Find the Roots/Zeros Using the Rational Roots Test 8x^4-72x^3+144x^2

$8 x^{4} - 72 x^{3} + 144 x^{2}$

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.

$p = \pm 0$

$q = \pm 1, \pm 2, \pm 4, \pm 8$

Step 2

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

$0$

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.

$8 {(0)}^{4} - 72 {(0)}^{3} + 144 {(0)}^{2}$

Step 4

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

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$8 \cdot 0 - 72 {(0)}^{3} + 144 {(0)}^{2}$

Multiply $8$ by $0$ .

$0 - 72 {(0)}^{3} + 144 {(0)}^{2}$

Raising $0$ to any positive power yields $0$ .

$0 - 72 \cdot 0 + 144 {(0)}^{2}$

Multiply $- 72$ by $0$ .

$0 + 0 + 144 {(0)}^{2}$

Raising $0$ to any positive power yields $0$ .

$0 + 0 + 144 \cdot 0$

Multiply $144$ by $0$ .

$0 + 0 + 0$

Simplify by adding numbers.

Add $0$ and $0$ .

$0 + 0$

Add $0$ and $0$ .

$0$

Step 5

Since $0$ is a known root, divide the polynomial by $x + 0$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{8 x^{4} - 72 x^{3} + 144 x^{2}}{x + 0}$

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.

$0$	$8$	$- 72$	$144$	$0$	$0$
	\xa0				\xa0

The first number in the dividend $(8)$ is put into the first position of the result area (below the horizontal line).

$0$	$8$	$- 72$	$144$	$0$	$0$
	\xa0				\xa0
	$8$

Multiply the newest entry in the result $(8)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(- 72)$ .

$0$	$8$	$- 72$	$144$	$0$	$0$
	\xa0	$0$			\xa0
	$8$

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

$0$	$8$	$- 72$	$144$	$0$	$0$
	\xa0	$0$			\xa0
	$8$	$- 72$

Multiply the newest entry in the result $(- 72)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(144)$ .

$0$	$8$	$- 72$	$144$	$0$	$0$
	\xa0	$0$	$0$		\xa0
	$8$	$- 72$

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

$0$	$8$	$- 72$	$144$	$0$	$0$
	\xa0	$0$	$0$		\xa0
	$8$	$- 72$	$144$

Multiply the newest entry in the result $(144)$ by the divisor $(0)$ and place the result of $(0)$ under the next term in the dividend $(0)$ .

$0$	$8$	$- 72$	$144$	$0$	$0$
	\xa0	$0$	$0$	$0$	\xa0
	$8$	$- 72$	$144$

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

$0$	$8$	$- 72$	$144$	$0$	$0$
	\xa0	$0$	$0$	$0$	\xa0
	$8$	$- 72$	$144$	$0$

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

$0$	$8$	$- 72$	$144$	$0$	$0$
	\xa0	$0$	$0$	$0$	$0$
	$8$	$- 72$	$144$	$0$

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

$0$	$8$	$- 72$	$144$	$0$	$0$
	\xa0	$0$	$0$	$0$	$0$
	$8$	$- 72$	$144$	$0$	$0$

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

$8 x^{3} + - 72 x^{2} + (144) x + 0$

Simplify the quotient polynomial.

$8 x^{3} - 72 x^{2} + 144 x$

Step 7

Factor $8 x$ out of $8 x^{3} - 72 x^{2} + 144 x$ .

Factor $8 x$ out of $8 x^{3}$ .

$(x + 0) (8 x (x^{2}) - 72 x^{2} + 144 x)$

Factor $8 x$ out of $- 72 x^{2}$ .

$(x + 0) (8 x (x^{2}) + 8 x (- 9 x) + 144 x)$

Factor $8 x$ out of $144 x$ .

$(x + 0) (8 x (x^{2}) + 8 x (- 9 x) + 8 x (18))$

Factor $8 x$ out of $8 x (x^{2}) + 8 x (- 9 x)$ .

$(x + 0) (8 x (x^{2} - 9 x) + 8 x (18))$

Factor $8 x$ out of $8 x (x^{2} - 9 x) + 8 x (18)$ .

$(x + 0) (8 x (x^{2} - 9 x + 18))$

Step 8

Factor.

Factor $x^{2} - 9 x + 18$ 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 $18$ and whose sum is $- 9$ .

$- 6, - 3$

Write the factored form using these integers.

$(x + 0) (8 x ((x - 6) (x - 3)))$

Remove unnecessary parentheses.

$(x + 0) (8 x (x - 6) (x - 3))$

Step 9

Factor the left side of the equation.

Factor $8 x^{2}$ out of $8 x^{4} - 72 x^{3} + 144 x^{2}$ .

Factor $8 x^{2}$ out of $8 x^{4}$ .

$8 x^{2} (x^{2}) - 72 x^{3} + 144 x^{2} = 0$

Factor $8 x^{2}$ out of $- 72 x^{3}$ .

$8 x^{2} (x^{2}) + 8 x^{2} (- 9 x) + 144 x^{2} = 0$

Factor $8 x^{2}$ out of $144 x^{2}$ .

$8 x^{2} (x^{2}) + 8 x^{2} (- 9 x) + 8 x^{2} (18) = 0$

Factor $8 x^{2}$ out of $8 x^{2} (x^{2}) + 8 x^{2} (- 9 x)$ .

$8 x^{2} (x^{2} - 9 x) + 8 x^{2} (18) = 0$

Factor $8 x^{2}$ out of $8 x^{2} (x^{2} - 9 x) + 8 x^{2} (18)$ .

$8 x^{2} (x^{2} - 9 x + 18) = 0$

Factor.

Factor $x^{2} - 9 x + 18$ 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 $18$ and whose sum is $- 9$ .

$- 6, - 3$

Write the factored form using these integers.

$8 x^{2} ((x - 6) (x - 3)) = 0$

Remove unnecessary parentheses.

$8 x^{2} (x - 6) (x - 3) = 0$

Step 10

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

$x^{2} = 0$

$x - 6 = 0$

$x - 3 = 0$

Step 11

Set $x^{2}$ equal to $0$ and solve for $x$ .

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Solve $x^{2} = 0$ for $x$ .

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

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

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

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

Step 12

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

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

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

Step 13

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 14

The final solution is all the values that make $8 x^{2} (x - 6) (x - 3) = 0$ true.

$x = 0, 6, 3$

Step 15

Find the Roots/Zeros Using the Rational Roots Test 8x^4-72x^3+144x^2

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