 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$2 x$
$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$19 x$	+	$30$
			-	$x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	-	$19 x$
					+	$2 x^{2}$	-	$4 x$

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

				$x^{2}$	+	$2 x$
$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$19 x$	+	$30$
			-	$x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	-	$19 x$
					-	$2 x^{2}$	+	$4 x$

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

				$x^{2}$	+	$2 x$
$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$19 x$	+	$30$
			-	$x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	-	$19 x$
					-	$2 x^{2}$	+	$4 x$
							-	$15 x$

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

				$x^{2}$	+	$2 x$
$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$19 x$	+	$30$
			-	$x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	-	$19 x$
					-	$2 x^{2}$	+	$4 x$
							-	$15 x$	+	$30$

Divide the highest order term in the dividend $- 15 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$2 x$	-	$15$
$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$19 x$	+	$30$
			-	$x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	-	$19 x$
					-	$2 x^{2}$	+	$4 x$
							-	$15 x$	+	$30$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$2 x$	-	$15$
$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$19 x$	+	$30$
			-	$x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	-	$19 x$
					-	$2 x^{2}$	+	$4 x$
							-	$15 x$	+	$30$
							-	$15 x$	+	$30$

The expression needs to be subtracted from the dividend, so change all the signs in $- 15 x + 30$

				$x^{2}$	+	$2 x$	-	$15$
$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$19 x$	+	$30$
			-	$x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	-	$19 x$
					-	$2 x^{2}$	+	$4 x$
							-	$15 x$	+	$30$
							+	$15 x$	-	$30$

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

				$x^{2}$	+	$2 x$	-	$15$
$x$	-	$2$		$x^{3}$	+	$0 x^{2}$	-	$19 x$	+	$30$
			-	$x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	-	$19 x$
					-	$2 x^{2}$	+	$4 x$
							-	$15 x$	+	$30$
							+	$15 x$	-	$30$
										$0$

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

$x^{2} + 2 x - 15$

Write $x^{3} - 19 x + 30$ as a set of factors.

$(x - 1) ((x - 2) (x^{2} + 2 x - 15)) = 0$

Factor $x^{2} + 2 x - 15$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 15$ and whose sum is $2$ .

$- 3, 5$

Write the factored form using these integers.

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

Remove unnecessary parentheses.

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

Remove unnecessary parentheses.

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

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

$x - 1 = 0$

$x - 2 = 0$

$x - 3 = 0$

$x + 5 = 0$

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

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Set $x + 5$ equal to $0$ and solve for $x$ .

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

The final solution is all the values that make $(x - 1) (x - 2) (x - 3) (x + 5) = 0$ true.

$x = 1, 2, 3, - 5$

Step 3

Find the Roots (Zeros) x^4-x^3-19x^2+49x-30

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