Question

The polynomial p(x)=2x^3-x^2-25x-12 has a known factor of (x+3). Rewrite p(x) as a product of linear factors.


P(x)=?

Answer 1

$\boxed{p(x)=(2x+1)(x-4)(x+3)}$

Explanation:

Here we have a cubic function expressed as:

$p(x)=2x^3-x^2-25x-12$

If $(x+3)$ is a factor of the polynomial, then we can rewrite it as:

$p(x)=(ax^2+bx+c)(x+3)$

So our goal is to find the other factors. So:

$\frac{p(x)}{x+3}=ax^2+bx+c$

Applying synthetic division, we have:

Step 1.

• You must write the problem in a division-like format (Figure 1)

Step 2.

• You must write down the first coefficient without changes (Figure 2)

Step 3.

• You must multiply the entry in the left part of the table by the last entry in the result row, which is under the horizontal line.

• Then you must add the obtained result to the next coefficient of the dividend. Next, write down the sum. (Figure 3)

Step 4.

• Do the same as step 3 (Figure 4)

Step 5.

• Do the same as step 3 (Figure 5)

So:

The quotient is:

$2x^2-7x-4$

And remainder is zero. Therefore:

$\frac{2 x^{3} - x^{2} - 25 x - 12}{x + 3}=2 x^{2} - 7 x - 4+\frac{0}{x + 3}=2 x^{2} - 7 x - 4$

But we can write:

$2 x^{2} - 7 x - 4 \\ \\ =\left(2x^2+x\right)+\left(-8x-4\right) \\ \\ =x(2x+1)-4\left(2x+1\right) \\ \\ =(2x+1)(x-4)$

So:

$\boxed{p(x)=(2x+1)(x-4)(x+3)}$

Learn more:

Long division: https://brainly.com/question/1303597

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Answer 2

Answer:

(2x+1)(x-4)(x+3)

Step-by-step explanation: