A trigonometric polynomial

Algebra Level 3

If $f(x)$ is a 4-degree polynomial with integer coefficients such that:

$f(\sin x)=f(\cos x)$

$f(1)=0$

$f(5)=-600$

find $f(2)$ .

1 0 -12 4π e

1 solution

Alberto Caldera Morante
Apr 11, 2018

$sen^{2} x+cos^{2} x=1$

$sen^{2} x=1-cos^{2} x$ and $cos^{2} x=1-sen^{2} x$

if we multiplicate in cross:

$sen^{2} x(1-sen^{2} x)=cos^{2} x(1-cos^{2} x)$

it is the same expression for sen x and for cos x so

$\textit{f(sen x)=f(cos x)})$ and I we substitute sen x or cos x for x the polynomial appears

$\textit{f(x)}=x^{2}(1-x^{2})$

$\textit{f(2)}=4(1-4)=-12$

What if f(x)=(x^4)[(1-x^2)^2] or f(x)=2(x^2)(1-x^2)?

X X - 3 years, 2 months ago

@X X ummm.........well, your given functions don't satisfy f(5)=-600...........

Aaghaz Mahajan - 2 years, 5 months ago