Write as a Set of Linear Factors f(x)=x^3+4x^2-2x-12

$f (x) = x^{3} + 4 x^{2} - 2 x - 12$

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.

$f (x) = p = \pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4$

$q = \pm 1$

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

$f (x) = \pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4$

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.

$f (x) = {(- 2)}^{3} + 4 {(- 2)}^{2} - 2 \cdot - 2 - 12$

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

$f (x) = - 8 + 4 {(- 2)}^{2} - 2 \cdot - 2 - 12$

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

$f (x) = - 8 + 4 \cdot 4 - 2 \cdot - 2 - 12$

Multiply $4$ by $4$ .

$f (x) = - 8 + 16 - 2 \cdot - 2 - 12$

Add $- 8$ and $16$ .

$f (x) = 8 - 2 \cdot - 2 - 12$

Multiply $- 2$ by $- 2$ .

$f (x) = 8 + 4 - 12$

Add $8$ and $4$ .

$f (x) = 12 - 12$

Subtract $12$ from $12$ .

$f (x) = 0$

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.

$f (x) = \frac{x^{3} + 4 x^{2} - 2 x - 12}{x + 2}$

Divide $x^{3} + 4 x^{2} - 2 x - 12$ 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 $- 6 x$ 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 $- 6 x - 12$

$f (x) =$

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.

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

Write $x^{3} + 4 x^{2} - 2 x - 12$ as a set of factors.

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

Write as a Set of Linear Factors f(x)=x^3+4x^2-2x-12

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