Need some help with this problem please.?

Question

f(x)= x^4 - 2x^3 + 6x^2 - 18x - 27

List all possible rational zeros, find all the zeros ( f(3)= 0 ), and express f(x) as a prioduct of linear factors.

can someone please help me with this problem? Thanks

Answer 1

Well they gave you one zero, so you can use synthetic substitution to reduce it:

3 | +01 -02 +06 -18 -27
--- +00 +03 +03 +27 +27
------------------------------
--- +01 +01 +09 +09 +00

So it becomes x^3 + x^2 + 9x + 9

Now this you can factor by noticing the similarity in coefficients:

x^2(x + 1) + 9(x + 1)
(x + 1)(x^2 + 9)

And x^2 + 9 has no real solutions but has 2 complex ones:

x^2 + 9 = 0
x^2 = -9
x = +/- 3i

So f(x) can be written as a product of linear factors:

f(x) = (x + 3)(x + 1)(x + 3i)(x - 3i)

The "real" zeros are -3 and -1. ALL zeros would also include 3i and -3i.

Answer 2

The polynomial has degree 4, so there should be four roots. The ratoinal zeros are 3, -1. The last two are imaginary: +/- 3i. Once you have the roots you can write the four linear factors:

(x - 3)(x + 1)(x - 3i) (x + 3i)