This is going to be a fourth degree polynomial because if x = 4, one of the factors is x - 4; if x = -2, then one of the factors is x + 2; and by the conjugate root theorem, if x = -1 + 2i, then -1-2i HAS to also be a root. If x = -1+2i, then the factor is (x-(-1+2i)). Simplfying that gives us (x+1-2i). Likewise, if x = -1-2i, then the factor is (x-(-1-2i)). Simplifying that gives us (x+1+2i). We will FOIL those complex factors first. Doing that we have [tex] x^2+x-2ix+x+1-2i+2ix+2i-4i^2 [/tex]. Once we simplify that down it's much easier to deal with than what it looks like right now. [tex] x^2+2x+1-4i^2 [/tex]. Since i^2 = -1, what we have in the end is [tex] x^2+2x+5 [/tex]. Now we will multiply that by x-4. [tex] (x^2+2x+5)(x-4)=x^3-2x^2-3x-20 [/tex]. Now finally we will multiply in the last factor of x+2. [tex] x^4-7x^2-26x-40 [/tex] is your final fourth degree polynomial.
Answer:
[tex]x^{4} - 7x^{2} -26x-40[/tex]
Step-by-step explanation:
The polynomial will be a fourth degree polynomial.
Let's take x = 4, then one of the factors is x - 4.
Then if x = -2, then the factor will be:
x + 2 = -2+ 2
x + 2 = 0
Then one of the polynomials is x + 2
Using the conjugate theorem:
if x = -1 + 2i, then -1-2i will be the root.
Then if x = -1+ 2i, then the factor of the function will be x- 1- 2i
Similarly, if x = -1-2i, then the factor x -(-1-2i)
This gives the final expression:
[tex]x^{4} -7x^{2} -26x-40[/tex]
