Find the Roots/Zeros Using the Rational Roots Test x^4-81

$x^{4} - 81$

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.

$p = \pm 1, \pm 3, \pm 9, \pm 27, \pm 81$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 3, \pm 9, \pm 27, \pm 81$

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.

${(3)}^{4} - 81$

Step 4

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

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

$81 - 81$

Subtract $81$ from $81$ .

$0$

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.

$\frac{x^{4} - 81}{x - 3}$

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.

$3$	$1$	$0$	$0$	$0$	$- 81$
	\xa0				\xa0

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

$3$	$1$	$0$	$0$	$0$	$- 81$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(3)$ and place the result of $(3)$ under the next term in the dividend $(0)$ .

$3$	$1$	$0$	$0$	$0$	$- 81$
	\xa0	$3$			\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.

$3$	$1$	$0$	$0$	$0$	$- 81$
	\xa0	$3$			\xa0
	$1$	$3$

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 $(0)$ .

$3$	$1$	$0$	$0$	$0$	$- 81$
	\xa0	$3$	$9$		\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.

$3$	$1$	$0$	$0$	$0$	$- 81$
	\xa0	$3$	$9$		\xa0
	$1$	$3$	$9$

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 $(0)$ .

$3$	$1$	$0$	$0$	$0$	$- 81$
	\xa0	$3$	$9$	$27$	\xa0
	$1$	$3$	$9$

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

$3$	$1$	$0$	$0$	$0$	$- 81$
	\xa0	$3$	$9$	$27$	\xa0
	$1$	$3$	$9$	$27$

Multiply the newest entry in the result $(27)$ by the divisor $(3)$ and place the result of $(81)$ under the next term in the dividend $(- 81)$ .

$3$	$1$	$0$	$0$	$0$	$- 81$
	\xa0	$3$	$9$	$27$	$81$
	$1$	$3$	$9$	$27$

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

$3$	$1$	$0$	$0$	$0$	$- 81$
	\xa0	$3$	$9$	$27$	$81$
	$1$	$3$	$9$	$27$	$0$

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

$1 x^{3} + 3 x^{2} + (9) x + 27$

Simplify the quotient polynomial.

$x^{3} + 3 x^{2} + 9 x + 27$

Step 7

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x - 3) (x^{3} + 3 x^{2}) + 9 x + 27$

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

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

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

Step 8

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

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

Step 9

Add $81$ to both sides of the equation.

$x^{4} = 81$

Step 10

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[4]{81}$

Step 11

Simplify $\pm \sqrt[4]{81}$ .

Rewrite $81$ as $3^{4}$ .

$x = \pm \sqrt[4]{3^{4}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 3$

Step 12

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 3$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 3$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 3, - 3$

Step 13

Find the Roots/Zeros Using the Rational Roots Test x^4-81

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