Find the Roots/Zeros Using the Rational Roots Test x^6-8x^5+12x^4+80x^3-416x^2+768x-512

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

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.

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

$q = \pm 1$

Step 2

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

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

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.

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

Step 4

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

Simplify each term.

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

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

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

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

Multiply $- 8$ by $32$ .

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

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

$64 - 256 + 12 \cdot 16 + 80 {(2)}^{3} - 416 {(2)}^{2} + 768 (2) - 512$

Multiply $12$ by $16$ .

$64 - 256 + 192 + 80 {(2)}^{3} - 416 {(2)}^{2} + 768 (2) - 512$

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

$64 - 256 + 192 + 80 \cdot 8 - 416 {(2)}^{2} + 768 (2) - 512$

Multiply $80$ by $8$ .

$64 - 256 + 192 + 640 - 416 {(2)}^{2} + 768 (2) - 512$

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

$64 - 256 + 192 + 640 - 416 \cdot 4 + 768 (2) - 512$

Multiply $- 416$ by $4$ .

$64 - 256 + 192 + 640 - 1664 + 768 (2) - 512$

Multiply $768$ by $2$ .

$64 - 256 + 192 + 640 - 1664 + 1536 - 512$

Simplify by adding and subtracting.

Subtract $256$ from $64$ .

$- 192 + 192 + 640 - 1664 + 1536 - 512$

Add $- 192$ and $192$ .

$0 + 640 - 1664 + 1536 - 512$

Add $0$ and $640$ .

$640 - 1664 + 1536 - 512$

Subtract $1664$ from $640$ .

$- 1024 + 1536 - 512$

Add $- 1024$ and $1536$ .

$512 - 512$

Subtract $512$ from $512$ .

$0$

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.

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

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.

$2$	$1$	$- 8$	$12$	$80$	$- 416$	$768$	$- 512$
	\xa0						\xa0

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

$2$	$1$	$- 8$	$12$	$80$	$- 416$	$768$	$- 512$
	\xa0						\xa0
	$1$

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 $(- 8)$ .

$2$	$1$	$- 8$	$12$	$80$	$- 416$	$768$	$- 512$
	\xa0	$2$					\xa0
	$1$