Find the Roots/Zeros Using the Rational Roots Test x^4-8x^3-30x^2+400x-875

$x^{4} - 8 x^{3} - 30 x^{2} + 400 x - 875$

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.

$p = \pm 1, \pm 5, \pm 7, \pm 25, \pm 35, \pm 125, \pm 175, \pm 875$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 5, \pm 7, \pm 25, \pm 35, \pm 125, \pm 175, \pm 875$

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.

${(5)}^{4} - 8 {(5)}^{3} - 30 {(5)}^{2} + 400 (5) - 875$

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 $4$ .

$625 - 8 {(5)}^{3} - 30 {(5)}^{2} + 400 (5) - 875$

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

$625 - 8 \cdot 125 - 30 {(5)}^{2} + 400 (5) - 875$

Multiply $- 8$ by $125$ .

$625 - 1000 - 30 {(5)}^{2} + 400 (5) - 875$

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

$625 - 1000 - 30 \cdot 25 + 400 (5) - 875$

Multiply $- 30$ by $25$ .

$625 - 1000 - 750 + 400 (5) - 875$

Multiply $400$ by $5$ .

$625 - 1000 - 750 + 2000 - 875$

Simplify by adding and subtracting.

Subtract $1000$ from $625$ .

$- 375 - 750 + 2000 - 875$

Subtract $750$ from $- 375$ .

$- 1125 + 2000 - 875$

Add $- 1125$ and $2000$ .

$875 - 875$

Subtract $875$ from $875$ .

$0$

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.

$\frac{x^{4} - 8 x^{3} - 30 x^{2} + 400 x - 875}{x - 5}$

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.

$5$	$1$	$- 8$	$- 30$	$400$	$- 875$
	\xa0				\xa0

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

$5$	$1$	$- 8$	$- 30$	$400$	$- 875$
	\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 $(- 8)$ .

$5$	$1$	$- 8$	$- 30$	$400$	$- 875$
	\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.

$5$	$1$	$- 8$	$- 30$	$400$	$- 875$
	\xa0	$5$			\xa0
	$1$	$- 3$

Multiply the newest entry in the result $(- 3)$ by the divisor $(5)$ and place the result of $(- 15)$ under the next term in the dividend $(- 30)$ .

$5$	$1$	$- 8$	$- 30$	$400$	$- 875$
	\xa0	$5$	$- 15$		\xa0
	$1$	$- 3$

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

$5$	$1$	$- 8$	$- 30$	$400$	$- 875$
	\xa0	$5$	$- 15$		\xa0
	$1$	$- 3$	$- 45$

Multiply the newest entry in the result $(- 45)$ by the divisor $(5)$ and place the result of $(- 225)$ under the next term in the dividend $(400)$ .

$5$	$1$	$- 8$	$- 30$	$400$	$- 875$
	\xa0	$5$	$- 15$	$- 225$	\xa0
	$1$	$- 3$	$- 45$

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

$5$	$1$	$- 8$	$- 30$	$400$	$- 875$
	\xa0	$5$	$- 15$	$- 225$	\xa0
	$1$	$- 3$	$- 45$	$175$

Multiply the newest entry in the result $(175)$ by the divisor $(5)$ and place the result of $(875)$ under the next term in the dividend $(- 875)$ .

$5$	$1$	$- 8$	$- 30$	$400$	$- 875$
	\xa0	$5$	$- 15$	$- 225$	$875$
	$1$	$- 3$	$- 45$	$175$

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

$5$	$1$	$- 8$	$- 30$	$400$	$- 875$
	\xa0	$5$	$- 15$	$- 225$	$875$
	$1$	$- 3$	$- 45$	$175$	$0$

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

$1 x^{3} + - 3 x^{2} + (- 45) x + 175$

Simplify the quotient polynomial.

$x^{3} - 3 x^{2} - 45 x + 175$

Step 7

Solve the equation to find any remaining roots.

Factor the left side of the equation.

Factor $x^{3} - 3 x^{2} - 45 x + 175$ 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.

$p = \pm 1, \pm 175, \pm 5, \pm 35, \pm 7, \pm 25 q = \pm 1$

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

$\pm 1, \pm 175, \pm 5, \pm 35, \pm 7, \pm 25$

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.

$5^{3} - 3 \cdot 5^{2} - 45 \cdot 5 + 175$

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

$125 - 3 \cdot 5^{2} - 45 \cdot 5 + 175$

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

$125 - 3 \cdot 25 - 45 \cdot 5 + 175$

Multiply $- 3$ by $25$ .

$125 - 75 - 45 \cdot 5 + 175$

Subtract $75$ from $125$ .

$50 - 45 \cdot 5 + 175$

Multiply $- 45$ by $5$ .

$50 - 225 + 175$

Subtract $225$ from $50$ .

$- 175 + 175$

Add $- 175$ and $175$ .

$0$

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.

$\frac{x^{3} - 3 x^{2} - 45 x + 175}{x - 5}$

Divide $x^{3} - 3 x^{2} - 45 x + 175$ 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 $0$ .

$x$

$5$

$x^{3}$

$3 x^{2}$

$45 x$

$175$

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

					$x^{2}$
	$x$	-	$5$		$x^{3}$	-	$3 x^{2}$	-	$45 x$	+	$175$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$5$		$x^{3}$	-	$3 x^{2}$	-	$45 x$	+	$175$
			+	$x^{3}$	-	$5 x^{2}$

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

				$x^{2}$
$x$	-	$5$		$x^{3}$	-	$3 x^{2}$	-	$45 x$	+	$175$
			-	$x^{3}$	+	$5 x^{2}$

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

				$x^{2}$
$x$	-	$5$		$x^{3}$	-	$3 x^{2}$	-	$45 x$	+	$175$
			-	$x^{3}$	+	$5 x^{2}$
					+	$2 x^{2}$

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

				$x^{2}$
$x$	-	$5$		$x^{3}$	-	$3 x^{2}$	-	$45 x$	+	$175$
			-	$x^{3}$	+	$5 x^{2}$
					+	$2 x^{2}$	-	$45 x$

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

				$x^{2}$	+	$2 x$
$x$	-	$5$		$x^{3}$	-	$3 x^{2}$	-	$45 x$	+	$175$
			-	$x^{3}$	+	$5 x^{2}$
					+	$2 x^{2}$	-	$45 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$2 x$
$x$	-	$5$		$x^{3}$	-	$3 x^{2}$	-	$45 x$	+	$175$
			-	$x^{3}$	+	$5 x^{2}$
					+	$2 x^{2}$	-	$45 x$
					+	$2 x^{2}$	-	$10 x$

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{2} - 10 x$

				$x^{2}$	+	$2 x$
$x$	-	$5$		$x^{3}$	-	$3 x^{2}$	-	$45 x$	+	$175$
			-	$x^{3}$	+	$5 x^{2}$
					+	$2 x^{2}$	-	$45 x$
					-	$2 x^{2}$	+	$10 x$

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

				$x^{2}$	+	$2 x$
$x$	-	$5$		$x^{3}$	-	$3 x^{2}$	-	$45 x$	+	$175$
			-	$x^{3}$	+	$5 x^{2}$
					+	$2 x^{2}$	-	$45 x$
					-	$2 x^{2}$	+	$10 x$
							-	$35 x$

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

				$x^{2}$	+	$2 x$
$x$	-	$5$		$x^{3}$	-	$3 x^{2}$	-	$45 x$	+	$175$
			-	$x^{3}$	+	$5 x^{2}$
					+	$2 x^{2}$	-	$45 x$
					-	$2 x^{2}$	+	$10 x$
							-	$35 x$	+	$175$

Divide the highest order term in the dividend $- 35 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$2 x$	-	$35$
$x$	-	$5$		$x^{3}$	-	$3 x^{2}$	-	$45 x$	+	$175$
			-	$x^{3}$	+	$5 x^{2}$
					+	$2 x^{2}$	-	$45 x$
					-	$2 x^{2}$	+	$10 x$
							-	$35 x$	+	$175$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$2 x$	-	$35$
$x$	-	$5$		$x^{3}$	-	$3 x^{2}$	-	$45 x$	+	$175$
			-	$x^{3}$	+	$5 x^{2}$
					+	$2 x^{2}$	-	$45 x$
					-	$2 x^{2}$	+	$10 x$
							-	$35 x$	+	$175$
							-	$35 x$	+	$175$

The expression needs to be subtracted from the dividend, so change all the signs in $- 35 x + 175$

				$x^{2}$	+	$2 x$	-	$35$
$x$	-	$5$		$x^{3}$	-	$3 x^{2}$	-	$45 x$	+	$175$
			-	$x^{3}$	+	$5 x^{2}$
					+	$2 x^{2}$	-	$45 x$
					-	$2 x^{2}$	+	$10 x$
							-	$35 x$	+	$175$
							+	$35 x$	-	$175$

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

				$x^{2}$	+	$2 x$	-	$35$
$x$	-	$5$		$x^{3}$	-	$3 x^{2}$	-	$45 x$	+	$175$
			-	$x^{3}$	+	$5 x^{2}$
					+	$2 x^{2}$	-	$45 x$
					-	$2 x^{2}$	+	$10 x$
							-	$35 x$	+	$175$
							+	$35 x$	-	$175$
										$0$

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

$x^{2} + 2 x - 35$

Write $x^{3} - 3 x^{2} - 45 x + 175$ as a set of factors.

$(x - 5) (x^{2} + 2 x - 35) = 0$

Factor $x^{2} + 2 x - 35$ 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 $- 35$ and whose sum is $2$ .

$- 5, 7$

Write the factored form using these integers.

$(x - 5) ((x - 5) (x + 7)) = 0$

Remove unnecessary parentheses.

$(x - 5) (x - 5) (x + 7) = 0$

Combine like factors.

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

$(x - 5) (x - 5) (x + 7) = 0$

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

$(x - 5) (x - 5) (x + 7) = 0$

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

${(x - 5)}^{1 + 1} (x + 7) = 0$

Add $1$ and $1$ .

${(x - 5)}^{2} (x + 7) = 0$

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

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

$x + 7 = 0$

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 $x + 7$ equal to $0$ and solve for $x$ .

Set $x + 7$ equal to $0$ .

$x + 7 = 0$

Subtract $7$ from both sides of the equation.

$x = - 7$

The final solution is all the values that make ${(x - 5)}^{2} (x + 7) = 0$ true.

$x = 5, - 7$

Step 8

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

$(x - 5) (x + 7)$

Step 9

These are the roots (zeros) of the polynomial $x^{4} - 8 x^{3} - 30 x^{2} + 400 x - 875$ .

$x = 5, - 7$

Step 10

Find the Roots/Zeros Using the Rational Roots Test x^4-8x^3-30x^2+400x-875

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