Write as a Set of Linear Factors F(x)=x^3-x^2-4x-6

$F (x) = x^{3} - x^{2} - 4 x - 6$

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.

$F (x) = p = \pm 1, \pm 6, \pm 2, \pm 3$

$q = \pm 1$

Step 2

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

$F (x) = \pm 1, \pm 6, \pm 2, \pm 3$

Step 3

Substitute $3$ and simplify the expression. In this case, the expression is equal to $0$ so $3$ is a root of the polynomial.

Substitute $3$ into the polynomial.

$F (x) = 3^{3} - 3^{2} - 4 \cdot 3 - 6$

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

$F (x) = 27 - 3^{2} - 4 \cdot 3 - 6$

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

$F (x) = 27 - 1 \cdot 9 - 4 \cdot 3 - 6$

Multiply $- 1$ by $9$ .

$F (x) = 27 - 9 - 4 \cdot 3 - 6$

Subtract $9$ from $27$ .

$F (x) = 18 - 4 \cdot 3 - 6$

Multiply $- 4$ by $3$ .

$F (x) = 18 - 12 - 6$

Subtract $12$ from $18$ .

$F (x) = 6 - 6$

Subtract $6$ from $6$ .

$F (x) = 0$

Step 4

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$F (x) = \frac{x^{3} - x^{2} - 4 x - 6}{x - 3}$

Step 5

Divide $x^{3} - x^{2} - 4 x - 6$ by $x - 3$ .

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 $x^{3} - 3 x^{2}$

$F (x) =$

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 $2 x^{2} - 6 x$

$F (x) =$

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 $2 x - 6$

$F (x) =$

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 $x^{3} - x^{2} - 4 x - 6$ as a set of factors.

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

Write as a Set of Linear Factors F(x)=x^3-x^2-4x-6

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