SOLUTION: Hi. I am supposed to create a 3rd degree polynomial with the roots of -4, 2+i, 2-i. I know I need to change it to show (x+4)(x-(2+i))(x-(2-i))and foil then distribute, but everytim

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: Hi. I am supposed to create a 3rd degree polynomial with the roots of -4, 2+i, 2-i. I know I need to change it to show (x+4)(x-(2+i))(x-(2-i))and foil then distribute, but everytim      Log On


   

Question 798748: Hi. I am supposed to create a 3rd degree polynomial with the roots of -4, 2+i, 2-i. I know I need to change it to show (x+4)(x-(2+i))(x-(2-i))and foil then distribute, but everytime I get confused on if i doing so correctly.
Found 2 solutions by MathLover1, josgarithmetic:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

%28x%2B4%29%28x-%282%2Bi%29%29%28x-%282-i%29%29.....good start, now just multiply
%28x%2B4%29%28x-2-i%29%28x-2%2Bi%29

%28x%5E2-2x-xi%2B4x-8-4i%29%28x-2%2Bi%29

%28x%5E2%2B2x-xi-4i-8%29%28x-2%2Bi%29

x%5E3%2B2x%5E2-x%5E2i-4xi-8x-2x%5E2-2x%2B2xi%2B8i%2B16%2Bx%5E2i%2B2xi%2Bxi%5E2-4i%5E2-8i+



x%5E3-10x+%2B16+%2Bx%28-1%29+-4%28-1%29++....recall that i%5E2=-1


x%5E3-10x+%2B16-x+%2B4++
x%5E3-11x+%2B20++.........your answer

+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+30%2C+x%5E3-11x+%2B20+%29+


Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Use just what you already learned about multiplying polynomials. You have the correct idea. Even you choice of associating is good. Now just do it!

Start with the complex factors first, simplify the result, and then multiply by the purely real binomial.

Try this for the complex two factors:
%28x-%282%2Bi%29%29%28x-%282-i%29%29
%28x-2-i%29%28x-2%2Bi%29
%28%28x-2%29-i%29%28%28x-2%29%2Bi%29, which will give you Difference of Two Squares,
%28x-2%29%5E2-i%5E2
x%5E2-4x%2B2-%28-1%29
x%5E2-4x%2B2%2B1
x%5E2-4x%2B3

Now you continue, performing %28x%2B4%29%28x%5E2-4x%2B3%29. Either do this using distributive property, or you can use the old-style "long multiplication" method as you learned in elementary school, but with the terms in the polynomial factors if you are comfortable/more comfortable using the method.
...
Alternatively, you could stop when you have the function in purely factored form instead of fully multiplying to get into general form.
(Meaning that your function is %28x%2B4%29%28x%5E2-4x%2B3%29 ).

The rest is for you to do.