Write as a Set of Linear Factors p(x)=x^5-4x^3-2x^2-4x

$p (x) = x^{5} - 4 x^{3} - 2 x^{2} - 4 x$

Step 1

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

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

$p (x) = x \cdot x^{4} - 4 x^{3} - 2 x^{2} - 4 x$

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

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

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

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

Factor $x$ out of $- 4 x$ .

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

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

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

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

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

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

$p (x) = x (x^{4} - 4 x^{2} - 2 x - 4)$

Step 2

Factor over the complex numbers.

Factor $x^{4} - 4 x^{2} - 2 x - 4$ 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.

$p (x) = p = \pm 1, \pm 4, \pm 2$

$q = \pm 1$

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

$p (x) = \pm 1, \pm 4, \pm 2$

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.

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

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

$p (x) = 16 - 4 {(- 2)}^{2} - 2 \cdot - 2 - 4$

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

$p (x) = 16 - 4 \cdot 4 - 2 \cdot - 2 - 4$

Multiply $- 4$ by $4$ .

$p (x) = 16 - 16 - 2 \cdot - 2 - 4$

Subtract $16$ from $16$ .

$p (x) = 0 - 2 \cdot - 2 - 4$

Multiply $- 2$ by $- 2$ .

$p (x) = 0 + 4 - 4$

Add $0$ and $4$ .

$p (x) = 4 - 4$

Subtract $4$ from $4$ .

$p (x) = 0$

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.

$p (x) = \frac{x^{4} - 4 x^{2} - 2 x - 4}{x + 2}$

Divide $x^{4} - 4 x^{2} - 2 x - 4$ 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 $0$ .

$p (x) =$

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

$p (x) =$

Multiply the new quotient term by the divisor.

$p (x) =$

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

$p (x) =$

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

$p (x) =$

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

$p (x) =$

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

$p (x) =$

Multiply the new quotient term by the divisor.

$p (x) =$

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

$p (x) =$

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

$p (x) =$

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

$p (x) =$

Divide the highest order term in the dividend $- 2 x$ by the highest order term in divisor $x$ .

$p (x) =$

Multiply the new quotient term by the divisor.

$p (x) =$

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

$p (x) =$

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

$p (x) =$

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

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

Write $x^{4} - 4 x^{2} - 2 x - 4$ as a set of factors.

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

Multiply $0$ by $x$ .

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

Factor.

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

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

Remove unnecessary parentheses.

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

Write as a Set of Linear Factors p(x)=x^5-4x^3-2x^2-4x

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