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

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

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.

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

$q = \pm 1$

Step 2

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

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

Step 3

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.

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

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

$f (x) = - 8 + 4 {(- 2)}^{2} + 2 \cdot - 2 - 4$

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

$f (x) = - 8 + 4 \cdot 4 + 2 \cdot - 2 - 4$

Multiply $4$ by $4$ .

$f (x) = - 8 + 16 + 2 \cdot - 2 - 4$

Add $- 8$ and $16$ .

$f (x) = 8 + 2 \cdot - 2 - 4$

Multiply $2$ by $- 2$ .

$f (x) = 8 - 4 - 4$

Subtract $4$ from $8$ .

$f (x) = 4 - 4$

Subtract $4$ from $4$ .

$f (x) = 0$

Step 4

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.

$f (x) = \frac{x^{3} + 4 x^{2} + 2 x - 4}{x + 2}$

Step 5

Divide $x^{3} + 4 x^{2} + 2 x - 4$ 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 $0$ .

$f (x) =$

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

$f (x) =$

Multiply the new quotient term by the divisor.

$f (x) =$

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

$f (x) =$

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

$f (x) =$

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

$f (x) =$

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

$f (x) =$

Multiply the new quotient term by the divisor.

$f (x) =$

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

$f (x) =$

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

$f (x) =$

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

$f (x) =$

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

$f (x) =$

Multiply the new quotient term by the divisor.

$f (x) =$

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

$f (x) =$

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

$f (x) =$

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

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

Step 6

Write $x^{3} + 4 x^{2} + 2 x - 4$ as a set of factors.

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

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

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