Find the Roots/Zeros Using the Rational Roots Test x^4-9x^3-15x^2+325x-750

$x^{4} - 9 x^{3} - 15 x^{2} + 325 x - 750$

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.

$p = \pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 10, \pm 15, \pm 25, \pm 30, \pm 50, \pm 75, \pm 125, \pm 150, \pm 250, \pm 375, \pm 750$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 10, \pm 15, \pm 25, \pm 30, \pm 50, \pm 75, \pm 125, \pm 150, \pm 250, \pm 375, \pm 750$

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.

${(5)}^{4} - 9 {(5)}^{3} - 15 {(5)}^{2} + 325 (5) - 750$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = 5$ is a root of the polynomial.

Simplify each term.

Raise $5$ to the power of $4$ .

$625 - 9 {(5)}^{3} - 15 {(5)}^{2} + 325 (5) - 750$

Raise $5$ to the power of $3$ .

$625 - 9 \cdot 125 - 15 {(5)}^{2} + 325 (5) - 750$

Multiply $- 9$ by $125$ .

$625 - 1125 - 15 {(5)}^{2} + 325 (5) - 750$

Raise $5$ to the power of $2$ .

$625 - 1125 - 15 \cdot 25 + 325 (5) - 750$

Multiply $- 15$ by $25$ .

$625 - 1125 - 375 + 325 (5) - 750$

Multiply $325$ by $5$ .

$625 - 1125 - 375 + 1625 - 750$

Simplify by adding and subtracting.

Subtract $1125$ from $625$ .

$- 500 - 375 + 1625 - 750$

Subtract $375$ from $- 500$ .

$- 875 + 1625 - 750$

Add $- 875$ and $1625$ .

$750 - 750$

Subtract $750$ from $750$ .

$0$

Step 5

Since $5$ is a known root, divide the polynomial by $x - 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{4} - 9 x^{3} - 15 x^{2} + 325 x - 750}{x - 5}$

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.

$5$	$1$	$- 9$	$- 15$	$325$	$- 750$
	\xa0				\xa0

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

$5$	$1$	$- 9$	$- 15$	$325$	$- 750$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(5)$ and place the result of $(5)$ under the next term in the dividend $(- 9)$ .

$5$	$1$	$- 9$	$- 15$	$325$	$- 750$
	\xa0	$5$			\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.

$5$	$1$	$- 9$	$- 15$	$325$	$- 750$
	\xa0	$5$			\xa0
	$1$	$- 4$

Multiply the newest entry in the result $(- 4)$ by the divisor $(5)$ and place the result of $(- 20)$ under the next term in the dividend $(- 15)$ .

$5$	$1$	$- 9$	$- 15$	$325$	$- 750$
	\xa0	$5$	$- 20$		\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.

$5$	$1$	$- 9$	$- 15$	$325$	$- 750$
	\xa0	$5$	$- 20$		\xa0
	$1$	$- 4$	$- 35$

Multiply the newest entry in the result $(- 35)$ by the divisor $(5)$ and place the result of $(- 175)$ under the next term in the dividend $(325)$ .

$5$	$1$	$- 9$	$- 15$	$325$	$- 750$
	\xa0	$5$	$- 20$	$- 175$	\xa0
	$1$	$- 4$	$- 35$

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

$5$	$1$	$- 9$	$- 15$	$325$	$- 750$
	\xa0	$5$	$- 20$	$- 175$	\xa0
	$1$	$- 4$	$- 35$	$150$

Multiply the newest entry in the result $(150)$ by the divisor $(5)$ and place the result of $(750)$ under the next term in the dividend $(- 750)$ .

$5$	$1$	$- 9$	$- 15$	$325$	$- 750$
	\xa0	$5$	$- 20$	$- 175$	$750$
	$1$	$- 4$	$- 35$	$150$

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

$5$	$1$	$- 9$	$- 15$	$325$	$- 750$
	\xa0	$5$	$- 20$	$- 175$	$750$
	$1$	$- 4$	$- 35$	$150$	$0$

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

$1 x^{3} + - 4 x^{2} + (- 35) x + 150$

Simplify the quotient polynomial.

$x^{3} - 4 x^{2} - 35 x + 150$

Step 7

Solve the equation to find any remaining roots.

Factor the left side of the equation.

Factor $x^{3} - 4 x^{2} - 35 x + 150$ 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.

$p = \pm 1, \pm 150, \pm 2, \pm 75, \pm 3, \pm 50, \pm 5, \pm 30, \pm 6, \pm 25, \pm 10, \pm 15 q = \pm 1$

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

$\pm 1, \pm 150, \pm 2, \pm 75, \pm 3, \pm 50, \pm 5, \pm 30, \pm 6, \pm 25, \pm 10, \pm 15$

Substitute $5$ and simplify the expression. In this case, the expression is equal to $0$ so $5$ is a root of the polynomial.

Substitute $5$ into the polynomial.

$5^{3} - 4 \cdot 5^{2} - 35 \cdot 5 + 150$

Raise $5$ to the power of $3$ .

$125 - 4 \cdot 5^{2} - 35 \cdot 5 + 150$

Raise $5$ to the power of $2$ .

$125 - 4 \cdot 25 - 35 \cdot 5 + 150$

Multiply $- 4$ by $25$ .

$125 - 100 - 35 \cdot 5 + 150$

Subtract $100$ from $125$ .

$25 - 35 \cdot 5 + 150$

Multiply $- 35$ by $5$ .

$25 - 175 + 150$

Subtract $175$ from $25$ .

$- 150 + 150$

Add $- 150$ and $150$ .

$0$

Since $5$ is a known root, divide the polynomial by $x - 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 4 x^{2} - 35 x + 150}{x - 5}$

Divide $x^{3} - 4 x^{2} - 35 x + 150$ by $x - 5$ .

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

$x$

$5$

$x^{3}$

$4 x^{2}$

$35 x$

$150$

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

					$x^{2}$
	$x$	-	$5$		$x^{3}$	-	$4 x^{2}$	-	$35 x$	+	$150$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$5$		$x^{3}$	-	$4 x^{2}$	-	$35 x$	+	$150$
			+	$x^{3}$	-	$5 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 5 x^{2}$

				$x^{2}$
$x$	-	$5$		$x^{3}$	-	$4 x^{2}$	-	$35 x$	+	$150$
			-	$x^{3}$	+	$5 x^{2}$

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

				$x^{2}$
$x$	-	$5$		$x^{3}$	-	$4 x^{2}$	-	$35 x$	+	$150$
			-	$x^{3}$	+	$5 x^{2}$
					+	$x^{2}$

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

				$x^{2}$
$x$	-	$5$		$x^{3}$	-	$4 x^{2}$	-	$35 x$	+	$150$
			-	$x^{3}$	+	$5 x^{2}$
					+	$x^{2}$	-	$35 x$

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

				$x^{2}$	+	$x$
$x$	-	$5$		$x^{3}$	-	$4 x^{2}$	-	$35 x$	+	$150$
			-	$x^{3}$	+	$5 x^{2}$
					+	$x^{2}$	-	$35 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$x$
$x$	-	$5$		$x^{3}$	-	$4 x^{2}$	-	$35 x$	+	$150$
			-	$x^{3}$	+	$5 x^{2}$
					+	$x^{2}$	-	$35 x$
					+	$x^{2}$	-	$5 x$

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

				$x^{2}$	+	$x$
$x$	-	$5$		$x^{3}$	-	$4 x^{2}$	-	$35 x$	+	$150$
			-	$x^{3}$	+	$5 x^{2}$
					+	$x^{2}$	-	$35 x$
					-	$x^{2}$	+	$5 x$

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

				$x^{2}$	+	$x$
$x$	-	$5$		$x^{3}$	-	$4 x^{2}$	-	$35 x$	+	$150$
			-	$x^{3}$	+	$5 x^{2}$
					+	$x^{2}$	-	$35 x$
					-	$x^{2}$	+	$5 x$
							-	$30 x$

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

				$x^{2}$	+	$x$
$x$	-	$5$		$x^{3}$	-	$4 x^{2}$	-	$35 x$	+	$150$
			-	$x^{3}$	+	$5 x^{2}$
					+	$x^{2}$	-	$35 x$
					-	$x^{2}$	+	$5 x$
							-	$30 x$	+	$150$

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

				$x^{2}$	+	$x$	-	$30$
$x$	-	$5$		$x^{3}$	-	$4 x^{2}$	-	$35 x$	+	$150$
			-	$x^{3}$	+	$5 x^{2}$
					+	$x^{2}$	-	$35 x$
					-	$x^{2}$	+	$5 x$
							-	$30 x$	+	$150$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$x$	-	$30$
$x$	-	$5$		$x^{3}$	-	$4 x^{2}$	-	$35 x$	+	$150$
			-	$x^{3}$	+	$5 x^{2}$
					+	$x^{2}$	-	$35 x$
					-	$x^{2}$	+	$5 x$
							-	$30 x$	+	$150$
							-	$30 x$	+	$150$

The expression needs to be subtracted from the dividend, so change all the signs in $- 30 x + 150$

				$x^{2}$	+	$x$	-	$30$
$x$	-	$5$		$x^{3}$	-	$4 x^{2}$	-	$35 x$	+	$150$
			-	$x^{3}$	+	$5 x^{2}$
					+	$x^{2}$	-	$35 x$
					-	$x^{2}$	+	$5 x$
							-	$30 x$	+	$150$
							+	$30 x$	-	$150$

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

				$x^{2}$	+	$x$	-	$30$
$x$	-	$5$		$x^{3}$	-	$4 x^{2}$	-	$35 x$	+	$150$
			-	$x^{3}$	+	$5 x^{2}$
					+	$x^{2}$	-	$35 x$
					-	$x^{2}$	+	$5 x$
							-	$30 x$	+	$150$
							+	$30 x$	-	$150$
										$0$

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

$x^{2} + x - 30$

Write $x^{3} - 4 x^{2} - 35 x + 150$ as a set of factors.

$(x - 5) (x^{2} + x - 30) = 0$

Factor $x^{2} + x - 30$ 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 $- 30$ and whose sum is $1$ .

$- 5, 6$

Write the factored form using these integers.

$(x - 5) ((x - 5) (x + 6)) = 0$

Remove unnecessary parentheses.

$(x - 5) (x - 5) (x + 6) = 0$

Combine like factors.

Raise $x - 5$ to the power of $1$ .

$(x - 5) (x - 5) (x + 6) = 0$

Raise $x - 5$ to the power of $1$ .

$(x - 5) (x - 5) (x + 6) = 0$

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

${(x - 5)}^{1 + 1} (x + 6) = 0$

Add $1$ and $1$ .

${(x - 5)}^{2} (x + 6) = 0$

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

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

$x + 6 = 0$

Set ${(x - 5)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(x - 5)}^{2}$ equal to $0$ .

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

Solve ${(x - 5)}^{2} = 0$ for $x$ .

Set the $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Set $x + 6$ equal to $0$ and solve for $x$ .

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

The final solution is all the values that make ${(x - 5)}^{2} (x + 6) = 0$ true.

$x = 5, - 6$

Step 8

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

$(x - 5) (x + 6)$

Step 9

These are the roots (zeros) of the polynomial $x^{4} - 9 x^{3} - 15 x^{2} + 325 x - 750$ .

$x = 5, - 6$

Step 10

Find the Roots/Zeros Using the Rational Roots Test x^4-9x^3-15x^2+325x-750

Related Questions

Support LCMGCF.com by disabling your adblocker.