Hello, I cant figure this problem out for the life of me. Could someone help me?

Find the polynomial P(x) of degree 4 with integer coefficients, and zeroes $\displaystyle 3-2i$ and 3 with 3, a zero of multiplicity of 2.

Thanks in advance...

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- Dec 5th 2009, 09:28 AMGodzillaFinding polynomials
Hello, I cant figure this problem out for the life of me. Could someone help me?

Find the polynomial P(x) of degree 4 with integer coefficients, and zeroes $\displaystyle 3-2i$ and 3 with 3, a zero of multiplicity of 2.

Thanks in advance... - Dec 5th 2009, 09:38 AMHaversine
- Dec 5th 2009, 09:42 AMGodzilla
can someone show me how to factor that? I am confused because of the (3-2i) and (3+2i)? Thanks!!

- Dec 5th 2009, 09:54 AMHaversine
This polynomial:

is actually a result of this factorization:

That is, if I know the roots are A, B, C, and D, I just set up a series of polynomial factors (x - A)(x - B)(x - C)(x - D).

There's nothing special about the (x - 3 + 2i) factor, for instance. All it's saying is that I want a term that equals zero when "3 - 2i" is substituted for x.

The other imaginary term pops in there because the complex conjugate will always be a root. - Dec 5th 2009, 11:00 AMHallsofIvy
The whole point is that $\displaystyle x^4-12x^3+58x^2-132x+117$ is

**from**the product (x-3+2i)(x-3- 2i)(x-3)(x-3). For this problem, at least, it is not a matter of "factoring" but of "multiplying"!

As far as (x-(3- 2i))(x-(3+2i))= (x-3+2i)(x-3-2i) is concerned, think of it as a "product of sum and difference= difference of squares": $\displaystyle ((x-3)- 2i)((x-3)+ 2i)= (x-3)^2- (2i)^2= x^3- 6x+ 9- 4= x^2- 6x+ 5$