Factor x^4-4x^3+8x^2-16x+16

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

Regroup terms.

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

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

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

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

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

$- 4 x^{2} (x) - 4 x^{2} (- 2) + x^{4} - 16 x + 16$

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

$- 4 x^{2} (x - 2) + x^{4} - 16 x + 16$

Factor $x^{4} - 16 x + 16$ 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.

$2^{4} - 16 \cdot 2 + 16$

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

$16 - 16 \cdot 2 + 16$

Multiply $- 16$ by $2$ .

$16 - 32 + 16$

Subtract $32$ from $16$ .

$- 16 + 16$

Add $- 16$ and $16$ .

$0$

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.

$\frac{x^{4} - 16 x + 16}{x - 2}$

Divide $x^{4} - 16 x + 16$ 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 $0$ .

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$16 x$

$16$

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

$x^{3}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$16 x$

$16$

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} - 2 x^{3}$

$x^{3}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

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

$x^{3}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

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

$x^{3}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

Divide the highest order term in the dividend $2 x^{3}$ by the highest order term in divisor $x$ .

$x^{3}$

$2 x^{2}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

Multiply the new quotient term by the divisor.

$x^{3}$

$2 x^{2}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{3} - 4 x^{2}$

$x^{3}$

$2 x^{2}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

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

$x^{3}$

$2 x^{2}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

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

$x^{3}$

$2 x^{2}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$16 x$

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

$x^{3}$

$2 x^{2}$

$4 x$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$16 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$2 x^{2}$

$4 x$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$16 x$

$4 x^{2}$

$8 x$

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

$x^{3}$

$2 x^{2}$

$4 x$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$16 x$

$4 x^{2}$

$8 x$

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

$x^{3}$

$2 x^{2}$

$4 x$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$16 x$

$4 x^{2}$

$8 x$

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

$x^{3}$

$2 x^{2}$

$4 x$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$16 x$

$4 x^{2}$

$8 x$

$16$

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

$x^{3}$

$2 x^{2}$

$4 x$

$8$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$16 x$

$4 x^{2}$

$8 x$

$16$

Multiply the new quotient term by the divisor.

$x^{3}$

$2 x^{2}$

$4 x$

$8$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$16 x$

$4 x^{2}$

$8 x$

$16$

$8 x$

$16$

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

$x^{3}$

$2 x^{2}$

$4 x$

$8$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$16 x$

$4 x^{2}$

$8 x$

$16$

$8 x$

$16$

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

$x^{3}$

$2 x^{2}$

$4 x$

$8$

$x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$16 x$

$16$

$x^{4}$

$2 x^{3}$

$0 x^{2}$

$2 x^{3}$

$4 x^{2}$

$16 x$

$4 x^{2}$

$8 x$

$16$

$8 x$

$16$

$0$

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

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

Write $x^{4} - 16 x + 16$ as a set of factors.

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

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

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

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

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

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

Add $- 4 x^{2}$ and $2 x^{2}$ .

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

Factor.

Rewrite $x^{3} - 2 x^{2} + 4 x - 8$ in a factored form.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

Factor the polynomial by factoring out the greatest common factor, $x - 2$ .

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

Remove unnecessary parentheses.

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

Combine exponents.

Raise $x - 2$ to the power of $1$ .

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

Raise $x - 2$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Factor x^4-4x^3+8x^2-16x+16

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