Precalculus Examples

$x^{6} + 16 x^{3} + 64$

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.

$p = \pm 1, \pm 2, \pm 4, \pm 8, \pm 16, \pm 32, \pm 64$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 2, \pm 4, \pm 8, \pm 16, \pm 32, \pm 64$

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.

${(- 2)}^{6} + 16 {(- 2)}^{3} + 64$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 2$ is a root of the polynomial.

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Step 4.1

Simplify each term.

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Step 4.1.1

Raise $- 2$ to the power of $6$ .

$64 + 16 {(- 2)}^{3} + 64$

Step 4.1.2

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

$64 + 16 \cdot - 8 + 64$

Step 4.1.3

Multiply $16$ by $- 8$ .

$64 - 128 + 64$

$64 - 128 + 64$

Step 4.2

Simplify by adding and subtracting.

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Step 4.2.1

Subtract $128$ from $64$ .

$- 64 + 64$

Step 4.2.2

Add $- 64$ and $64$ .

$0$

$0$

$0$

Step 5

Since $- 2$ is a known root, divide the polynomial by $x + 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{6} + 16 x^{3} + 64}{x + 2}$

Step 6

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

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Step 6.1

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

$- 2$	$1$	$0$	$0$	$16$	$0$	$0$	$64$

Step 6.2

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

$- 2$	$1$	$0$	$0$	$16$	$0$	$0$	$64$

	$1$

Step 6.3

Multiply the newest entry in the result $(1)$ by the divisor $(- 2)$ and place the result of $(- 2)$ under the next term in the dividend $(0)$ .

$- 2$	$1$	$0$	$0$	$16$	$0$	$0$	$64$
		$- 2$
	$1$

Step 6.4

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

$- 2$	$1$	$0$	$0$	$16$	$0$	$0$	$64$
		$- 2$
	$1$	$- 2$

Step 6.5

Multiply the newest entry in the result $(- 2)$ by the divisor $(- 2)$ and place the result of $(4)$ under the next term in the dividend $(0)$ .

$- 2$	$1$	$0$	$0$	$16$	$0$	$0$	$64$
		$- 2$	$4$
	$1$	$- 2$

Step 6.6

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

$- 2$	$1$	$0$	$0$	$16$	$0$	$0$	$64$
		$- 2$	$4$
	$1$	$- 2$	$4$

Step 6.7

Multiply the newest entry in the result $(4)$ by the divisor $(- 2)$ and place the result of $(- 8)$ under the next term in the dividend $(16)$ .

$- 2$	$1$	$0$	$0$	$16$	$0$	$0$	$64$
		$- 2$	$4$	$- 8$
	$1$	$- 2$	$4$

Step 6.8

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

$- 2$	$1$	$0$	$0$	$16$	$0$	$0$	$64$
		$- 2$	$4$	$- 8$
	$1$	$- 2$	$4$	$8$

Step 6.9

Multiply the newest entry in the result $(8)$ by the divisor $(- 2)$ and place the result of $(- 16)$ under the next term in the dividend $(0)$ .

$- 2$	$1$	$0$	$0$	$16$	$0$	$0$	$64$
		$- 2$	$4$	$- 8$	$- 16$
	$1$	$- 2$	$4$	$8$

Step 6.10

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

$- 2$	$1$	$0$	$0$	$16$	$0$	$0$	$64$
		$- 2$	$4$	$- 8$	$- 16$
	$1$	$- 2$	$4$	$8$	$- 16$

Step 6.11

Multiply the newest entry in the result $(- 16)$ by the divisor $(- 2)$ and place the result of $(32)$ under the next term in the dividend $(0)$ .

$- 2$	$1$	$0$	$0$	$16$	$0$	$0$	$64$
		$- 2$	$4$	$- 8$	$- 16$	$32$
	$1$	$- 2$	$4$	$8$	$- 16$

Step 6.12

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

$- 2$	$1$	$0$	$0$	$16$	$0$	$0$	$64$
		$- 2$	$4$	$- 8$	$- 16$	$32$
	$1$	$- 2$	$4$	$8$	$- 16$	$32$

Step 6.13

Multiply the newest entry in the result $(32)$ by the divisor $(- 2)$ and place the result of $(- 64)$ under the next term in the dividend $(64)$ .

$- 2$	$1$	$0$	$0$	$16$	$0$	$0$	$64$
		$- 2$	$4$	$- 8$	$- 16$	$32$	$- 64$
	$1$	$- 2$	$4$	$8$	$- 16$	$32$

Step 6.14

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

$- 2$	$1$	$0$	$0$	$16$	$0$	$0$	$64$
		$- 2$	$4$	$- 8$	$- 16$	$32$	$- 64$
	$1$	$- 2$	$4$	$8$	$- 16$	$32$	$0$

Step 6.15

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

$1 x^{5} + - 2 x^{4} + 4 x^{3} + 8 x^{2} + (- 16) x + 32$

Step 6.16

Simplify the quotient polynomial.

$x^{5} - 2 x^{4} + 4 x^{3} + 8 x^{2} - 16 x + 32$

$x^{5} - 2 x^{4} + 4 x^{3} + 8 x^{2} - 16 x + 32$

Step 7

Solve the equation to find any remaining roots.

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Step 7.1

Factor the left side of the equation.

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Step 7.1.1

Regroup terms.

$x^{5} - 2 x^{4} + 4 x^{3} + 8 x^{2} - 16 x + 32 = 0$

Step 7.1.2

Factor $x^{3}$ out of $x^{5} - 2 x^{4} + 4 x^{3}$ .

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Step 7.1.2.1

Factor $x^{3}$ out of $x^{5}$ .

$x^{3} x^{2} - 2 x^{4} + 4 x^{3} + 8 x^{2} - 16 x + 32 = 0$

Step 7.1.2.2

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

$x^{3} x^{2} + x^{3} (- 2 x) + 4 x^{3} + 8 x^{2} - 16 x + 32 = 0$

Step 7.1.2.3

Factor $x^{3}$ out of $4 x^{3}$ .

$x^{3} x^{2} + x^{3} (- 2 x) + x^{3} \cdot 4 + 8 x^{2} - 16 x + 32 = 0$

Step 7.1.2.4

Factor $x^{3}$ out of $x^{3} x^{2} + x^{3} (- 2 x)$ .

$x^{3} (x^{2} - 2 x) + x^{3} \cdot 4 + 8 x^{2} - 16 x + 32 = 0$

Step 7.1.2.5

Factor $x^{3}$ out of $x^{3} (x^{2} - 2 x) + x^{3} \cdot 4$ .

$x^{3} (x^{2} - 2 x + 4) + 8 x^{2} - 16 x + 32 = 0$

$x^{3} (x^{2} - 2 x + 4) + 8 x^{2} - 16 x + 32 = 0$

Step 7.1.3

Factor $8$ out of $8 x^{2} - 16 x + 32$ .

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Step 7.1.3.1

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

$x^{3} (x^{2} - 2 x + 4) + 8 (x^{2}) - 16 x + 32 = 0$

Step 7.1.3.2

Factor $8$ out of $- 16 x$ .

$x^{3} (x^{2} - 2 x + 4) + 8 (x^{2}) + 8 (- 2 x) + 32 = 0$

Step 7.1.3.3

Factor $8$ out of $32$ .

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

Step 7.1.3.4

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

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

Step 7.1.3.5

Factor $8$ out of $8 (x^{2} - 2 x) + 8 \cdot 4$ .

$x^{3} (x^{2} - 2 x + 4) + 8 (x^{2} - 2 x + 4) = 0$

$x^{3} (x^{2} - 2 x + 4) + 8 (x^{2} - 2 x + 4) = 0$

Step 7.1.4

Factor $x^{2} - 2 x + 4$ out of $x^{3} (x^{2} - 2 x + 4) + 8 (x^{2} - 2 x + 4)$ .

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Step 7.1.4.1

Factor $x^{2} - 2 x + 4$ out of $x^{3} (x^{2} - 2 x + 4)$ .

$(x^{2} - 2 x + 4) x^{3} + 8 (x^{2} - 2 x + 4) = 0$

Step 7.1.4.2

Factor $x^{2} - 2 x + 4$ out of $8 (x^{2} - 2 x + 4)$ .

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

Step 7.1.4.3

Factor $x^{2} - 2 x + 4$ out of $(x^{2} - 2 x + 4) x^{3} + (x^{2} - 2 x + 4) \cdot 8$ .

$(x^{2} - 2 x + 4) (x^{3} + 8) = 0$

$(x^{2} - 2 x + 4) (x^{3} + 8) = 0$

Step 7.1.5

Rewrite $8$ as $2^{3}$ .

$(x^{2} - 2 x + 4) (x^{3} + 2^{3}) = 0$

Step 7.1.6

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 2$ .

$(x^{2} - 2 x + 4) ((x + 2) (x^{2} - x \cdot 2 + 2^{2})) = 0$

Step 7.1.7

Factor.

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Step 7.1.7.1

Simplify.

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Step 7.1.7.1.1

Multiply $2$ by $- 1$ .

$(x^{2} - 2 x + 4) ((x + 2) (x^{2} - 2 x + 2^{2})) = 0$

Step 7.1.7.1.2

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

$(x^{2} - 2 x + 4) ((x + 2) (x^{2} - 2 x + 4)) = 0$

$(x^{2} - 2 x + 4) ((x + 2) (x^{2} - 2 x + 4)) = 0$

Step 7.1.7.2

Remove unnecessary parentheses.

$(x^{2} - 2 x + 4) (x + 2) (x^{2} - 2 x + 4) = 0$

$(x^{2} - 2 x + 4) (x + 2) (x^{2} - 2 x + 4) = 0$

Step 7.1.8

Combine exponents.

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Step 7.1.8.1

Raise $x^{2} - 2 x + 4$ to the power of $1$ .

$(x^{2} - 2 x + 4) (x^{2} - 2 x + 4) (x + 2) = 0$

Step 7.1.8.2

Raise $x^{2} - 2 x + 4$ to the power of $1$ .

$(x^{2} - 2 x + 4) (x^{2} - 2 x + 4) (x + 2) = 0$

Step 7.1.8.3

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

${(x^{2} - 2 x + 4)}^{1 + 1} (x + 2) = 0$

Step 7.1.8.4

Add $1$ and $1$ .

${(x^{2} - 2 x + 4)}^{2} (x + 2) = 0$

${(x^{2} - 2 x + 4)}^{2} (x + 2) = 0$

${(x^{2} - 2 x + 4)}^{2} (x + 2) = 0$

Step 7.2

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

${(x^{2} - 2 x + 4)}^{2} = 0$

$x + 2 = 0$

Step 7.3

Set ${(x^{2} - 2 x + 4)}^{2}$ equal to $0$ and solve for $x$ .

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Step 7.3.1

Set ${(x^{2} - 2 x + 4)}^{2}$ equal to $0$ .

${(x^{2} - 2 x + 4)}^{2} = 0$

Step 7.3.2

Solve ${(x^{2} - 2 x + 4)}^{2} = 0$ for $x$ .

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Step 7.3.2.1

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

$x^{2} - 2 x + 4 = \pm \sqrt{0}$

Step 7.3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 7.3.2.2.1

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

$x^{2} - 2 x + 4 = \pm \sqrt{0^{2}}$

Step 7.3.2.2.2

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

$x^{2} - 2 x + 4 = \pm 0$

Step 7.3.2.2.3

Plus or minus $0$ is $0$ .

$x^{2} - 2 x + 4 = 0$

$x^{2} - 2 x + 4 = 0$

Step 7.3.2.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 7.3.2.4

Substitute the values $a = 1$ , $b = - 2$ , and $c = 4$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot 4)}}{2 \cdot 1}$

Step 7.3.2.5

Simplify.

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Step 7.3.2.5.1

Simplify the numerator.

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Step 7.3.2.5.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 7.3.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 7.3.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Step 7.3.2.5.1.2.2

Multiply $- 4$ by $4$ .

$x = \frac{2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

$x = \frac{2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 7.3.2.5.1.3

Subtract $16$ from $4$ .

$x = \frac{2 \pm \sqrt{- 12}}{2 \cdot 1}$

Step 7.3.2.5.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Step 7.3.2.5.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Step 7.3.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 7.3.2.5.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 7.3.2.5.1.7.1

Factor $4$ out of $12$ .

$x = \frac{2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 7.3.2.5.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

$x = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 7.3.2.5.1.8

Pull terms out from under the radical.

$x = \frac{2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Step 7.3.2.5.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

$x = \frac{2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

Step 7.3.2.5.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 i \sqrt{3}}{2}$

Step 7.3.2.5.3

Simplify $\frac{2 \pm 2 i \sqrt{3}}{2}$ .

$x = 1 \pm i \sqrt{3}$

$x = 1 \pm i \sqrt{3}$

Step 7.3.2.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 7.3.2.6.1

Simplify the numerator.

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Step 7.3.2.6.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 7.3.2.6.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 7.3.2.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Step 7.3.2.6.1.2.2

Multiply $- 4$ by $4$ .

$x = \frac{2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

$x = \frac{2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 7.3.2.6.1.3

Subtract $16$ from $4$ .

$x = \frac{2 \pm \sqrt{- 12}}{2 \cdot 1}$

Step 7.3.2.6.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Step 7.3.2.6.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Step 7.3.2.6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 7.3.2.6.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 7.3.2.6.1.7.1

Factor $4$ out of $12$ .

$x = \frac{2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 7.3.2.6.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

$x = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 7.3.2.6.1.8

Pull terms out from under the radical.

$x = \frac{2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Step 7.3.2.6.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

$x = \frac{2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

Step 7.3.2.6.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 i \sqrt{3}}{2}$

Step 7.3.2.6.3

Simplify $\frac{2 \pm 2 i \sqrt{3}}{2}$ .

$x = 1 \pm i \sqrt{3}$

Step 7.3.2.6.4

Change the $\pm$ to $+$ .

$x = 1 + i \sqrt{3}$

$x = 1 + i \sqrt{3}$

Step 7.3.2.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 7.3.2.7.1

Simplify the numerator.

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Step 7.3.2.7.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 7.3.2.7.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 7.3.2.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Step 7.3.2.7.1.2.2

Multiply $- 4$ by $4$ .

$x = \frac{2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

$x = \frac{2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 7.3.2.7.1.3

Subtract $16$ from $4$ .

$x = \frac{2 \pm \sqrt{- 12}}{2 \cdot 1}$

Step 7.3.2.7.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Step 7.3.2.7.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Step 7.3.2.7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 7.3.2.7.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 7.3.2.7.1.7.1

Factor $4$ out of $12$ .

$x = \frac{2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 7.3.2.7.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

$x = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 7.3.2.7.1.8

Pull terms out from under the radical.

$x = \frac{2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Step 7.3.2.7.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

$x = \frac{2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

Step 7.3.2.7.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 i \sqrt{3}}{2}$

Step 7.3.2.7.3

Simplify $\frac{2 \pm 2 i \sqrt{3}}{2}$ .

$x = 1 \pm i \sqrt{3}$

Step 7.3.2.7.4

Change the $\pm$ to $-$ .

$x = 1 - i \sqrt{3}$

$x = 1 - i \sqrt{3}$

Step 7.3.2.8

The final answer is the combination of both solutions.

$x = 1 + i \sqrt{3}, 1 - i \sqrt{3}$

$x = 1 + i \sqrt{3}, 1 - i \sqrt{3}$

$x = 1 + i \sqrt{3}, 1 - i \sqrt{3}$

Step 7.4

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 7.4.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 7.4.2

Subtract $2$ from both sides of the equation.

$x = - 2$

$x = - 2$

Step 7.5

The final solution is all the values that make ${(x^{2} - 2 x + 4)}^{2} (x + 2) = 0$ true.

$x = 1 + i \sqrt{3}, 1 - i \sqrt{3}, - 2$

$x = 1 + i \sqrt{3}, 1 - i \sqrt{3}, - 2$

Step 8

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

$(x + 2) (x - 1 - i \sqrt{3}) (x - 1 + \sqrt{3} i)$

Step 9

These are the roots (zeros) of the polynomial $x^{6} + 16 x^{3} + 64$ .

$x = - 2, 1 + i \sqrt{3}, 1 - i \sqrt{3}$

Step 10

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$