Please tell me the polynomials and why are they polynomials. Thank you for your time.
Which of the following are polynomial functions?
f(x) = 0
f(x) = -9
f(x) = 3 + 1
f(x) = 1/2 + 8
f(x) = 32 + 2
f(x) = 45/84
The Correct Answer and Explanation is:
Let’s examine each of the following functions to determine which are polynomial functions and explain why.
What is a Polynomial Function?
A polynomial function is a mathematical expression involving a sum of powers of the variable x with real number coefficients. A general form is: f(x)=anxn+an−1xn−1+…+a1x+a0f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0
Where:
- an,an−1,…,a0a_n, a_{n-1}, \ldots, a_0 are real numbers (called coefficients),
- nn is a non-negative integer (no negative or fractional exponents),
- The variable xx is raised to whole-number powers only.
Let’s evaluate each function:
- f(x) = 0
✅ Polynomial
This is a zero polynomial, which is valid because it can be written as 0x00x^0. It has no variable terms, but 0 is considered a constant polynomial. - f(x) = -9
✅ Polynomial
This is a constant polynomial, which is valid and has degree 0 (no variable, just a constant). Constants are polynomials. - f(x) = 3 + 1
✅ Polynomial
This simplifies to f(x)=4f(x) = 4, a constant. As above, constants are polynomials of degree 0. - f(x) = 1/2 + 8
✅ Polynomial
This simplifies to f(x)=8.5f(x) = 8.5 or 172\frac{17}{2}. It is a constant polynomial — again, valid. - f(x) = 32 + 2
✅ Polynomial
This simplifies to f(x)=34f(x) = 34. Constant again — a polynomial. - f(x) = 45/84
✅ Polynomial
This simplifies to f(x)=1528f(x) = \frac{15}{28}, a constant. Any real number constant is a polynomial of degree 0.
✅ Final Answer: All of them are polynomial functions.
Why Are They Polynomials?
Even though these functions don’t involve a variable (like xx), they are still polynomials. Polynomials include:
- Zero (like f(x)=0f(x) = 0),
- Constants (like f(x)=−9f(x) = -9, f(x)=172f(x) = \frac{17}{2}, or any real number),
- Or expressions like f(x)=axn+bxn−1+…f(x) = ax^n + bx^{n-1} + \ldots (none of which appear here, but would still qualify).
There are no variables raised to a negative or fractional exponent, no variables inside square roots, and all the coefficients are real numbers — these are the key rules that define a polynomial. Since all examples here are constants and constants are polynomials (specifically, of degree 0), they are all valid polynomial functions.
Let me know if you want examples of non-polynomials for contrast!