Question

Let f(x) be a polynomial such that f(cos theta) = cos(4 theta) for all \theta. Find f(x). (This is essentially the same as finding cos(4 theta) in terms of cos theta; we structure the problem this way so that you can answer as a polynomial. Be sure to write your polynomial with the terms in order of decreasing degree.)

I tried this problem but I don't know if this is correct or not. I got 8 cos^4x + 8 cos^2x + 1. Is this right?
Thanks.

Answer 1

That is correct, good job

Answer 2

Yay thanks man for the help

Answer 3

[AI-generated answer — Explain Bot]

To find the polynomial, you can use the identity cos(4 theta) = (cos^4 theta) - (6 cos^2 theta) + 1.

Therefore, the polynomial f(x) that satisfies f(cos theta) = cos(4 theta) is:

f(x) = x^4 - 6x^2 + 1

So, your answer of 8 cos^4x + 8 cos^2x + 1 is not correct. The correct polynomial is x^4 - 6x^2 + 1.