Find the complete linear factorization of f(x) and list all real zeros.

Question

Let f(x)=3x^3-2x^2-7x-2.

Answer 1

First set up the equation in the form of ax³+bx²+cx+d

d in this case is equal to -2. Then, factor out -2 in order to find the first root of the equation. The factors for -2 are 1, -2, -1, and 2. Plug in all 4 factors into the equation to find which answer equals 0. Both 2 and -1 make the equation equal to 0 to plugged in. This tells us 2 of our factors for the equation are (x-2) and (x+1). Now simply multiply these two factors and then use long division to find the third factor.

3x³-2x²-7x-2/(x-2)(x+1) = (3x+1) after long division. Therefore the three factors are (x-2), (x+1), and (3x+1), making the zeros 2, -1. and -1/3

Answer 2

Use the Rational Root Theorem to determine a root, -1 works

By Descartes Rule of Signs at most 1 positive root.

Then use suppressed polynomial to determine other roots, 2 and -1/3

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

Answer 3

Dear Jillian,

I am assuming that you have studied the Rational Root Theorem. Have you?

To be able to factor this function, you need to find one rational root by using such a theorem.

I have recorded a video to show you how to do this, but it is still processing.

In any case, this theorem will show you that "2" is a zero or a root of this function.

The next step would be to use the synthetic division to divide the function by (x - 2).

You will find that the quotient of such division is 3x² + 4x + 1.

Thus, f(x) = (x - 2)(3x² + 4x + 1).

Last step, factor the quadratic function.

If my video uploads you will see all the steps.

I would love to help you with these content, if you need more help, we can meet one another for a tutoring session.