Write as a Set of Linear Factors f(x)=x^3-5x^2+2x+12

$f (x) = x^{3} - 5 x^{2} + 2 x + 12$

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.

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

$q = \pm 1$

Step 2

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

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

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.

$f (x) = 3^{3} - 5 \cdot 3^{2} + 2 \cdot 3 + 12$

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

$f (x) = 27 - 5 \cdot 3^{2} + 2 \cdot 3 + 12$

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

$f (x) = 27 - 5 \cdot 9 + 2 \cdot 3 + 12$

Multiply $- 5$ by $9$ .

$f (x) = 27 - 45 + 2 \cdot 3 + 12$

Subtract $45$ from $27$ .

$f (x) = - 18 + 2 \cdot 3 + 12$

Multiply $2$ by $3$ .

$f (x) = - 18 + 6 + 12$

Add $- 18$ and $6$ .

$f (x) = - 12 + 12$

Add $- 12$ and $12$ .

$f (x) = 0$

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.

$f (x) = \frac{x^{3} - 5 x^{2} + 2 x + 12}{x - 3}$

Step 5

Divide $x^{3} - 5 x^{2} + 2 x + 12$ 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 $- 4 x$ 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 $- 4 x + 12$

$f (x) =$

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.

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

Step 6

Write $x^{3} - 5 x^{2} + 2 x + 12$ as a set of factors.

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

Write as a Set of Linear Factors f(x)=x^3-5x^2+2x+12

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