X^3-x^2-4x+4

Question

How do you solve this it has to equal to zero

Answer 1

P(x) = x³ - x² - 4x + 4 

The zeros are the x's such that P(x) = 0

 

P(1) = 0 which implies P(x) divisible by (x - 1); a careful look quickly suggests an easy factoring of P(x)

 

x²(x - 1) - 4(x - 1) = (x - 1)(x² - 2²) = (x - 1)(x - 2)(x + 2)

 

The zeros are -2 ; 1 and 2

Answer 2

x³ - x² - 4x + 4 = 0

 

By the Rational Zero Theorem, the possible rational zeros are 

±(1, 2, 4).

 

By trying the various possibilities, we find that 2 is a root. So, by the Factor Theorem, x - 2 is a factor.  

 

Divide synthetically by x - 2:

 

2⌋    1         -1        -4         4

                     2         2         -4 

       1            1        -2         0

 

x³ - x² - 4x + 4 = (x - 2)(x² + x - 2)

 

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

 

Setting each factor equal to zero, the roots are 2, -2, and 1.