Write as a Set of Linear Factors f(x)=2x^3+7x^2-28x+12

$f (x) = 2 x^{3} + 7 x^{2} - 28 x + 12$

Step 1

Factor $2 x^{3} + 7 x^{2} - 28 x + 12$ 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.

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

$q = \pm 1, \pm 2$

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

$f (x) = \pm 1, \pm 0.5, \pm 12, \pm 6, \pm 2, \pm 3, \pm 1.5, \pm 4$

Substitute $0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $0.5$ is a root of the polynomial.

Substitute $0.5$ into the polynomial.

$f (x) = 2 \cdot {0.5}^{3} + 7 \cdot {0.5}^{2} - 28 \cdot 0.5 + 12$

Raise $0.5$ to the power of $3$ .

$f (x) = 2 \cdot 0.125 + 7 \cdot {0.5}^{2} - 28 \cdot 0.5 + 12$

Multiply $2$ by $0.125$ .

$f (x) = 0.25 + 7 \cdot {0.5}^{2} - 28 \cdot 0.5 + 12$

Raise $0.5$ to the power of $2$ .

$f (x) = 0.25 + 7 \cdot 0.25 - 28 \cdot 0.5 + 12$

Multiply $7$ by $0.25$ .

$f (x) = 0.25 + 1.75 - 28 \cdot 0.5 + 12$

Add $0.25$ and $1.75$ .

$f (x) = 2 - 28 \cdot 0.5 + 12$

Multiply $- 28$ by $0.5$ .

$f (x) = 2 - 14 + 12$

Subtract $14$ from $2$ .

$f (x) = - 12 + 12$

Add $- 12$ and $12$ .

$f (x) = 0$

Since $0.5$ is a known root, divide the polynomial by $2 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$f (x) = \frac{2 x^{3} + 7 x^{2} - 28 x + 12}{2 x - 1}$

Divide $2 x^{3} + 7 x^{2} - 28 x + 12$ by $2 x - 1$ .

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

$f (x) =$

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

$f (x) =$

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 $8 x^{2}$ by the highest order term in divisor $2 x$ .

$f (x) =$

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 $8 x^{2} - 4 x$

$f (x) =$

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 $- 24 x$ by the highest order term in divisor $2 x$ .

$f (x) =$

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 $- 24 x + 12$

$f (x) =$

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.

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

Write $2 x^{3} + 7 x^{2} - 28 x + 12$ as a set of factors.

$f (x) = (2 x - 1) (x^{2} + 4 x - 12)$

Step 2

Factor $x^{2} + 4 x - 12$ using the AC method.

Consider the form $x^{2} + i x + c$ . Find a pair of integers whose product is $c$ and whose sum is $i$ . In this case, whose product is $- 12$ and whose sum is $4$ .

$f (x) = - 2, 6$

Write the factored form using these integers.

$f (x) = (2 x - 1) ((x - 2) (x + 6))$

Remove unnecessary parentheses.

$f (x) = (2 x - 1) (x - 2) (x + 6)$

Write as a Set of Linear Factors f(x)=2x^3+7x^2-28x+12

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