Factor Theorem: What is the correct way to use it?

[matthew1982]

1. Use the factor theorem to factorise completely f(x)= x³+x²-4x-4.
Hence solve the equation x³+x²-4x-4=0

f(x) = x³ + x² - 4x - 4

f(x) = (x³ + x²) - (4x + 4)

f(x) = x²(x + 1) - 4(x + 1)

f(x) = (x + 1)(x² - 4)

f(x) = (x + 1)(x² - 2²)

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

x³ + x² - 4x - 4 = 0

(x + 1)(x + 2)(x - 2) = 0

Hi I was wondering if someone can help me with this assignment question, I did it and handed it into my tutor and this is the feedback he gave me. He said I haven't applied the factor theorum correctly and factorised it without using factor theorum where have I gone wrong?

(These are my tutors comment)
The start.
Initially you had f(x) =x³+x²-4x-4.
Then checking you find f(-1) = 0.
Hence f(x)=(x+1)g(x) and work out what g is.
And repeat for g(x).
It's not the easiest way to do it. And your method would probably be the best way to go about it but the task says "Use the factor theorem".

 

[eumyang]

The factor theorem states that (x-k) is a factor of a polynomial f(x) iff f(k) = 0. The problem is that you haven't been finding k's so that f(k) = 0, so you haven't been using the factor theorem.

 

[matthew1982]

The factor theorem states that (x-k) is a factor of a polynomial f(x) iff f(k) = 0. The problem is that you haven't been finding k's so that f(k) = 0, so you haven't been using the factor theorem.

So how do I go about finding k? I think you use the letter "K" where we use the letter "a" (x-a). So how do i go about finding this?

 

[Mark44]

You can use the Rational Root Theorem to determine the potential roots of your polynomial. If p/q is a rational root, then by the Rational Root Theorem, p has to divide -4 (the constant term) and q has to divide 1 (the coefficient of the highest degree term).

Possibilities for p are {±1, ±2, ±4}.
Possibilities for q are {±1}.
Then what are the possible roots?