Definition
\[ n \in \mathbb{N} \] \[ a_n,\ a_{n-1},\ a_{n-2},\ldots,a_0 \in \mathbb{R} \] \[ P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0 \]every defined function from R to R is called polynomial
\[ a_nx^n,\ a_{n-1}x^{n-1},\ldots,a_1x,\ a_0 \]terminology of polynomial
\[ a_n,\ a_{n-1},\ldots,a_1,\ a_0 \]coefficient of polynomial
\[ a_n \]leading coefficient of polynomial
\[ a_0 \]fixed term of polynomial
\[ n \]Degree of polynomial (greatest exponent of x)
\[ d[P(x)]=derP(x)=n \]A function which results from natural exponents of x is called polynomial.
1. Which of the following functions are polynomial?
a) \(f(x)=2\)
b) \(f(x)=3x^2-5x+7\)
c) \(f(x)=0\)
d) \(f(x)=\sqrt{2}\)
e) \(f(x)=\sqrt{x}\)
f) \(f(x)=\sqrt{3}x^2\)
g) \(f(x)=\samefrac{1}{x^2}\)
h) \(f(x)=3x^2-5x\)
i) \(f(x)=x^3+\sin x\)
j) \(f(x)=4x^3-7x^2+3x+1\)
k) \(f(x)=x^2-\log x\)
2. What are the degrees of the given polynomials?
a) \(P(x)=2x^2-5x+7\)
b) \(P(x)=x+7x^2-5x^3\)
c) \(P(x)=7x-1\)
d) \(P(x)=9\)
e) \(P(x)=0\)