Find the Roots/Zeros Using the Rational Roots Test f(x)=x^3-11x^2+31x-21

$f (x) = x^{3} - 11 x^{2} + 31 x - 21$

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.

$p = \pm 1, \pm 3, \pm 7, \pm 21$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 3, \pm 7, \pm 21$

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.

${(1)}^{3} - 11 {(1)}^{2} + 31 (1) - 21$

Step 4

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

Simplify each term.

One to any power is one.

$1 - 11 {(1)}^{2} + 31 (1) - 21$

One to any power is one.

$1 - 11 \cdot 1 + 31 (1) - 21$

Multiply $- 11$ by $1$ .

$1 - 11 + 31 (1) - 21$

Multiply $31$ by $1$ .

$1 - 11 + 31 - 21$

Simplify by adding and subtracting.

Subtract $11$ from $1$ .

$- 10 + 31 - 21$

Add $- 10$ and $31$ .

$21 - 21$

Subtract $21$ from $21$ .

$0$

Step 5

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.

$\frac{x^{3} - 11 x^{2} + 31 x - 21}{x - 1}$

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.

$1$	$1$	$- 11$	$31$	$- 21$
	\xa0			\xa0

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

$1$	$1$	$- 11$	$31$	$- 21$
	\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 $(- 11)$ .

$1$	$1$	$- 11$	$31$	$- 21$
	\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.

$1$	$1$	$- 11$	$31$	$- 21$
	\xa0	$1$		\xa0
	$1$	$- 10$

Multiply the newest entry in the result $(- 10)$ by the divisor $(1)$ and place the result of $(- 10)$ under the next term in the dividend $(31)$ .

$1$	$1$	$- 11$	$31$	$- 21$
	\xa0	$1$	$- 10$	\xa0
	$1$	$- 10$

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

$1$	$1$	$- 11$	$31$	$- 21$
	\xa0	$1$	$- 10$	\xa0
	$1$	$- 10$	$21$

Multiply the newest entry in the result $(21)$ by the divisor $(1)$ and place the result of $(21)$ under the next term in the dividend $(- 21)$ .

$1$	$1$	$- 11$	$31$	$- 21$
	\xa0	$1$	$- 10$	$21$
	$1$	$- 10$	$21$

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

$1$	$1$	$- 11$	$31$	$- 21$
	\xa0	$1$	$- 10$	$21$
	$1$	$- 10$	$21$	$0$

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

$1 x^{2} + - 10 x + 21$

Simplify the quotient polynomial.

$x^{2} - 10 x + 21$

Step 7

Factor $x^{2} - 10 x + 21$ 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 $21$ and whose sum is $- 10$ .

$- 7, - 3$

Write the factored form using these integers.

$(x - 1) + (x - 7) (x - 3)$

$(x - 1) (x - 7) (x - 3)$

Step 8

Factor the left side of the equation.

Factor $x^{3} - 11 x^{2} + 31 x - 21$ 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.

$p = \pm 1, \pm 21, \pm 3, \pm 7 q = \pm 1$

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

$\pm 1, \pm 21, \pm 3, \pm 7$

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.

$1^{3} - 11 \cdot 1^{2} + 31 \cdot 1 - 21$

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

$1 - 11 \cdot 1^{2} + 31 \cdot 1 - 21$

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

$1 - 11 \cdot 1 + 31 \cdot 1 - 21$

Multiply $- 11$ by $1$ .

$1 - 11 + 31 \cdot 1 - 21$

Subtract $11$ from $1$ .

$- 10 + 31 \cdot 1 - 21$

Multiply $31$ by $1$ .

$- 10 + 31 - 21$

Add $- 10$ and $31$ .

$21 - 21$

Subtract $21$ from $21$ .

$0$

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.

$\frac{x^{3} - 11 x^{2} + 31 x - 21}{x - 1}$

Divide $x^{3} - 11 x^{2} + 31 x - 21$ 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 $0$ .

$x$

$1$

$x^{3}$

$11 x^{2}$

$31 x$

$21$

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

					$x^{2}$
	$x$	-	$1$		$x^{3}$	-	$11 x^{2}$	+	$31 x$	-	$21$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$11 x^{2}$	+	$31 x$	-	$21$
			+	$x^{3}$	-	$x^{2}$

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$11 x^{2}$	+	$31 x$	-	$21$
			-	$x^{3}$	+	$x^{2}$

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$11 x^{2}$	+	$31 x$	-	$21$
			-	$x^{3}$	+	$x^{2}$
					-	$10 x^{2}$

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	-	$11 x^{2}$	+	$31 x$	-	$21$
			-	$x^{3}$	+	$x^{2}$
					-	$10 x^{2}$	+	$31 x$

Divide the highest order term in the dividend $- 10 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$10 x$
$x$	-	$1$		$x^{3}$	-	$11 x^{2}$	+	$31 x$	-	$21$
			-	$x^{3}$	+	$x^{2}$
					-	$10 x^{2}$	+	$31 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$10 x$
$x$	-	$1$		$x^{3}$	-	$11 x^{2}$	+	$31 x$	-	$21$
			-	$x^{3}$	+	$x^{2}$
					-	$10 x^{2}$	+	$31 x$
					-	$10 x^{2}$	+	$10 x$

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

				$x^{2}$	-	$10 x$
$x$	-	$1$		$x^{3}$	-	$11 x^{2}$	+	$31 x$	-	$21$
			-	$x^{3}$	+	$x^{2}$
					-	$10 x^{2}$	+	$31 x$
					+	$10 x^{2}$	-	$10 x$

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

				$x^{2}$	-	$10 x$
$x$	-	$1$		$x^{3}$	-	$11 x^{2}$	+	$31 x$	-	$21$
			-	$x^{3}$	+	$x^{2}$
					-	$10 x^{2}$	+	$31 x$
					+	$10 x^{2}$	-	$10 x$
							+	$21 x$

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

				$x^{2}$	-	$10 x$
$x$	-	$1$		$x^{3}$	-	$11 x^{2}$	+	$31 x$	-	$21$
			-	$x^{3}$	+	$x^{2}$
					-	$10 x^{2}$	+	$31 x$
					+	$10 x^{2}$	-	$10 x$
							+	$21 x$	-	$21$

Divide the highest order term in the dividend $21 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$10 x$	+	$21$
$x$	-	$1$		$x^{3}$	-	$11 x^{2}$	+	$31 x$	-	$21$
			-	$x^{3}$	+	$x^{2}$
					-	$10 x^{2}$	+	$31 x$
					+	$10 x^{2}$	-	$10 x$
							+	$21 x$	-	$21$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$10 x$	+	$21$
$x$	-	$1$		$x^{3}$	-	$11 x^{2}$	+	$31 x$	-	$21$
			-	$x^{3}$	+	$x^{2}$
					-	$10 x^{2}$	+	$31 x$
					+	$10 x^{2}$	-	$10 x$
							+	$21 x$	-	$21$
							+	$21 x$	-	$21$

The expression needs to be subtracted from the dividend, so change all the signs in $21 x - 21$

				$x^{2}$	-	$10 x$	+	$21$
$x$	-	$1$		$x^{3}$	-	$11 x^{2}$	+	$31 x$	-	$21$
			-	$x^{3}$	+	$x^{2}$
					-	$10 x^{2}$	+	$31 x$
					+	$10 x^{2}$	-	$10 x$
							+	$21 x$	-	$21$
							-	$21 x$	+	$21$

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

				$x^{2}$	-	$10 x$	+	$21$
$x$	-	$1$		$x^{3}$	-	$11 x^{2}$	+	$31 x$	-	$21$
			-	$x^{3}$	+	$x^{2}$
					-	$10 x^{2}$	+	$31 x$
					+	$10 x^{2}$	-	$10 x$
							+	$21 x$	-	$21$
							-	$21 x$	+	$21$
										$0$

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

$x^{2} - 10 x + 21$

Write $x^{3} - 11 x^{2} + 31 x - 21$ as a set of factors.

$(x - 1) (x^{2} - 10 x + 21) = 0$

Factor $x^{2} - 10 x + 21$ 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 $21$ and whose sum is $- 10$ .

$- 7, - 3$

Write the factored form using these integers.

$(x - 1) ((x - 7) (x - 3)) = 0$

Remove unnecessary parentheses.

$(x - 1) (x - 7) (x - 3) = 0$

Step 9

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

$x - 1 = 0$

$x - 7 = 0$

$x - 3 = 0$

Step 10

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.

$x = 1$

Step 11

Set $x - 7$ equal to $0$ and solve for $x$ .

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

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

Step 12

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.

$x = 3$

Step 13

The final solution is all the values that make $(x - 1) (x - 7) (x - 3) = 0$ true.

$x = 1, 7, 3$

Step 14

Find the Roots/Zeros Using the Rational Roots Test f(x)=x^3-11x^2+31x-21

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