Finding the polynomial of a degree?

Question

Find a polynomial of degree 4 with real coefficients, whose roots include -2,5 and 3-2i. (Note, it must be in factored form. But the answer should not have i.)

I am aware that -2 becomes x-2 and 5 becomes x+5. But I'm not sure what to do after that sadly.

Answer 1

You have two mistakes already.

A root is a number for the variable that makes the factor equal to 0. Now if you plug in -2 for x in (x-2), the (x-2) will NOT be 0. We all know that 2 - 2 is 0 so -2 -2 will not be 0 (it will be -4). Instead try the factor (x+2). Now if you replace x with -2, then x+2 will be 0!

Every time you have a root, let's call it r you will have a factor (x-r). So if r=-2, then the factor will be (x- -2)=(x+2).

If 5 is a root, then (x-5) is the factor. Note that 5-5 =0. So 5 does NOT become (x+5) (since 5+5 is not 0!)

If 3-2i is a root, then 3+2i is another root. They come in conjugate pairs.

So two more factors will be (x-(3-2i)) and (x-(3+2i)). Rewrite (x-(3-2i)) as ((x-3)+2i) and rewrite (x-(3+2i)) as ((x-3)-2i). Now we multiply ((x-3)+2i)((x-3)-2i) = (x-3)^2 -(2i)^2 = x^2-6x+9-4i^2 =x^2-6x+9+4=x^2-6x+13.

The final is is (x+2)(x-5)(x^2-6x+13)