I need help with math please?

Question

what do they mean by "find all zeros of f(x), and write a linear factorization of f(x)"
and can you please explain how to solve
"3-2i is a zero of f(x) = x^4 - 6x^3 +11x^2 +12x -26"

Answer 1

If (3 - 2i) is a factor, then so is its conjugate pair, (3 + 2i).
To write as linear factors, you just put (x - factor)
In this case:
[x - (3 - 2i)] and [x - (3 + 2i)]

To find the 2 remaining factors, I would multiply these 2 complex roots together and then synthetically divide your polynomial by the result. Your result will be a quadratic polynomial and you can then apply the quadratic formula.

good luck!

Answer 2

'Find all the zeros' simply means find all of the values of x for which f(x) = 0. If you have all of them (call them a, b, c, etc.), then f(x) can be written as
(x-a)*(x-b)*(x-c)...........

Not sure what's to solve in the 2'nd problem. Plug 3-2i in for x and verify that the polynomial goes to zero. Then (since it has real coefficients) another root would be the conjugate of 3-2i or 3+2i. That means that their product (x-(3-2i)) and (x-(3+2i)) would divide f(x) leaving you with a quadratic that you could get zeros from by factoring or by using the quadratic formula.

HTH

Doug