Which of the following expressions are polynomials?

Question

Explain your answers.

(a) 3x^2-1/2x+2^-1

(b) 5x^2-x^1/2+1/4

(c) 5x^5+5^x+1

(d) 4x^2-rt2

(e) x^4+x^-3+2

Answer 1

Polynomials are of the form

ax^n + bx^(n - 1) + cx^(n - 2) + ...

where a, b, and c are constants, and where n is an integer greater than or equal to 0.

a) might be a polynomial, if the second term is (1/2)x and not
1/[2x] {This is where bracketing is significant.}

b) Not a polynomial, because of the x^(1/2).

c) Not a polynomial, due to the exponential 5^x.

d) 4x^2 - sqrt(2) is a polynomial; the sqrt(2) is just a constant.

e) Not a polynomial, because of the x^(-3).

Answer 2

The degree of a polynomial is always a whole number. If it is a negative number or a fraction it cannot be a polynomial. So
1. Not a polynomial because of 2^-1
2. Not a polynomial because of x^1/2
3. It can be a polynomial if x in 5^x +1 is a whole number
4. Not a polynomial because rt2 = 2^1/2
5. Not a polynomial.

Answer 3

(a) not a polynomial because it involves a negative power of x
(b) is not a polynomial because not all of the power of x are integers
(c) is not a polynomial because it contains x in an exponent
(d) is a polynomial
(e) is not a polynomial because it has a negative power of x (x^-3)