Find the Roots/Zeros Using the Rational Roots Test 9x^4-48x^3+46x^2+48x+9

$9 x^{4} - 48 x^{3} + 46 x^{2} + 48 x + 9$

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.

$p = \pm 1, \pm 3, \pm 9$

$q = \pm 1, \pm 3, \pm 9$

Step 2

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

$\pm 1, \pm \frac{1}{3}, \pm \frac{1}{9}, \pm 3, \pm 9$

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.

$9 {(3)}^{4} - 48 {(3)}^{3} + 46 {(3)}^{2} + 48 (3) + 9$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = 3$ is a root of the polynomial.

Simplify each term.

Raise $3$ to the power of $4$ .

$9 \cdot 81 - 48 {(3)}^{3} + 46 {(3)}^{2} + 48 (3) + 9$

Multiply $9$ by $81$ .

$729 - 48 {(3)}^{3} + 46 {(3)}^{2} + 48 (3) + 9$

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

$729 - 48 \cdot 27 + 46 {(3)}^{2} + 48 (3) + 9$

Multiply $- 48$ by $27$ .

$729 - 1296 + 46 {(3)}^{2} + 48 (3) + 9$

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

$729 - 1296 + 46 \cdot 9 + 48 (3) + 9$

Multiply $46$ by $9$ .

$729 - 1296 + 414 + 48 (3) + 9$

Multiply $48$ by $3$ .

$729 - 1296 + 414 + 144 + 9$

Simplify by adding and subtracting.

Subtract $1296$ from $729$ .

$- 567 + 414 + 144 + 9$

Add $- 567$ and $414$ .

$- 153 + 144 + 9$

Add $- 153$ and $144$ .

$- 9 + 9$

Add $- 9$ and $9$ .

$0$

Step 5

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{9 x^{4} - 48 x^{3} + 46 x^{2} + 48 x + 9}{x - 3}$

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.

$3$	$9$	$- 48$	$46$	$48$	$9$
	\xa0				\xa0

The first number in the dividend $(9)$ is put into the first position of the result area (below the horizontal line).

$3$	$9$	$- 48$	$46$	$48$	$9$
	\xa0				\xa0
	$9$

Multiply the newest entry in the result $(9)$ by the divisor $(3)$ and place the result of $(27)$ under the next term in the dividend $(- 48)$ .

$3$	$9$	$- 48$	$46$	$48$	$9$
	\xa0	$27$			\xa0
	$9$

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

$3$	$9$	$- 48$	$46$	$48$	$9$
	\xa0	$27$			\xa0
	$9$	$- 21$

Multiply the newest entry in the result $(- 21)$ by the divisor $(3)$ and place the result of $(- 63)$ under the next term in the dividend $(46)$ .

$3$	$9$	$- 48$	$46$	$48$	$9$
	\xa0	$27$	$- 63$		\xa0
	$9$	$- 21$

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

$3$	$9$	$- 48$	$46$	$48$	$9$
	\xa0	$27$	$- 63$		\xa0
	$9$	$- 21$	$- 17$

Multiply the newest entry in the result $(- 17)$ by the divisor $(3)$ and place the result of $(- 51)$ under the next term in the dividend $(48)$ .

$3$	$9$	$- 48$	$46$	$48$	$9$
	\xa0	$27$	$- 63$	$- 51$	\xa0
	$9$	$- 21$	$- 17$

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

$3$	$9$	$- 48$	$46$	$48$	$9$
	\xa0	$27$	$- 63$	$- 51$	\xa0
	$9$	$- 21$	$- 17$	$- 3$

Multiply the newest entry in the result $(- 3)$ by the divisor $(3)$ and place the result of $(- 9)$ under the next term in the dividend $(9)$ .

$3$	$9$	$- 48$	$46$	$48$	$9$
	\xa0	$27$	$- 63$	$- 51$	$- 9$
	$9$	$- 21$	$- 17$	$- 3$

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

$3$	$9$	$- 48$	$46$	$48$	$9$
	\xa0	$27$	$- 63$	$- 51$	$- 9$
	$9$	$- 21$	$- 17$	$- 3$	$0$

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

$9 x^{3} + - 21 x^{2} + (- 17) x - 3$

Simplify the quotient polynomial.

$9 x^{3} - 21 x^{2} - 17 x - 3$

Step 7

Solve the equation to find any remaining roots.

Factor the left side of the equation.

Factor $9 x^{3} - 21 x^{2} - 17 x - 3$ 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.

$p = \pm 1, \pm 3 q = \pm 1, \pm 9, \pm 3$

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

$\pm 1, \pm 0.\overline{1}, \pm 0.\overline{3}, \pm 3$

Substitute $- 0.\overline{3}$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.\overline{3}$ is a root of the polynomial.

Substitute $- 0.\overline{3}$ into the polynomial.

$9 {(- 0.\overline{3})}^{3} - 21 {(- 0.\overline{3})}^{2} - 17 \cdot - 0.\overline{3} - 3$

Raise $- 0.\overline{3}$ to the power of $3$ .

$9 \cdot - 0.\overline{037} - 21 {(- 0.\overline{3})}^{2} - 17 \cdot - 0.\overline{3} - 3$

Multiply $9$ by $- 0.\overline{037}$ .

$- 0.\overline{3} - 21 {(- 0.\overline{3})}^{2} - 17 \cdot - 0.\overline{3} - 3$

Raise $- 0.\overline{3}$ to the power of $2$ .

$- 0.\overline{3} - 21 \cdot 0.\overline{1} - 17 \cdot - 0.\overline{3} - 3$

Multiply $- 21$ by $0.\overline{1}$ .

$- 0.\overline{3} - 2.\overline{3} - 17 \cdot - 0.\overline{3} - 3$

Subtract $2.\overline{3}$ from $- 0.\overline{3}$ .

$- 2.\overline{6} - 17 \cdot - 0.\overline{3} - 3$

Multiply $- 17$ by $- 0.\overline{3}$ .

$- 2.\overline{6} + 5.\overline{6} - 3$

Add $- 2.\overline{6}$ and $5.\overline{6}$ .

$3 - 3$

Subtract $3$ from $3$ .

$0$

Since $- 0.\overline{3}$ is a known root, divide the polynomial by $3 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{9 x^{3} - 21 x^{2} - 17 x - 3}{3 x + 1}$

Divide $9 x^{3} - 21 x^{2} - 17 x - 3$ by $3 x + 1$ .

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

$3 x$

$1$

$9 x^{3}$

$21 x^{2}$

$17 x$

$3$

Divide the highest order term in the dividend $9 x^{3}$ by the highest order term in divisor $3 x$ .

					$3 x^{2}$
	$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	-	$17 x$	-	$3$

Multiply the new quotient term by the divisor.

				$3 x^{2}$
$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	-	$17 x$	-	$3$
			+	$9 x^{3}$	+	$3 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $9 x^{3} + 3 x^{2}$

				$3 x^{2}$
$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	-	$17 x$	-	$3$
			-	$9 x^{3}$	-	$3 x^{2}$

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

				$3 x^{2}$
$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	-	$17 x$	-	$3$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$24 x^{2}$

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

				$3 x^{2}$
$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	-	$17 x$	-	$3$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$24 x^{2}$	-	$17 x$

Divide the highest order term in the dividend $- 24 x^{2}$ by the highest order term in divisor $3 x$ .

				$3 x^{2}$	-	$8 x$
$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	-	$17 x$	-	$3$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$24 x^{2}$	-	$17 x$

Multiply the new quotient term by the divisor.

				$3 x^{2}$	-	$8 x$
$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	-	$17 x$	-	$3$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$24 x^{2}$	-	$17 x$
					-	$24 x^{2}$	-	$8 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 24 x^{2} - 8 x$

				$3 x^{2}$	-	$8 x$
$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	-	$17 x$	-	$3$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$24 x^{2}$	-	$17 x$
					+	$24 x^{2}$	+	$8 x$

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

				$3 x^{2}$	-	$8 x$
$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	-	$17 x$	-	$3$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$24 x^{2}$	-	$17 x$
					+	$24 x^{2}$	+	$8 x$
							-	$9 x$

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

				$3 x^{2}$	-	$8 x$
$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	-	$17 x$	-	$3$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$24 x^{2}$	-	$17 x$
					+	$24 x^{2}$	+	$8 x$
							-	$9 x$	-	$3$

Divide the highest order term in the dividend $- 9 x$ by the highest order term in divisor $3 x$ .

				$3 x^{2}$	-	$8 x$	-	$3$
$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	-	$17 x$	-	$3$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$24 x^{2}$	-	$17 x$
					+	$24 x^{2}$	+	$8 x$
							-	$9 x$	-	$3$

Multiply the new quotient term by the divisor.

				$3 x^{2}$	-	$8 x$	-	$3$
$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	-	$17 x$	-	$3$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$24 x^{2}$	-	$17 x$
					+	$24 x^{2}$	+	$8 x$
							-	$9 x$	-	$3$
							-	$9 x$	-	$3$

The expression needs to be subtracted from the dividend, so change all the signs in $- 9 x - 3$

				$3 x^{2}$	-	$8 x$	-	$3$
$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	-	$17 x$	-	$3$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$24 x^{2}$	-	$17 x$
					+	$24 x^{2}$	+	$8 x$
							-	$9 x$	-	$3$
							+	$9 x$	+	$3$

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

				$3 x^{2}$	-	$8 x$	-	$3$
$3 x$	+	$1$		$9 x^{3}$	-	$21 x^{2}$	-	$17 x$	-	$3$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$24 x^{2}$	-	$17 x$
					+	$24 x^{2}$	+	$8 x$
							-	$9 x$	-	$3$
							+	$9 x$	+	$3$
										$0$

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

$3 x^{2} - 8 x - 3$

Write $9 x^{3} - 21 x^{2} - 17 x - 3$ as a set of factors.

$(3 x + 1) (3 x^{2} - 8 x - 3) = 0$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 3 = - 9$ and whose sum is $b = - 8$ .

Factor $- 8$ out of $- 8 x$ .

$(3 x + 1) (3 x^{2} - 8 x - 3) = 0$

Rewrite $- 8$ as $1$ plus $- 9$

$(3 x + 1) (3 x^{2} + (1 - 9) x - 3) = 0$

Apply the distributive property.

$(3 x + 1) (3 x^{2} + 1 x - 9 x - 3) = 0$

Multiply $x$ by $1$ .

$(3 x + 1) (3 x^{2} + x - 9 x - 3) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(3 x + 1) ((3 x^{2} + x) - 9 x - 3) = 0$

Factor out the greatest common factor (GCF) from each group.

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

Factor the polynomial by factoring out the greatest common factor, $3 x + 1$ .

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

Remove unnecessary parentheses.

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

Combine like factors.

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

$x - 3 = 0$

Set ${(3 x + 1)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(3 x + 1)}^{2}$ equal to $0$ .

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

Solve ${(3 x + 1)}^{2} = 0$ for $x$ .

Set the $3 x + 1$ equal to $0$ .

$3 x + 1 = 0$

Solve for $x$ .

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Divide each term in $3 x = - 1$ by $3$ and simplify.

Divide each term in $3 x = - 1$ by $3$ .

$\frac{3 x}{3} = \frac{- 1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 1}{3}$

Divide $x$ by $1$ .

$x = \frac{- 1}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{3}$

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.

$x = 3$

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

$x = - \frac{1}{3}, 3$

Step 8

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

$(x - 3) (x + \frac{1}{3})$

Step 9

These are the roots (zeros) of the polynomial $9 x^{4} - 48 x^{3} + 46 x^{2} + 48 x + 9$ .

$x = 3, - \frac{1}{3}$

Step 10

Find the Roots/Zeros Using the Rational Roots Test 9x^4-48x^3+46x^2+48x+9

Related Questions

Support LCMGCF.com by disabling your adblocker.