Edward A. answered • 01/14/23
High School Whiz Kid Grown Up--I've even tutored my grandchildren
Michael,
I assume the problem wants you to creat a polynomial with 3 roots, two of which are -6 and 1+ i.
Do you know about pairs of convex roots, called “complex conjugates? If you have one complex root, you will also have its conjugate.
What is a complex conjugate? If your root is ax + ib, then the other root is ax - ib, and vice versa.
so that makes two roots, and 6 makes the third root, so you’ve got them all.
First, put the monomials together:
(x - 6)(x + i)(x - i)
if you need it in standard form, do two things:
(1) multiply the conjugates together
( x + i)(x - i)
(2) then multiply the result by (x - 6)
if you have any questions, please answer here.
Edward A.
Ok Michael, the first thing was to realize that, since (1+ i) is a root, another root is (1 - i). So I hope I explained that, and that understood that we can write the expression as (x - 6)(x -(1 + i))(x - (1 - i)) (Notice, my original answer was inaccurate, I treated them as I and - i, but actually the problem states that you have 1+ i, therefore you have also 1 - i. I’m doing it correctly now). I’m guessing that here is where you said you don’t know how to do that—you don’t know how to multiply these monomials. Start with the complex roots, because they turn into something simpler. ( x - (1 + i)) * (x - (1 - i)) In algebra class, they teach to multiply First (F) Outer (O) Inner (I) Last (L), also known as FOIL. F: multiply both first terms. They are both “x”, so the result is F): x*x or “x squared” O: multiply the outer term, that means the first term of the left expression (x) by the last term of the second ( - (1 - i)), result is O): - x + ix I: multiply the inner terms, that means the second part of the left ( - (1 + i)) by the first term on the right (x), resulting in I): - x - ix L: multiply the last parts (-(1-i)) * ( - (1 + i)). This is simply 1 - i*i I hope you know that “i” means “the square root of -1” Therefore, 1 - i * i = 1 - ( - 1) = 1 + 1 = 2 Add up the F, O , I, and L items x * x - x + ix - x - ix + 2 The ix terms add to zero, so this yields (x*x - 2x + 2) Now multiply that by (x - 6). That is X * (x*x -2x + 2) -6 * (x*x -2x + 2) Multiply these out and collect terms. If you have more question, please ask and tell how far you’ve gotten.01/14/23
Mark M.
01/15/23
Edward A.
Oops, thanks Mark!01/15/23
Michael W.
The thing is i dont know how to do that01/14/23