Factor f(x)=x^6+4x^4-41x^2+36

$f (x) = x^{6} + 4 x^{4} - 41 x^{2} + 36$

Step 1

Factor $x^{6} + 4 x^{4} - 41 x^{2} + 36$ 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.

$p = \pm 1, \pm 36, \pm 2, \pm 18, \pm 3, \pm 12, \pm 4, \pm 9, \pm 6 q = \pm 1$

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

$\pm 1, \pm 36, \pm 2, \pm 18, \pm 3, \pm 12, \pm 4, \pm 9, \pm 6$

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

Substitute $- 1$ into the polynomial.

${(- 1)}^{6} + 4 {(- 1)}^{4} - 41 {(- 1)}^{2} + 36$

Raise $- 1$ to the power of $6$ .

$1 + 4 {(- 1)}^{4} - 41 {(- 1)}^{2} + 36$

Raise $- 1$ to the power of $4$ .

$1 + 4 \cdot 1 - 41 {(- 1)}^{2} + 36$

Multiply $4$ by $1$ .

$1 + 4 - 41 {(- 1)}^{2} + 36$

Add $1$ and $4$ .

$5 - 41 {(- 1)}^{2} + 36$

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

$5 - 41 \cdot 1 + 36$

Multiply $- 41$ by $1$ .

$5 - 41 + 36$

Subtract $41$ from $5$ .

$- 36 + 36$

Add $- 36$ and $36$ .

$0$

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{6} + 4 x^{4} - 41 x^{2} + 36}{x + 1}$

Divide $x^{6} + 4 x^{4} - 41 x^{2} + 36$ by $x + 1$ .

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

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

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

$x^{5}$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

Multiply the new quotient term by the divisor.

$x^{5}$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

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

$x^{5}$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

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

$x^{5}$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

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

$x^{5}$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

$4 x^{4}$

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

$x^{5}$

$x^{4}$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

$4 x^{4}$

Multiply the new quotient term by the divisor.

$x^{5}$

$x^{4}$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

$4 x^{4}$

$x^{5}$

$x^{4}$

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

$x^{5}$

$x^{4}$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

$4 x^{4}$

$x^{5}$

$x^{4}$

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

$x^{5}$

$x^{4}$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

$4 x^{4}$

$x^{5}$

$x^{4}$

$5 x^{4}$

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

$x^{5}$

$x^{4}$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

$4 x^{4}$

$x^{5}$

$x^{4}$

$5 x^{4}$

$0 x^{3}$

Divide the highest order term in the dividend $5 x^{4}$ by the highest order term in divisor $x$ .

$x^{5}$

$x^{4}$

$5 x^{3}$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

$4 x^{4}$

$x^{5}$

$x^{4}$

$5 x^{4}$

$0 x^{3}$

Multiply the new quotient term by the divisor.

$x^{5}$

$x^{4}$

$5 x^{3}$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

$4 x^{4}$

$x^{5}$

$x^{4}$

$5 x^{4}$

$0 x^{3}$

$5 x^{4}$

$5 x^{3}$

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{4} + 5 x^{3}$

$x^{5}$

$x^{4}$

$5 x^{3}$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

$4 x^{4}$

$x^{5}$

$x^{4}$

$5 x^{4}$

$0 x^{3}$

$5 x^{4}$

$5 x^{3}$

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

$x^{5}$

$x^{4}$

$5 x^{3}$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

$4 x^{4}$

$x^{5}$

$x^{4}$

$5 x^{4}$

$0 x^{3}$

$5 x^{4}$

$5 x^{3}$

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

$x^{5}$

$x^{4}$

$5 x^{3}$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

$4 x^{4}$

$x^{5}$

$x^{4}$

$5 x^{4}$

$0 x^{3}$

$5 x^{4}$

$5 x^{3}$

$41 x^{2}$

Divide the highest order term in the dividend $- 5 x^{3}$ by the highest order term in divisor $x$ .

$x^{5}$

$x^{4}$

$5 x^{3}$

$5 x^{2}$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

$4 x^{4}$

$x^{5}$

$x^{4}$

$5 x^{4}$

$0 x^{3}$

$5 x^{4}$

$5 x^{3}$

$41 x^{2}$

Multiply the new quotient term by the divisor.

$x^{5}$

$x^{4}$

$5 x^{3}$

$5 x^{2}$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

$4 x^{4}$

$x^{5}$

$x^{4}$

$5 x^{4}$

$0 x^{3}$

$5 x^{4}$

$5 x^{3}$

$41 x^{2}$

$5 x^{3}$

$5 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x^{3} - 5 x^{2}$

$x^{5}$

$x^{4}$

$5 x^{3}$

$5 x^{2}$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

$4 x^{4}$

$x^{5}$

$x^{4}$

$5 x^{4}$

$0 x^{3}$

$5 x^{4}$

$5 x^{3}$

$41 x^{2}$

$5 x^{3}$

$5 x^{2}$

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

$x^{5}$

$x^{4}$

$5 x^{3}$

$5 x^{2}$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

$4 x^{4}$

$x^{5}$

$x^{4}$

$5 x^{4}$

$0 x^{3}$

$5 x^{4}$

$5 x^{3}$

$41 x^{2}$

$5 x^{3}$

$5 x^{2}$

$36 x^{2}$

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

$x^{5}$

$x^{4}$

$5 x^{3}$

$5 x^{2}$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

$4 x^{4}$

$x^{5}$

$x^{4}$

$5 x^{4}$

$0 x^{3}$

$5 x^{4}$

$5 x^{3}$

$41 x^{2}$

$5 x^{3}$

$5 x^{2}$

$36 x^{2}$

$0 x$

Divide the highest order term in the dividend $- 36 x^{2}$ by the highest order term in divisor $x$ .

$x^{5}$

$x^{4}$

$5 x^{3}$

$5 x^{2}$

$36 x$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

$4 x^{4}$

$x^{5}$

$x^{4}$

$5 x^{4}$

$0 x^{3}$

$5 x^{4}$

$5 x^{3}$

$41 x^{2}$

$5 x^{3}$

$5 x^{2}$

$36 x^{2}$

$0 x$

Multiply the new quotient term by the divisor.

$x^{5}$

$x^{4}$

$5 x^{3}$

$5 x^{2}$

$36 x$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

$4 x^{4}$

$x^{5}$

$x^{4}$

$5 x^{4}$

$0 x^{3}$

$5 x^{4}$

$5 x^{3}$

$41 x^{2}$

$5 x^{3}$

$5 x^{2}$

$36 x^{2}$

$0 x$

$36 x^{2}$

$36 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 36 x^{2} - 36 x$

$x^{5}$

$x^{4}$

$5 x^{3}$

$5 x^{2}$

$36 x$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

$4 x^{4}$

$x^{5}$

$x^{4}$

$5 x^{4}$

$0 x^{3}$

$5 x^{4}$

$5 x^{3}$

$41 x^{2}$

$5 x^{3}$

$5 x^{2}$

$36 x^{2}$

$0 x$

$36 x^{2}$

$36 x$

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

$x^{5}$

$x^{4}$

$5 x^{3}$

$5 x^{2}$

$36 x$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

$4 x^{4}$

$x^{5}$

$x^{4}$

$5 x^{4}$

$0 x^{3}$

$5 x^{4}$

$5 x^{3}$

$41 x^{2}$

$5 x^{3}$

$5 x^{2}$

$36 x^{2}$

$0 x$

$36 x^{2}$

$36 x$

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

$x^{5}$

$x^{4}$

$5 x^{3}$

$5 x^{2}$

$36 x$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

$4 x^{4}$

$x^{5}$

$x^{4}$

$5 x^{4}$

$0 x^{3}$

$5 x^{4}$

$5 x^{3}$

$41 x^{2}$

$5 x^{3}$

$5 x^{2}$

$36 x^{2}$

$0 x$

$36 x^{2}$

$36 x$

$36$

Divide the highest order term in the dividend $36 x$ by the highest order term in divisor $x$ .

$x^{5}$

$x^{4}$

$5 x^{3}$

$5 x^{2}$

$36 x$

$36$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

$4 x^{4}$

$x^{5}$

$x^{4}$

$5 x^{4}$

$0 x^{3}$

$5 x^{4}$

$5 x^{3}$

$41 x^{2}$

$5 x^{3}$

$5 x^{2}$

$36 x^{2}$

$0 x$

$36 x^{2}$

$36 x$

$36$

Multiply the new quotient term by the divisor.

$x^{5}$

$x^{4}$

$5 x^{3}$

$5 x^{2}$

$36 x$

$36$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

$4 x^{4}$

$x^{5}$

$x^{4}$

$5 x^{4}$

$0 x^{3}$

$5 x^{4}$

$5 x^{3}$

$41 x^{2}$

$5 x^{3}$

$5 x^{2}$

$36 x^{2}$

$0 x$

$36 x^{2}$

$36 x$

$36$

$36 x$

$36$

The expression needs to be subtracted from the dividend, so change all the signs in $36 x + 36$

$x^{5}$

$x^{4}$

$5 x^{3}$

$5 x^{2}$

$36 x$

$36$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

$4 x^{4}$

$x^{5}$

$x^{4}$

$5 x^{4}$

$0 x^{3}$

$5 x^{4}$

$5 x^{3}$

$41 x^{2}$

$5 x^{3}$

$5 x^{2}$

$36 x^{2}$

$0 x$

$36 x^{2}$

$36 x$

$36$

$36 x$

$36$

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

$x^{5}$

$x^{4}$

$5 x^{3}$

$5 x^{2}$

$36 x$

$36$

$x$

$1$

$x^{6}$

$0 x^{5}$

$4 x^{4}$

$0 x^{3}$

$41 x^{2}$

$0 x$

$36$

$x^{6}$

$x^{5}$

$4 x^{4}$

$x^{5}$

$x^{4}$

$5 x^{4}$

$0 x^{3}$

$5 x^{4}$

$5 x^{3}$

$41 x^{2}$

$5 x^{3}$

$5 x^{2}$

$36 x^{2}$

$0 x$

$36 x^{2}$

$36 x$

$36$

$36 x$

$36$

$0$

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

$x^{5} - x^{4} + 5 x^{3} - 5 x^{2} - 36 x + 36$

Write $x^{6} + 4 x^{4} - 41 x^{2} + 36$ as a set of factors.

$f (x) = (x + 1) (x^{5} - x^{4} + 5 x^{3} - 5 x^{2} - 36 x + 36)$

Step 2

Regroup terms.

$f (x) = (x + 1) (x^{5} + 5 x^{3} - 36 x - x^{4} - 5 x^{2} + 36)$

Step 3

Factor $x$ out of $x^{5} + 5 x^{3} - 36 x$ .

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

$f (x) = (x + 1) (x \cdot x^{4} + 5 x^{3} - 36 x - x^{4} - 5 x^{2} + 36)$

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

$f (x) = (x + 1) (x \cdot x^{4} + x (5 x^{2}) - 36 x - x^{4} - 5 x^{2} + 36)$

Factor $x$ out of $- 36 x$ .

$f (x) = (x + 1) (x \cdot x^{4} + x (5 x^{2}) + x \cdot - 36 - x^{4} - 5 x^{2} + 36)$

Factor $x$ out of $x \cdot x^{4} + x (5 x^{2})$ .

$f (x) = (x + 1) (x (x^{4} + 5 x^{2}) + x \cdot - 36 - x^{4} - 5 x^{2} + 36)$

Factor $x$ out of $x (x^{4} + 5 x^{2}) + x \cdot - 36$ .

$f (x) = (x + 1) (x (x^{4} + 5 x^{2} - 36) - x^{4} - 5 x^{2} + 36)$

Step 4

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

$f (x) = (x + 1) (x ({(x^{2})}^{2} + 5 x^{2} - 36) - x^{4} - 5 x^{2} + 36)$

Step 5

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

$f (x) = (x + 1) (x (u^{2} + 5 u - 36) - x^{4} - 5 x^{2} + 36)$

Step 6

Factor $u^{2} + 5 u - 36$ using the AC method.

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 $- 36$ and whose sum is $5$ .

$- 4, 9$

Write the factored form using these integers.

$f (x) = (x + 1) (x ((u - 4) (u + 9)) - x^{4} - 5 x^{2} + 36)$

Step 7

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

$f (x) = (x + 1) (x ((x^{2} - 4) (x^{2} + 9)) - x^{4} - 5 x^{2} + 36)$

Step 8

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

$f (x) = (x + 1) (x ((x^{2} - 2^{2}) (x^{2} + 9)) - x^{4} - 5 x^{2} + 36)$

Step 9

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 $b = 2$ .

$f (x) = (x + 1) (x ((x + 2) (x - 2) (x^{2} + 9)) - x^{4} - 5 x^{2} + 36)$

Remove unnecessary parentheses.

$f (x) = (x + 1) (x (x + 2) (x - 2) (x^{2} + 9) - x^{4} - 5 x^{2} + 36)$

Step 10

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

$f (x) = (x + 1) (x (x + 2) (x - 2) (x^{2} + 9) - {(x^{2})}^{2} - 5 x^{2} + 36)$

Step 11

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

$f (x) = (x + 1) (x (x + 2) (x - 2) (x^{2} + 9) - u^{2} - 5 u + 36)$

Step 12

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 $a \cdot c = - 1 \cdot 36 = - 36$ and whose sum is $b = - 5$ .

Factor $- 5$ out of $- 5 u$ .

$f (x) = (x + 1) (x (x + 2) (x - 2) (x^{2} + 9) - u^{2} - 5 u + 36)$

Rewrite $- 5$ as $4$ plus $- 9$

$f (x) = (x + 1) (x (x + 2) (x - 2) (x^{2} + 9) - u^{2} + (4 - 9) u + 36)$

Apply the distributive property.

$f (x) = (x + 1) (x (x + 2) (x - 2) (x^{2} + 9) - u^{2} + 4 u - 9 u + 36)$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$f (x) = (x + 1) (x (x + 2) (x - 2) (x^{2} + 9) - u^{2} + 4 u - 9 u + 36)$

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

$f (x) = (x + 1) (x (x + 2) (x - 2) (x^{2} + 9) + u (- u + 4) + 9 (- u + 4))$

Factor the polynomial by factoring out the greatest common factor, $- u + 4$ .

$f (x) = (x + 1) (x (x + 2) (x - 2) (x^{2} + 9) + (- u + 4) (u + 9))$

Step 13

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

$f (x) = (x + 1) (x (x + 2) (x - 2) (x^{2} + 9) + (- x^{2} + 4) (x^{2} + 9))$

Step 14

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

$f (x) = (x + 1) (x (x + 2) (x - 2) (x^{2} + 9) + (- x^{2} + 2^{2}) (x^{2} + 9))$

Step 15

Reorder $- x^{2}$ and $2^{2}$ .

$f (x) = (x + 1) (x (x + 2) (x - 2) (x^{2} + 9) + (2^{2} - x^{2}) (x^{2} + 9))$

Step 16

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

$f (x) = (x + 1) (x (x + 2) (x - 2) (x^{2} + 9) + (2 + x) (2 - x) (x^{2} + 9))$

Step 17

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

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

$f (x) = (x + 1) ((x^{2} + 9) (x (x + 2) (x - 2)) + (2 + x) (2 - x) (x^{2} + 9))$

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

$f (x) = (x + 1) ((x^{2} + 9) (x (x + 2) (x - 2)) + (x^{2} + 9) ((2 + x) (2 - x)))$

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

$f (x) = (x + 1) ((x^{2} + 9) (x (x + 2) (x - 2) + (2 + x) (2 - x)))$

Step 18

Apply the distributive property.

$f (x) = (x + 1) ((x^{2} + 9) ((x \cdot x + x \cdot 2) (x - 2) + (2 + x) (2 - x)))$

Step 19

Multiply $x$ by $x$ .

$f (x) = (x + 1) ((x^{2} + 9) ((x^{2} + x \cdot 2) (x - 2) + (2 + x) (2 - x)))$

Step 20

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

$f (x) = (x + 1) ((x^{2} + 9) ((x^{2} + 2 x) (x - 2) + (2 + x) (2 - x)))$

Step 21

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

Apply the distributive property.

$f (x) = (x + 1) ((x^{2} + 9) (x^{2} (x - 2) + 2 x (x - 2) + (2 + x) (2 - x)))$

Apply the distributive property.

$f (x) = (x + 1) ((x^{2} + 9) (x^{2} x + x^{2} \cdot - 2 + 2 x (x - 2) + (2 + x) (2 - x)))$

Apply the distributive property.

$f (x) = (x + 1) ((x^{2} + 9) (x^{2} x + x^{2} \cdot - 2 + 2 x \cdot x + 2 x \cdot - 2 + (2 + x) (2 - x)))$

Step 22

Simplify and combine like terms.

Simplify each term.

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

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

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

$f (x) = (x + 1) ((x^{2} + 9) (x^{2} x + x^{2} \cdot - 2 + 2 x \cdot x + 2 x \cdot - 2 + (2 + x) (2 - x)))$

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

$f (x) = (x + 1) ((x^{2} + 9) (x^{2 + 1} + x^{2} \cdot - 2 + 2 x \cdot x + 2 x \cdot - 2 + (2 + x) (2 - x)))$

Add $2$ and $1$ .

$f (x) = (x + 1) ((x^{2} + 9) (x^{3} + x^{2} \cdot - 2 + 2 x \cdot x + 2 x \cdot - 2 + (2 + x) (2 - x)))$

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

$f (x) = (x + 1) ((x^{2} + 9) (x^{3} - 2 x^{2} + 2 x \cdot x + 2 x \cdot - 2 + (2 + x) (2 - x)))$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$f (x) = (x + 1) ((x^{2} + 9) (x^{3} - 2 x^{2} + 2 (x \cdot x) + 2 x \cdot - 2 + (2 + x) (2 - x)))$

Multiply $x$ by $x$ .

$f (x) = (x + 1) ((x^{2} + 9) (x^{3} - 2 x^{2} + 2 x^{2} + 2 x \cdot - 2 + (2 + x) (2 - x)))$

Multiply $- 2$ by $2$ .

$f (x) = (x + 1) ((x^{2} + 9) (x^{3} - 2 x^{2} + 2 x^{2} - 4 x + (2 + x) (2 - x)))$

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

$f (x) = (x + 1) ((x^{2} + 9) (x^{3} + 0 - 4 x + (2 + x) (2 - x)))$

Add $x^{3}$ and $0$ .

$f (x) = (x + 1) ((x^{2} + 9) (x^{3} - 4 x + (2 + x) (2 - x)))$

Step 23

Expand $(2 + x) (2 - x)$ using the FOIL Method.

Apply the distributive property.

$f (x) = (x + 1) ((x^{2} + 9) (x^{3} - 4 x + 2 (2 - x) + x (2 - x)))$

Apply the distributive property.

$f (x) = (x + 1) ((x^{2} + 9) (x^{3} - 4 x + 2 \cdot 2 + 2 (- x) + x (2 - x)))$

Apply the distributive property.

$f (x) = (x + 1) ((x^{2} + 9) (x^{3} - 4 x + 2 \cdot 2 + 2 (- x) + x \cdot 2 + x (- x)))$

Step 24

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

$f (x) = (x + 1) ((x^{2} + 9) (x^{3} - 4 x + 4 + 2 (- x) + x \cdot 2 + x (- x)))$

Multiply $- 1$ by $2$ .

$f (x) = (x + 1) ((x^{2} + 9) (x^{3} - 4 x + 4 - 2 x + x \cdot 2 + x (- x)))$

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

$f (x) = (x + 1) ((x^{2} + 9) (x^{3} - 4 x + 4 - 2 x + 2 x + x (- x)))$

Rewrite using the commutative property of multiplication.

$f (x) = (x + 1) ((x^{2} + 9) (x^{3} - 4 x + 4 - 2 x + 2 x - x \cdot x))$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$f (x) = (x + 1) ((x^{2} + 9) (x^{3} - 4 x + 4 - 2 x + 2 x - (x \cdot x)))$

Multiply $x$ by $x$ .

$f (x) = (x + 1) ((x^{2} + 9) (x^{3} - 4 x + 4 - 2 x + 2 x - x^{2}))$

Add $- 2 x$ and $2 x$ .

$f (x) = (x + 1) ((x^{2} + 9) (x^{3} - 4 x + 4 + 0 - x^{2}))$

Add $4$ and $0$ .

$f (x) = (x + 1) ((x^{2} + 9) (x^{3} - 4 x + 4 - x^{2}))$

Step 25

Reorder terms.

$f (x) = (x + 1) ((x^{2} + 9) (x^{3} - x^{2} - 4 x + 4))$

Step 26

Factor.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$f (x) = (x + 1) ((x^{2} + 9) ((x^{3} - x^{2}) - 4 x + 4))$

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

$f (x) = (x + 1) ((x^{2} + 9) (x^{2} (x - 1) - 4 (x - 1)))$

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

$f (x) = (x + 1) ((x^{2} + 9) ((x - 1) (x^{2} - 4)))$

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

$f (x) = (x + 1) ((x^{2} + 9) ((x - 1) (x^{2} - 2^{2})))$

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 $b = 2$ .

$f (x) = (x + 1) ((x^{2} + 9) ((x - 1) ((x + 2) (x - 2))))$

Remove unnecessary parentheses.

$f (x) = (x + 1) ((x^{2} + 9) ((x - 1) (x + 2) (x - 2)))$

Remove unnecessary parentheses.

$f (x) = (x + 1) ((x^{2} + 9) (x - 1) (x + 2) (x - 2))$

Remove unnecessary parentheses.

$f (x) = (x + 1) (x^{2} + 9) (x - 1) (x + 2) (x - 2)$

Factor f(x)=x^6+4x^4-41x^2+36

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