Find the Roots/Zeros Using the Rational Roots Test f(x)=x^4-3x^3-7x^2+15x+18

$f (x) = x^{4} - 3 x^{3} - 7 x^{2} + 15 x + 18$

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.

$p = \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18$

$q = \pm 1$

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

$\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18$

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.

${(- 1)}^{4} - 3 {(- 1)}^{3} - 7 {(- 1)}^{2} + 15 (- 1) + 18$

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

Simplify each term.

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

$1 - 3 {(- 1)}^{3} - 7 {(- 1)}^{2} + 15 (- 1) + 18$

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

$1 - 3 \cdot - 1 - 7 {(- 1)}^{2} + 15 (- 1) + 18$

Multiply $- 3$ by $- 1$ .

$1 + 3 - 7 {(- 1)}^{2} + 15 (- 1) + 18$

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

$1 + 3 - 7 \cdot 1 + 15 (- 1) + 18$

Multiply $- 7$ by $1$ .

$1 + 3 - 7 + 15 (- 1) + 18$

Multiply $15$ by $- 1$ .

$1 + 3 - 7 - 15 + 18$

Simplify by adding and subtracting.

Add $1$ and $3$ .

$4 - 7 - 15 + 18$

Subtract $7$ from $4$ .

$- 3 - 15 + 18$

Subtract $15$ from $- 3$ .

$- 18 + 18$

Add $- 18$ and $18$ .

$0$

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.

$\frac{x^{4} - 3 x^{3} - 7 x^{2} + 15 x + 18}{x + 1}$

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.

$- 1$	$1$	$- 3$	$- 7$	$15$	$18$
	\xa0				\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$- 1$	$1$	$- 3$	$- 7$	$15$	$18$
	\xa0				\xa0
	$1$

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 $(- 3)$ .

$- 1$	$1$	$- 3$	$- 7$	$15$	$18$
	\xa0	$- 1$			\xa0
	$1$

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

$- 1$	$1$	$- 3$	$- 7$	$15$	$18$
	\xa0	$- 1$			\xa0
	$1$	$- 4$

Multiply the newest entry in the result $(- 4)$ by the divisor $(- 1)$ and place the result of $(4)$ under the next term in the dividend $(- 7)$ .

$- 1$	$1$	$- 3$	$- 7$	$15$	$18$
	\xa0	$- 1$	$4$		\xa0
	$1$	$- 4$

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

$- 1$	$1$	$- 3$	$- 7$	$15$	$18$
	\xa0	$- 1$	$4$		\xa0
	$1$	$- 4$	$- 3$

Multiply the newest entry in the result $(- 3)$ by the divisor $(- 1)$ and place the result of $(3)$ under the next term in the dividend $(15)$ .

$- 1$	$1$	$- 3$	$- 7$	$15$	$18$
	\xa0	$- 1$	$4$	$3$	\xa0
	$1$	$- 4$	$- 3$

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

$- 1$	$1$	$- 3$	$- 7$	$15$	$18$
	\xa0	$- 1$	$4$	$3$	\xa0
	$1$	$- 4$	$- 3$	$18$

Multiply the newest entry in the result $(18)$ by the divisor $(- 1)$ and place the result of $(- 18)$ under the next term in the dividend $(18)$ .

$- 1$	$1$	$- 3$	$- 7$	$15$	$18$
	\xa0	$- 1$	$4$	$3$	$- 18$
	$1$	$- 4$	$- 3$	$18$

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

$- 1$	$1$	$- 3$	$- 7$	$15$	$18$
	\xa0	$- 1$	$4$	$3$	$- 18$
	$1$	$- 4$	$- 3$	$18$	$0$

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

$1 x^{3} + - 4 x^{2} + (- 3) x + 18$

Simplify the quotient polynomial.

$x^{3} - 4 x^{2} - 3 x + 18$

Solve the equation to find any remaining roots.

Factor the left side of the equation.

Factor $x^{3} - 4 x^{2} - 3 x + 18$ 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.

$p = \pm 1, \pm 18, \pm 2, \pm 9, \pm 3, \pm 6 q = \pm 1$

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

$\pm 1, \pm 18, \pm 2, \pm 9, \pm 3, \pm 6$

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.

${(- 2)}^{3} - 4 {(- 2)}^{2} - 3 \cdot - 2 + 18$

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

$- 8 - 4 {(- 2)}^{2} - 3 \cdot - 2 + 18$

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

$- 8 - 4 \cdot 4 - 3 \cdot - 2 + 18$

Multiply $- 4$ by $4$ .

$- 8 - 16 - 3 \cdot - 2 + 18$

Subtract $16$ from $- 8$ .

$- 24 - 3 \cdot - 2 + 18$

Multiply $- 3$ by $- 2$ .

$- 24 + 6 + 18$

Add $- 24$ and $6$ .

$- 18 + 18$

Add $- 18$ and $18$ .

$0$

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.

$\frac{x^{3} - 4 x^{2} - 3 x + 18}{x + 2}$

Divide $x^{3} - 4 x^{2} - 3 x + 18$ 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 $0$ .

$x$

$2$

$x^{3}$

$4 x^{2}$

$3 x$

$18$

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

					$x^{2}$
	$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$3 x$	+	$18$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$3 x$	+	$18$
			+	$x^{3}$	+	$2 x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$3 x$	+	$18$
			-	$x^{3}$	-	$2 x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$3 x$	+	$18$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$

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

				$x^{2}$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$3 x$	+	$18$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$3 x$

Divide the highest order term in the dividend $- 6 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$6 x$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$3 x$	+	$18$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$3 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$6 x$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$3 x$	+	$18$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$3 x$
					-	$6 x^{2}$	-	$12 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{2} - 12 x$

				$x^{2}$	-	$6 x$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$3 x$	+	$18$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$3 x$
					+	$6 x^{2}$	+	$12 x$

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

				$x^{2}$	-	$6 x$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$3 x$	+	$18$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$3 x$
					+	$6 x^{2}$	+	$12 x$
							+	$9 x$

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

				$x^{2}$	-	$6 x$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$3 x$	+	$18$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$3 x$
					+	$6 x^{2}$	+	$12 x$
							+	$9 x$	+	$18$

Divide the highest order term in the dividend $9 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$6 x$	+	$9$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$3 x$	+	$18$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$3 x$
					+	$6 x^{2}$	+	$12 x$
							+	$9 x$	+	$18$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$6 x$	+	$9$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$3 x$	+	$18$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$3 x$
					+	$6 x^{2}$	+	$12 x$
							+	$9 x$	+	$18$
							+	$9 x$	+	$18$

The expression needs to be subtracted from the dividend, so change all the signs in $9 x + 18$

				$x^{2}$	-	$6 x$	+	$9$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$3 x$	+	$18$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$3 x$
					+	$6 x^{2}$	+	$12 x$
							+	$9 x$	+	$18$
							-	$9 x$	-	$18$

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

				$x^{2}$	-	$6 x$	+	$9$
$x$	+	$2$		$x^{3}$	-	$4 x^{2}$	-	$3 x$	+	$18$
			-	$x^{3}$	-	$2 x^{2}$
					-	$6 x^{2}$	-	$3 x$
					+	$6 x^{2}$	+	$12 x$
							+	$9 x$	+	$18$
							-	$9 x$	-	$18$
										$0$

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

$x^{2} - 6 x + 9$

Write $x^{3} - 4 x^{2} - 3 x + 18$ as a set of factors.

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

Factor using the perfect square rule.

Rewrite $9$ as $3^{2}$ .

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

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

$6 x = 2 \cdot x \cdot 3$

Rewrite the polynomial.

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

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 3$ .

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

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

$x + 2 = 0$

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

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.

$x = - 2$

Set ${(x - 3)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(x - 3)}^{2}$ equal to $0$ .

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

Solve ${(x - 3)}^{2} = 0$ for $x$ .

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

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

$x = - 2, 3$

The polynomial can be written as a set of linear factors.

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

These are the roots (zeros) of the polynomial $x^{4} - 3 x^{3} - 7 x^{2} + 15 x + 18$ .

$x = - 1, - 2, 3$

Find the Roots/Zeros Using the Rational Roots Test f(x)=x^4-3x^3-7x^2+15x+18

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