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

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

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.

$p = \pm 0$

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

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

$0$

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.

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

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

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Simplify each term.

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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 zeros.

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Add $0$ and $0$ .

$0 + 0$

Add $0$ and $0$ .

$0$

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.

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

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

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Place the numbers representing the divisor and the dividend into a division-like configuration.

$0$	$8$	$- 72$	$144$	$0$	$0$

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

$0$	$8$	$- 72$	$144$	$0$	$0$

	$8$

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 $(- 72)$ .

$0$	$8$	$- 72$	$144$	$0$	$0$
		$0$
	$8$

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

$0$	$8$	$- 72$	$144$	$0$	$0$
		$0$
	$8$	$- 72$

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 $(144)$ .

$0$	$8$	$- 72$	$144$	$0$	$0$
		$0$	$0$
	$8$	$- 72$

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

$0$	$8$	$- 72$	$144$	$0$	$0$
		$0$	$0$
	$8$	$- 72$	$144$

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 $(0)$ .

$0$	$8$	$- 72$	$144$	$0$	$0$
		$0$	$0$	$0$
	$8$	$- 72$	$144$

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

$0$	$8$	$- 72$	$144$	$0$	$0$
		$0$	$0$	$0$
	$8$	$- 72$	$144$	$0$

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 $(0)$ .

$0$	$8$	$- 72$	$144$	$0$	$0$
		$0$	$0$	$0$	$0$
	$8$	$- 72$	$144$	$0$

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

$0$	$8$	$- 72$	$144$	$0$	$0$
		$0$	$0$	$0$	$0$
	$8$	$- 72$	$144$	$0$	$0$

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.

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

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

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

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

Factor.

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Factor $x^{2} - 9 x + 18$ using the AC method.

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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.

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

Factor the left side of the equation.

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

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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.

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Factor $x^{2} - 9 x + 18$ using the AC method.

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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.

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

Divide each term by $8$ and simplify.

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Divide each term in $8 x^{2} (x - 6) (x - 3) = 0$ by $8$ .

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

Simplify $\frac{8 x^{2} (x - 6) (x - 3)}{8}$ .

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Cancel the common factor of $8$ .

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Cancel the common factor.

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

Divide $(x^{2} (x - 6)) (x - 3)$ by $1$ .

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

Apply the distributive property.

$(x^{2} x + x^{2} \cdot - 6) (x - 3) = \frac{0}{8}$

Multiply $x^{2}$ by $x$ by adding the exponents.

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Multiply $x^{2}$ by $x$ .

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

$(x^{2} x^{1} + x^{2} \cdot - 6) (x - 3) = \frac{0}{8}$

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

$(x^{2 + 1} + x^{2} \cdot - 6) (x - 3) = \frac{0}{8}$

Add $2$ and $1$ .

$(x^{3} + x^{2} \cdot - 6) (x - 3) = \frac{0}{8}$

Move $- 6$ to the left of $x^{2}$ .

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

Expand $(x^{3} - 6 x^{2}) (x - 3)$ using the FOIL Method.

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Apply the distributive property.

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

Apply the distributive property.

$x^{3} x + x^{3} \cdot - 3 - 6 x^{2} (x - 3) = \frac{0}{8}$

Apply the distributive property.

$x^{3} x + x^{3} \cdot - 3 - 6 x^{2} x - 6 x^{2} \cdot - 3 = \frac{0}{8}$

Simplify and combine like terms.

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Simplify each term.

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Multiply $x^{3}$ by $x$ by adding the exponents.

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Multiply $x^{3}$ by $x$ .

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

$x^{3} x^{1} + x^{3} \cdot - 3 - 6 x^{2} x - 6 x^{2} \cdot - 3 = \frac{0}{8}$

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

$x^{3 + 1} + x^{3} \cdot - 3 - 6 x^{2} x - 6 x^{2} \cdot - 3 = \frac{0}{8}$

Add $3$ and $1$ .

$x^{4} + x^{3} \cdot - 3 - 6 x^{2} x - 6 x^{2} \cdot - 3 = \frac{0}{8}$

Move $- 3$ to the left of $x^{3}$ .

$x^{4} - 3 \cdot x^{3} - 6 x^{2} x - 6 x^{2} \cdot - 3 = \frac{0}{8}$

Multiply $x^{2}$ by $x$ by adding the exponents.

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Move $x$ .

$x^{4} - 3 x^{3} - 6 (x \cdot x^{2}) - 6 x^{2} \cdot - 3 = \frac{0}{8}$

Multiply $x$ by $x^{2}$ .

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

$x^{4} - 3 x^{3} - 6 (x^{1} x^{2}) - 6 x^{2} \cdot - 3 = \frac{0}{8}$

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

$x^{4} - 3 x^{3} - 6 x^{1 + 2} - 6 x^{2} \cdot - 3 = \frac{0}{8}$

Add $1$ and $2$ .

$x^{4} - 3 x^{3} - 6 x^{3} - 6 x^{2} \cdot - 3 = \frac{0}{8}$

Multiply $- 3$ by $- 6$ .

$x^{4} - 3 x^{3} - 6 x^{3} + 18 x^{2} = \frac{0}{8}$

Subtract $6 x^{3}$ from $- 3 x^{3}$ .

$x^{4} - 9 x^{3} + 18 x^{2} = \frac{0}{8}$

Divide $0$ by $8$ .

$x^{4} - 9 x^{3} + 18 x^{2} = 0$

Factor the left side of the equation.

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Factor $x^{2}$ out of $x^{4} - 9 x^{3} + 18 x^{2}$ .

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

$x^{2} x^{2} - 9 x^{3} + 18 x^{2} = 0$

Factor $x^{2}$ out of $- 9 x^{3}$ .

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

Factor $x^{2}$ out of $18 x^{2}$ .

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

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

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

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

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

Factor.

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Factor $x^{2} - 9 x + 18$ using the AC method.

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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.

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

Remove unnecessary parentheses.

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

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

$x^{2} = 0$

$x - 6 = 0$

$x - 3 = 0$

Set the first factor equal to $0$ and solve.

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Set the first factor equal to $0$ .

$x^{2} = 0$

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

$x = \pm \sqrt{0}$

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

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Simplify the right side of the equation.

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

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

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

$x = \pm 0$

$\pm 0$ is equal to $0$ .

$x = 0$

Set the next factor equal to $0$ and solve.

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Set the next factor equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

Set the next factor equal to $0$ and solve.

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Set the next factor 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 $x^{2} (x - 6) (x - 3) = 0$ true.

$x = 0, 6, 3$

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

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