Answer to Question #158683 in Algebra for moon

Question

Determine if each function is a polynomial function or not. Determine the leading term, the constant term, and the degrees of each polynomial function.

1. p(z) = z² + 3z + 2017
2. q(x) = 3x/2 + ⁴√x
3. r(x) = x(x - 1) (x -2)²
4. f(s) = 2017^-3 + 2s^-4 - 3s⁸
5. A =  πr²
6. A(e) = 6e²

Answer 1

A polynomial in the variable $x$ is an expression of the form

$$a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$$

where $a_0,a_1,...a_n$ are real numbers, and $n$ is a nonnegative integer. 

1. $p(z)=z^2+3z+2017$

$p(z)$ is a polynomial function in the variable $z$

Leading term is $z^2,$ constant term is $2017,$ degree is $2.$

2. $q(x)=3x/2+\sqrt[4]{x}$

$q(x)$ is not a polynomial function in the variable $x$

3. $r(x)=x(x-1)(x-2)^2$

$r(x)$ is a polynomial function in the variable $x$

Leading term is $x^4,$ constant term is $0,$ degree is $4.$

4. $f(s)=2017^{-3}+2s^{-4}-3s^8$

$f(s)$ is not a polynomial function in the variable $s$

5. $A=\pi r^2$

$A(r)$ is a polynomial function in the variable $r$

Leading term is $\pi r^2,$ constant term is $0,$ degree is $2.$

6. $A(e)=6e^2$

$A(e)$ is a polynomial function in the variable $e$

Leading term is $6e^2,$ constant term is $0,$ degree is $0.$

If $e$ is a mathemical constant (Euler's number), then A is number and is a polynomial function of degree 0 in the any variable .

Leading term is $6e^2,$ constant term is $6e^2,$ degree is $0.$