Find the Roots (Zeros) f(x)=x^6-8x^5+12x^4+80x^3-416x^2+768x-512

$f (x) = x^{6} - 8 x^{5} + 12 x^{4} + 80 x^{3} - 416 x^{2} + 768 x - 512$

Step 1

Set $x^{6} - 8 x^{5} + 12 x^{4} + 80 x^{3} - 416 x^{2} + 768 x - 512$ equal to $0$ .

$x^{6} - 8 x^{5} + 12 x^{4} + 80 x^{3} - 416 x^{2} + 768 x - 512 = 0$

Step 2

Solve for $x$ .

Factor the left side of the equation.

Factor $x^{6} - 8 x^{5} + 12 x^{4} + 80 x^{3} - 416 x^{2} + 768 x - 512$ 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.

$p = \pm 1, \pm 512, \pm 2, \pm 256, \pm 4, \pm 128, \pm 8, \pm 64, \pm 16, \pm 32 q = \pm 1$

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

$\pm 1, \pm 512, \pm 2, \pm 256, \pm 4, \pm 128, \pm 8, \pm 64, \pm 16, \pm 32$

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

Substitute $2$ into the polynomial.

$2^{6} - 8 \cdot 2^{5} + 12 \cdot 2^{4} + 80 \cdot 2^{3} - 416 \cdot 2^{2} + 768 \cdot 2 - 512$

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

$64 - 8 \cdot 2^{5} + 12 \cdot 2^{4} + 80 \cdot 2^{3} - 416 \cdot 2^{2} + 768 \cdot 2 - 512$

Raise $2$ to the power of $5$ .

$64 - 8 \cdot 32 + 12 \cdot 2^{4} + 80 \cdot 2^{3} - 416 \cdot 2^{2} + 768 \cdot 2 - 512$

Multiply $- 8$ by $32$ .

$64 - 256 + 12 \cdot 2^{4} + 80 \cdot 2^{3} - 416 \cdot 2^{2} + 768 \cdot 2 - 512$

Subtract $256$ from $64$ .

$- 192 + 12 \cdot 2^{4} + 80 \cdot 2^{3} - 416 \cdot 2^{2} + 768 \cdot 2 - 512$

Raise $2$ to the power of $4$ .

$- 192 + 12 \cdot 16 + 80 \cdot 2^{3} - 416 \cdot 2^{2} + 768 \cdot 2 - 512$

Multiply $12$ by $16$ .

$- 192 + 192 + 80 \cdot 2^{3} - 416 \cdot 2^{2} + 768 \cdot 2 - 512$

Add $- 192$ and $192$ .

$0 + 80 \cdot 2^{3} - 416 \cdot 2^{2} + 768 \cdot 2 - 512$

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

$0 + 80 \cdot 8 - 416 \cdot 2^{2} + 768 \cdot 2 - 512$

Multiply $80$ by $8$ .

$0 + 640 - 416 \cdot 2^{2} + 768 \cdot 2 - 512$

Add $0$ and $640$ .

$640 - 416 \cdot 2^{2} + 768 \cdot 2 - 512$

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

$640 - 416 \cdot 4 + 768 \cdot 2 - 512$

Multiply $- 416$ by $4$ .

$640 - 1664 + 768 \cdot 2 - 512$

Subtract $1664$ from $640$ .

$- 1024 + 768 \cdot 2 - 512$

Multiply $768$ by $2$ .

$- 1024 + 1536 - 512$

Add $- 1024$ and $1536$ .

$512 - 512$

Subtract $512$ from $512$ .

$0$

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.

$\frac{x^{6} - 8 x^{5} + 12 x^{4} + 80 x^{3} - 416 x^{2} + 768 x - 512}{x - 2}$

Divide $x^{6} - 8 x^{5} + 12 x^{4} + 80 x^{3} - 416 x^{2} + 768 x - 512$ by $x - 2$ .

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

$x$

$2$

$x^{6}$

$8 x^{5}$

$12 x^{4}$

$80 x^{3}$

$416 x^{2}$

$768 x$

$512$

Divide the highest order term in the dividend $x^{6}$ by the highest order term in divisor $x$ .

$x^{5}$

$x$

$2$

$x^{6}$

$8 x^{5}$

$12 x^{4}$

$80 x^{3}$

$416 x^{2}$

$768 x$

$512$

Multiply the new quotient term by the divisor.

$x^{5}$

$x$

$2$

$x^{6}$

$8 x^{5}$

$12 x^{4}$

$80 x^{3}$

$416 x^{2}$

$768 x$

$512$

$x^{6}$

$2 x^{5}$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{6} - 2 x^{5}$

$x^{5}$

$x$

$2$

$x^{6}$

$8 x^{5}$

$12 x^{4}$

$80 x^{3}$

$416 x^{2}$

$768 x$

$512$

$x^{6}$

$2 x^{5}$

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

$x^{5}$

$x$

$2$

$x^{6}$

$8 x^{5}$

$12 x^{4}$

$80 x^{3}$

$416 x^{2}$

$768 x$

$512$

$x^{6}$

$2 x^{5}$

$6 x^{5}$

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

$x^{5}$

$x$

$2$

$x^{6}$

$8 x^{5}$

$12 x^{4}$

$80 x^{3}$

$416 x^{2}$

$768 x$

$512$

$x^{6}$

$2 x^{5}$

$6 x^{5}$

$12 x^{4}$

Divide the highest order term in the dividend $- 6 x^{5}$ by the highest order term in divisor $x$ .

$x^{5}$

$6 x^{4}$

$x$

$2$

$x^{6}$

$8 x^{5}$

$12 x^{4}$

$80 x^{3}$

$416 x^{2}$

$768 x$

$512$

$x^{6}$

$2 x^{5}$

$6 x^{5}$

$12 x^{4}$

Multiply the new quotient term by the divisor.

$x^{5}$

$6 x^{4}$

$x$

$2$

$x^{6}$

$8 x^{5}$

$12 x^{4}$

$80 x^{3}$

$416 x^{2}$

$768 x$

$512$

$x^{6}$

$2 x^{5}$

$6 x^{5}$

$12 x^{4}$

$6 x^{5}$

$12 x^{4}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{5} + 12 x^{4}$

$x^{5}$

$6 x^{4}$

$x$

$2$

$x^{6}$

$8 x^{5}$

$12 x^{4}$

$80 x^{3}$

$416 x^{2}$

$768 x$

$512$

$x^{6}$

$2 x^{5}$

$6 x^{5}$

$12 x^{4}$

$6 x^{5}$

$12 x^{4}$

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

$x^{5}$

$6 x^{4}$

$x$

$2$

$x^{6}$

$8 x^{5}$

$12 x^{4}$

$80 x^{3}$

$416 x^{2}$

$768 x$

$512$

$x^{6}$

$2 x^{5}$

$6 x^{5}$

$12 x^{4}$

$6 x^{5}$

$12 x^{4}$

$0$

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

$x^{5}$

$6 x^{4}$

$x$

$2$

$x^{6}$

$8 x^{5}$

$12 x^{4}$

$80 x^{3}$

$416 x^{2}$

$768 x$

$512$

$x^{6}$

$2 x^{5}$

$6 x^{5}$

$12 x^{4}$

$6 x^{5}$

$12 x^{4}$

$0$

$80 x^{3}$

$416 x^{2}$

Divide the highest order term in the dividend $80 x^{3}$ by the highest order term in divisor $x$ .

$x^{5}$

$6 x^{4}$

$0 x^{3}$

$80 x^{2}$

$x$

$2$

$x^{6}$

$8 x^{5}$

$12 x^{4}$

$80 x^{3}$

$416 x^{2}$

$768 x$

$512$

$x^{6}$

$2 x^{5}$

$6 x^{5}$

$12 x^{4}$

$6 x^{5}$

$12 x^{4}$

$0$

$80 x^{3}$

$416 x^{2}$

Multiply the new quotient term by the divisor.

$x^{5}$

$6 x^{4}$

$0 x^{3}$

$80 x^{2}$

$x$

$2$

$x^{6}$

$8 x^{5}$

$12 x^{4}$

$80 x^{3}$

$416 x^{2}$

$768 x$

$512$

$x^{6}$

$2 x^{5}$

$6 x^{5}$

$12 x^{4}$

$6 x^{5}$

$12 x^{4}$

$0$

$80 x^{3}$

$416 x^{2}$

$80 x^{3}$

$160 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $80 x^{3} - 160 x^{2}$

$x^{5}$

$6 x^{4}$

$0 x^{3}$

$80 x^{2}$

$x$

$2$

$x^{6}$

$8 x^{5}$

$12 x^{4}$

$80 x^{3}$

$416 x^{2}$

$768 x$

$512$

$x^{6}$

$2 x^{5}$

$6 x^{5}$

$12 x^{4}$

$6 x^{5}$

$12 x^{4}$

$0$

$80 x^{3}$

$416 x^{2}$

$80 x^{3}$

$160 x^{2}$

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

$x^{5}$

$6 x^{4}$

$0 x^{3}$

$80 x^{2}$

$x$

$2$

$x^{6}$

$8 x^{5}$

$12 x^{4}$

$80 x^{3}$

$416 x^{2}$

$768 x$

$512$

$x^{6}$

$2 x^{5}$

$6 x^{5}$

$12 x^{4}$

$6 x^{5}$

$12 x^{4}$

$0$

$80 x^{3}$

$416 x^{2}$

$80 x^{3}$

$160 x^{2}$

$256 x^{2}$

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

$x^{5}$

$6 x^{4}$

$0 x^{3}$

$80 x^{2}$

$x$

$2$

$x^{6}$

$8 x^{5}$

$12 x^{4}$

$80 x^{3}$

$416 x^{2}$

$768 x$

$512$

$x^{6}$

$2 x^{5}$

$6 x^{5}$

$12 x^{4}$

$6 x^{5}$

$12 x^{4}$

$0$

$80 x^{3}$

$416 x^{2}$

$80 x^{3}$

$160 x^{2}$

$256 x^{2}$

$768 x$

Divide the highest order term in the dividend $- 256 x^{2}$ by the highest order term in divisor $x$ .

$x^{5}$

$6 x^{4}$

$0 x^{3}$

$80 x^{2}$

$256 x$

$x$

$2$

$x^{6}$

$8 x^{5}$

$12 x^{4}$

$80 x^{3}$

$416 x^{2}$

$768 x$

$512$

$x^{6}$

$2 x^{5}$

$6 x^{5}$

$12 x^{4}$

$6 x^{5}$

$12 x^{4}$

$0$

$80 x^{3}$

$416 x^{2}$

$80 x^{3}$

$160 x^{2}$

$256 x^{2}$

$768 x$

Multiply the new quotient term by the divisor.

$x^{5}$

$6 x^{4}$

$0 x^{3}$

$80 x^{2}$

$256 x$

$x$

$2$

$x^{6}$

$8 x^{5}$

$12 x^{4}$

$80 x^{3}$

$416 x^{2}$

$768 x$

$512$

$x^{6}$

$2 x^{5}$

$6 x^{5}$

$12 x^{4}$

$6 x^{5}$

$12 x^{4}$

$0$

$80 x^{3}$

$416 x^{2}$

$80 x^{3}$

$160 x^{2}$

$256 x^{2}$

$768 x$

$256 x^{2}$

$512 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 256 x^{2} + 512 x$

$x^{5}$

$6 x^{4}$

$0 x^{3}$

$80 x^{2}$

$256 x$

$x$

$2$

$x^{6}$

$8 x^{5}$

$12 x^{4}$

$80 x^{3}$

$416 x^{2}$

$768 x$

$512$

$x^{6}$

$2 x^{5}$

$6 x^{5}$

$12 x^{4}$

$6 x^{5}$

$12 x^{4}$

$0$

$80 x^{3}$

$416 x^{2}$

$80 x^{3}$

$160 x^{2}$

$256 x^{2}$

$768 x$

$256 x^{2}$

$512 x$

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

$x^{5}$

$6 x^{4}$

$0 x^{3}$

$80 x^{2}$

$256 x$

$x$

$2$

$x^{6}$

$8 x^{5}$

$12 x^{4}$

$80 x^{3}$

$416 x^{2}$

$768 x$

$512$

$x^{6}$

$2 x^{5}$

$6 x^{5}$

$12 x^{4}$

$6 x^{5}$

$12 x^{4}$

$0$

$80 x^{3}$

$416 x^{2}$

$80 x^{3}$

$160 x^{2}$

$256 x^{2}$

$768 x$

$256 x^{2}$

$512 x$

$256 x$

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

$x^{5}$

$6 x^{4}$

$0 x^{3}$

$80 x^{2}$

$256 x$

$x$

$2$

$x^{6}$

$8 x^{5}$

$12 x^{4}$

$80 x^{3}$

$416 x^{2}$

$768 x$

$512$

$x^{6}$

$2 x^{5}$

$6 x^{5}$

$12 x^{4}$

$6 x^{5}$

$12 x^{4}$

$0$

$80 x^{3}$

$416 x^{2}$

$80 x^{3}$

$160 x^{2}$

$256 x^{2}$

$768 x$

$256 x^{2}$

$512 x$

$256 x$

$512$

Divide the highest order term in the dividend $256 x$ by the highest order term in divisor $x$ .

$x^{5}$

$6 x^{4}$

$0 x^{3}$

$80 x^{2}$

$256 x$

$256$

$x$

$2$

$x^{6}$

$8 x^{5}$

$12 x^{4}$

$80 x^{3}$

$416 x^{2}$

$768 x$

$512$

$x^{6}$

$2 x^{5}$

$6 x^{5}$

$12 x^{4}$

$6 x^{5}$

$12 x^{4}$

$0$

$80 x^{3}$

$416 x^{2}$

$80 x^{3}$

$160 x^{2}$

$256 x^{2}$

$768 x$

$256 x^{2}$

$512 x$

$256 x$

$512$

Multiply the new quotient term by the divisor.

$x^{5}$

$6 x^{4}$

$0 x^{3}$

$80 x^{2}$

$256 x$

$256$

$x$

$2$

$x^{6}$

$8 x^{5}$

$12 x^{4}$

$80 x^{3}$

$416 x^{2}$

$768 x$

$512$

$x^{6}$

$2 x^{5}$

$6 x^{5}$

$12 x^{4}$

$6 x^{5}$

$12 x^{4}$

$0$

$80 x^{3}$

$416 x^{2}$

$80 x^{3}$

$160 x^{2}$

$256 x^{2}$

$768 x$

$256 x^{2}$

$512 x$

$256 x$

$512$

$256 x$

$512$

The expression needs to be subtracted from the dividend, so change all the signs in $256 x - 512$

$x^{5}$

$6 x^{4}$

$0 x^{3}$

$80 x^{2}$

$256 x$

$256$

$x$

$2$

$x^{6}$

$8 x^{5}$

$12 x^{4}$

$80 x^{3}$

$416 x^{2}$

$768 x$

$512$

$x^{6}$

$2 x^{5}$

$6 x^{5}$

$12 x^{4}$

$6 x^{5}$

$12 x^{4}$

$0$

$80 x^{3}$

$416 x^{2}$

$80 x^{3}$

$160 x^{2}$

$256 x^{2}$

$768 x$

$256 x^{2}$

$512 x$

$256 x$

$512$

$256 x$

$512$

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

$x^{5}$

$6 x^{4}$

$0 x^{3}$

$80 x^{2}$

$256 x$

$256$

$x$

$2$

$x^{6}$

$8 x^{5}$

$12 x^{4}$

$80 x^{3}$

$416 x^{2}$

$768 x$

$512$

$x^{6}$

$2 x^{5}$

$6 x^{5}$

$12 x^{4}$

$6 x^{5}$

$12 x^{4}$

$0$

$80 x^{3}$

$416 x^{2}$

$80 x^{3}$

$160 x^{2}$

$256 x^{2}$

$768 x$

$256 x^{2}$

$512 x$

$256 x$

$512$

$256 x$

$512$

$0$

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

$x^{5} - 6 x^{4} + 0 x^{3} + 80 x^{2} - 256 x + 256$

Write $x^{6} - 8 x^{5} + 12 x^{4} + 80 x^{3} - 416 x^{2} + 768 x - 512$ as a set of factors.

$(x - 2) (x^{5} - 6 x^{4} + 0 x^{3} + 80 x^{2} - 256 x + 256) = 0$

Regroup terms.

$(x - 2) (x^{5} + 0 x^{3} - 256 x - 6 x^{4} + 80 x^{2} + 256) = 0$

Factor $x$ out of $x^{5} + 0 x^{3} - 256 x$ .

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

$(x - 2) (x \cdot x^{4} + 0 x^{3} - 256 x - 6 x^{4} + 80 x^{2} + 256) = 0$

Factor $x$ out of $0 x^{3}$ .

$(x - 2) (x \cdot x^{4} + x (0 x^{2}) - 256 x - 6 x^{4} + 80 x^{2} + 256) = 0$

Factor $x$ out of $- 256 x$ .

$(x - 2) (x \cdot x^{4} + x (0 x^{2}) + x \cdot - 256 - 6 x^{4} + 80 x^{2} + 256) = 0$

Factor $x$ out of $x \cdot x^{4} + x (0 x^{2})$ .

$(x - 2) (x (x^{4} + 0 x^{2}) + x \cdot - 256 - 6 x^{4} + 80 x^{2} + 256) = 0$

Factor $x$ out of $x (x^{4} + 0 x^{2}) + x \cdot - 256$ .

$(x - 2) (x (x^{4} + 0 x^{2} - 256) - 6 x^{4} + 80 x^{2} + 256) = 0$

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$(x - 2) (x ({(x^{2})}^{2} + 0 x^{2} - 256) - 6 x^{4} + 80 x^{2} + 256) = 0$

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$(x - 2) (x (u^{2} - 256) - 6 x^{4} + 80 x^{2} + 256) = 0$

Rewrite $256$ as $16^{2}$ .

$(x - 2) (x (u^{2} - 16^{2}) - 6 x^{4} + 80 x^{2} + 256) = 0$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = u$ and $b = 16$ .

$(x - 2) (x ((u + 16) (u - 16)) - 6 x^{4} + 80 x^{2} + 256) = 0$

Replace all occurrences of $u$ with $x^{2}$ .

$(x - 2) (x ((x^{2} + 16) (x^{2} - 16)) - 6 x^{4} + 80 x^{2} + 256) = 0$

Rewrite $16$ as $4^{2}$ .

$(x - 2) (x ((x^{2} + 16) (x^{2} - 4^{2})) - 6 x^{4} + 80 x^{2} + 256) = 0$

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

$(x - 2) (x ((x^{2} + 16) ((x + 4) (x - 4))) - 6 x^{4} + 80 x^{2} + 256) = 0$

Remove unnecessary parentheses.

$(x - 2) (x ((x^{2} + 16) (x + 4) (x - 4)) - 6 x^{4} + 80 x^{2} + 256) = 0$

Remove unnecessary parentheses.

$(x - 2) (x (x^{2} + 16) (x + 4) (x - 4) - 6 x^{4} + 80 x^{2} + 256) = 0$

Factor $2$ out of $- 6 x^{4} + 80 x^{2} + 256$ .

Factor $2$ out of $- 6 x^{4}$ .

$(x - 2) (x (x^{2} + 16) (x + 4) (x - 4) + 2 (- 3 x^{4}) + 80 x^{2} + 256) = 0$

Factor $2$ out of $80 x^{2}$ .

$(x - 2) (x (x^{2} + 16) (x + 4) (x - 4) + 2 (- 3 x^{4}) + 2 (40 x^{2}) + 256) = 0$

Factor $2$ out of $256$ .

$(x - 2) (x (x^{2} + 16) (x + 4) (x - 4) + 2 (- 3 x^{4}) + 2 (40 x^{2}) + 2 (128)) = 0$

Factor $2$ out of $2 (- 3 x^{4}) + 2 (40 x^{2})$ .

$(x - 2) (x (x^{2} + 16) (x + 4) (x - 4) + 2 (- 3 x^{4} + 40 x^{2}) + 2 (128)) = 0$

Factor $2$ out of $2 (- 3 x^{4} + 40 x^{2}) + 2 (128)$ .

$(x - 2) (x (x^{2} + 16) (x + 4) (x - 4) + 2 (- 3 x^{4} + 40 x^{2} + 128)) = 0$

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$(x - 2) (x (x^{2} + 16) (x + 4) (x - 4) + 2 (- 3 {(x^{2})}^{2} + 40 x^{2} + 128)) = 0$

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$(x - 2) (x (x^{2} + 16) (x + 4) (x - 4) + 2 (- 3 u^{2} + 40 u + 128)) = 0$

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 $a \cdot c = - 3 \cdot 128 = - 384$ and whose sum is $b = 40$ .

Factor $40$ out of $40 u$ .

$(x - 2) (x (x^{2} + 16) (x + 4) (x - 4) + 2 (- 3 u^{2} + 40 (u) + 128)) = 0$

Rewrite $40$ as $- 8$ plus $48$

$(x - 2) (x (x^{2} + 16) (x + 4) (x - 4) + 2 (- 3 u^{2} + (- 8 + 48) u + 128)) = 0$

Apply the distributive property.

$(x - 2) (x (x^{2} + 16) (x + 4) (x - 4) + 2 (- 3 u^{2} - 8 u + 48 u + 128)) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x - 2) (x (x^{2} + 16) (x + 4) (x - 4) + 2 ((- 3 u^{2} - 8 u) + 48 u + 128)) = 0$

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

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

Factor the polynomial by factoring out the greatest common factor, $- 3 u - 8$ .

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

Replace all occurrences of $u$ with $x^{2}$ .

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

Rewrite $16$ as $4^{2}$ .

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

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

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

Remove unnecessary parentheses.

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

Remove unnecessary parentheses.

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

Factor $(x + 4) (x - 4)$ out of $x (x^{2} + 16) (x + 4) (x - 4) + 2 (- 3 x^{2} - 8) (x + 4) (x - 4)$ .

Factor $(x + 4) (x - 4)$ out of $x (x^{2} + 16) (x + 4) (x - 4)$ .

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

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

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

Factor $(x + 4) (x - 4)$ out of $(x + 4) (x - 4) (x (x^{2} + 16)) + (x + 4) (x - 4) (2 (- 3 x^{2} - 8))$ .

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

Apply the distributive property.

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

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

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

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

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

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

$(x - 2) ((x + 4) (x - 4) (x^{1 + 2} + x \cdot 16 + 2 (- 3 x^{2} - 8))) = 0$

Add $1$ and $2$ .

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

Move $16$ to the left of $x$ .

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

Apply the distributive property.

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

Multiply $- 3$ by $2$ .

$(x - 2) ((x + 4) (x - 4) (x^{3} + 16 x - 6 x^{2} + 2 \cdot - 8)) = 0$

Multiply $2$ by $- 8$ .

$(x - 2) ((x + 4) (x - 4) (x^{3} + 16 x - 6 x^{2} - 16)) = 0$

Reorder terms.

$(x - 2) ((x + 4) (x - 4) (x^{3} - 6 x^{2} + 16 x - 16)) = 0$

Factor.

Factor $x^{3} - 6 x^{2} + 16 x - 16$ 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.

$p = \pm 1, \pm 16, \pm 2, \pm 8, \pm 4 q = \pm 1$

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

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

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

Substitute $2$ into the polynomial.

$2^{3} - 6 \cdot 2^{2} + 16 \cdot 2 - 16$

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

$8 - 6 \cdot 2^{2} + 16 \cdot 2 - 16$

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

$8 - 6 \cdot 4 + 16 \cdot 2 - 16$

Multiply $- 6$ by $4$ .

$8 - 24 + 16 \cdot 2 - 16$

Subtract $24$ from $8$ .

$- 16 + 16 \cdot 2 - 16$

Multiply $16$ by $2$ .

$- 16 + 32 - 16$

Add $- 16$ and $32$ .

$16 - 16$

Subtract $16$ from $16$ .

$0$

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.

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

Divide $x^{3} - 6 x^{2} + 16 x - 16$ by $x - 2$ .

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

$x$

$2$

$x^{3}$

$6 x^{2}$

$16 x$

$16$

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

					$x^{2}$
	$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$16 x$	-	$16$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$16 x$	-	$16$
			+	$x^{3}$	-	$2 x^{2}$

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$16 x$	-	$16$
			-	$x^{3}$	+	$2 x^{2}$

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$16 x$	-	$16$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$16 x$	-	$16$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$16 x$

Divide the highest order term in the dividend $- 4 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$4 x$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$16 x$	-	$16$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$16 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$16 x$	-	$16$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$16 x$
					-	$4 x^{2}$	+	$8 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{2} + 8 x$

				$x^{2}$	-	$4 x$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$16 x$	-	$16$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$16 x$
					+	$4 x^{2}$	-	$8 x$

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

				$x^{2}$	-	$4 x$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$16 x$	-	$16$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$16 x$
					+	$4 x^{2}$	-	$8 x$
							+	$8 x$

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

				$x^{2}$	-	$4 x$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$16 x$	-	$16$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$16 x$
					+	$4 x^{2}$	-	$8 x$
							+	$8 x$	-	$16$

Divide the highest order term in the dividend $8 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$4 x$	+	$8$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$16 x$	-	$16$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$16 x$
					+	$4 x^{2}$	-	$8 x$
							+	$8 x$	-	$16$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$	+	$8$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$16 x$	-	$16$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$16 x$
					+	$4 x^{2}$	-	$8 x$
							+	$8 x$	-	$16$
							+	$8 x$	-	$16$

The expression needs to be subtracted from the dividend, so change all the signs in $8 x - 16$

				$x^{2}$	-	$4 x$	+	$8$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$16 x$	-	$16$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$16 x$
					+	$4 x^{2}$	-	$8 x$
							+	$8 x$	-	$16$
							-	$8 x$	+	$16$

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

				$x^{2}$	-	$4 x$	+	$8$
$x$	-	$2$		$x^{3}$	-	$6 x^{2}$	+	$16 x$	-	$16$
			-	$x^{3}$	+	$2 x^{2}$
					-	$4 x^{2}$	+	$16 x$
					+	$4 x^{2}$	-	$8 x$
							+	$8 x$	-	$16$
							-	$8 x$	+	$16$
										$0$

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

$x^{2} - 4 x + 8$

Write $x^{3} - 6 x^{2} + 16 x - 16$ as a set of factors.

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

Remove unnecessary parentheses.

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

Remove unnecessary parentheses.

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

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

$x - 2 = 0$

$x + 4 = 0$

$x - 4 = 0$

$x - 2 = 0$

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

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

Set $x^{2} - 4 x + 8$ equal to $0$ and solve for $x$ .

Set $x^{2} - 4 x + 8$ equal to $0$ .

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

Solve $x^{2} - 4 x + 8 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 4$ , and $c = 8$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot 8)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 8}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 8$ .

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 8}}{2 \cdot 1}$

Multiply $- 4$ by $8$ .

$x = \frac{4 \pm \sqrt{16 - 32}}{2 \cdot 1}$

Subtract $32$ from $16$ .

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

Rewrite $- 16$ as $- 1 (16)$ .

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

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

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

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

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

Rewrite $16$ as $4^{2}$ .

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

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

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

Move $4$ to the left of $i$ .

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

Multiply $2$ by $1$ .

$x = \frac{4 \pm 4 i}{2}$

Simplify $\frac{4 \pm 4 i}{2}$ .

$x = 2 \pm 2 i$

The final answer is the combination of both solutions.

$x = 2 + 2 i, 2 - 2 i$

The final solution is all the values that make $(x - 2) (x + 4) (x - 4) (x - 2) (x^{2} - 4 x + 8) = 0$ true.

$x = 2, - 4, 4, 2 + 2 i, 2 - 2 i$

Step 3

Find the Roots (Zeros) f(x)=x^6-8x^5+12x^4+80x^3-416x^2+768x-512

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