Given f(x)= -x^4 + 3x^2 + 2x. Is x^2 + 2x + 1 a factor of f(x)? Find all Roots of f(x).

Question

Sketch a complete graph of f(x). Label all roots, x-intercepts and y-intercept clearly.

Answer 1

f(x) = -x(x³ - 3x - 2)

 

By inspection, we see that -1 is a root of x³ - 3x - 2

 

Divide synthetically by x - (-1):

 

   -1⌋    1       0       -3        -2

        _______-1____1______2

           1       -1      -2         0

 

So, f(x) = -x(x+1)(x²-x-2) = -x(x+1)(x+1)(x-2)

 

The roots of f(x) are 0, 2 and -1.  The roots 0 and 2 each have multiplicity one while the root -1 has multiplicity two.  

 

Since (x+1)(x+1) = x²+2x+1, x²+2x+1 is a factor of f(x).  

 

The x-intercepts are (0,0), (2,0), and (-1,0)

The y-intercept is (0,0)