f(x)=x^4-2x^3-5x^2+8x+4

Question

i need help with solving this equation

Answer 1

If this function has rational zeroes they will be factors of 4 which has only 1, 2, 4 as its factors. The first thing to do is to substitute these factors (in positive or negative form) for x in f(x):

f(1)=6, f(-1)=-6, f(2)=0, f(-2)=0, f(4)=84, f(-4)=276. Clearly x=-2 or 2 is a zero.

The factors corresponding to these zeroes are (x+2) and (x-2) so we can use synthetic division to divide by each:

-2 | 1 -2 -5    8    4

      1 -2  8   -6 | -4

2 |  1 -4  3    2 |  0 = x³-4x²+3x+2

      1  2 -4 | -2

      1 -2 -1 |  0 = x²-2x-1.

Finally solve the quadratic:

x²-2x=1, complete the square:

x²-2x+1=2,

(x-1)²=2, x-1=±√2, x=1+√2, 1-√2.

Zeroes are: -2, 2, 1+√2, 1-√2.