Question

I don’t remember how I get the answer to this. I have the answer key though. Please help, I have a test in an hour

Answer 1

Correct answer: f(x) = (x - 2) · (x - 1 - 3 i) · ( x - 1 + 3 i)

Explanation:

Given:

f(x) = x³ - 4x² + 14x - 20 polynomial on the third degree

If 2 is zero it means that the given polynomial is divisible by the binomial (x - 2)

(x³ - 4x² + 14x - 20) / (x - 2) = x² - 2x + 10

We will now factorize this polynomial:

x² - 2x + 10 = x² - 2x + 1 + 9 = (x - 1)² - ( -9) = (x - 1)² - ( -1 · 3²)

and i is an imaginary constant and i² = - 1

x² - 2x + 10 = (x - 1)² - ( 3 i )² = (x - 1 - 3 i) · ( x - 1 + 3 i)

x² - 2x + 10 = (x - 1 - 3 i) · ( x - 1 + 3 i)

where (x - 1 - 3 i) and ( x - 1 + 3 i) are conjugate complex numbers

and finally we get :

x³ - 4x² + 14x - 2 = (x -2) · (x - 1 - 3 i) · ( x - 1 + 3 i)

God is with you!!!

Answer 2

f(x) = (x -2)(x -1+3i)(x -1-3i)

Explanation:

You can use synthetic division to find the remaining quadratic factor in the cubic. Then any of the usual means of solving the quadratic will help you find its linear factors.

In the attached, I show the synthetic division, the factoring to real numbers, and the solution that finds the complex linear factors by completing the square.

Of course, you know that for zeros a, b, and c, the linear factors are ...

f(x) = (x -a)(x -b)(x -c)

Here, we have a=2, b=1-3i, c=1+3i.

f(x) = (x -2)(x -1+3i)(x -1-3i)