solve problem

Question

Factor the polynomial F(x). Then solve the equation f(x)=0

 

f(x)=x⁴-5x³-9x²+81-108

 

The factorization of F(x) is f(x)=

 

The solutions to f(x)=0 are=

Answer 1

Assuming that the second last term is 81 x, this quartic can be factored with aid of a graphing calculator such as the TI-84.

 

Just plot the function and do a little bit of exploration.   It is fairly easy to see that there are zeros at x = -4 and x = 3.  A little more exploration shows that the zero at x = 3 is a triple zero.  Thus the function factors as

 

  f(x) = (x+4) (x-3)³

 

This can be checked by expanding this form and checking.

 

A calculus approach would also reveal the triple zero at x =3 because there is a point of inflection at x =3.