Thread: polynomials and zeros

How do you solve these?? Have been puzzling over them since Monday. I need to pass this 8 hrs from now. Thanks!

1) R is a root of f(x)=x^2+x+1; what is the remainder when P(x)= x^3333-x^333+x^33-x^3+3 is divided by x/r?

2) What is the domain of the 1-to-1 function f(x)= 2^x/2 - 3; find the inverse function and its range. Then, show that the composite functions f(f^1(x)) and f^1(f(x)) are both equal to x.

Originally Posted by paladin123

1) R is a root of f(x)=x^2+x+1; what is the remainder when P(x)= x^3333-x^333+x^33-x^3+3 is divided by x/r?
.

What is R?
By Quadradic Forumula,
x=[(-1)+/-isqrt(3)]/2

Let us say that,
R=(-1/2)+(sqrt(3)/2)i
The "-" should give the same result.

Now express in polar coordinates,
cos(5pi/6)+i*sin(5pi/6)

Now you can easily compute what 3333,333,33,3
Given when raised to their power through de-Moiver's theorem.

2) What is the domain of the 1-to-1 function f(x)= 2^x/2 - 3; find the inverse function and its range. Then, show that the composite functions f(f^1(x)) and f^1(f(x)) are both equal to x.

Use paranthesis!