Factor the polynomial f(x)=x^3-x^2-4x+4 completely

Question

List all possible roots of f(x) show work.

Answer 1

f(x)=x³- x²- 4x + 4

 

The possible real roots are the factors of the constant, 4, divided by the factors

of the coefficient of the highest powered variable, 1

Possible real roots:  ±{1, 2, 4} which gives us 6 possibilities.

Total number of roots is 3

 

Factor x² from 1st two terms and -4 from last 2 terms

 

f(x) = x²(x-1) - 4(x-1)

f(x) = (x²-4)(x-1)

 

Note that x²-4 is difference of two squares... (x²-2²)

a² - b² = (a-b)(a+b)

 

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

 

The roots are {2, -2, 1}