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
Check work
The Correct Answer and Explanation is:
Let’s analyze which of the following functions are polynomial functions, based on the image you provided.
✅ Polynomial Functions:
A polynomial function is a function that:
- Involves only non-negative integer exponents of the variable.
- Has real-number coefficients.
- Has no variables in denominators, no negative or fractional exponents, and no non-algebraic functions like exponential, logarithmic, or trigonometric functions.
✅ These ARE polynomial functions:
- f(x)=0f(x) = 0
- This is the zero polynomial. It satisfies all conditions: no variable terms, constant is 0.
- f(x)=−9f(x) = -9
- Constant function. Degree = 0. This is a valid polynomial.
- f(x)=3x+1f(x) = 3x + 1
- Linear polynomial (degree 1). Coefficients are real and exponents are integers ≥ 0.
- f(x)=x7−32×6−πx3+4584f(x) = x^7 – 32x^6 – \pi x^3 + \frac{45}{84}
- Despite π\pi being irrational and 4584\frac{45}{84} being a fraction, both are real numbers. All exponents are non-negative integers.
❌ These are NOT polynomial functions:
- f(x)=x1/2−x+8f(x) = x^{1/2} – x + 8
- Not a polynomial: x1/2x^{1/2} is a fractional exponent.
- f(x)=−4x−3+5x−1+7−18x2f(x) = -4x^{-3} + 5x^{-1} + 7 – 18x^2
- Not a polynomial: negative exponents like x−3x^{-3}, x−1x^{-1}.
- f(x)=(x+1)(x−1)+ex−exf(x) = (x+1)(x-1) + e^x – e^x
- Simplifies to (x+1)(x−1)(x+1)(x-1), which is polynomial, but the presence of exponential terms (exe^x) disqualifies it as a pure polynomial.
- f(x)=x2−3x+2x−2f(x) = \frac{x^2 – 3x + 2}{x – 2}
- Although the numerator is polynomial, division by a variable disqualifies it unless it simplifies fully to a polynomial, which this doesn’t over the entire domain.
✅ Final Answer: The following are polynomial functions:
- f(x)=0f(x) = 0
- f(x)=−9f(x) = -9
- f(x)=3x+1f(x) = 3x + 1
- f(x)=x7−32×6−πx3+4584f(x) = x^7 – 32x^6 – \pi x^3 + \frac{45}{84}
📚 Explanation
Polynomial functions are algebraic expressions that consist of variables raised to whole number powers and multiplied by real-number coefficients. The defining traits include: (1) exponents on variables must be non-negative integers (e.g., 0, 1, 2…), (2) coefficients can be any real numbers (including fractions and irrational numbers like π\pi), and (3) the expression must not include square roots of variables, negative exponents, variables in denominators, or non-algebraic functions like exponential or logarithmic functions.
In the given options, the functions f(x)=0f(x) = 0, f(x)=−9f(x) = -9, and f(x)=3x+1f(x) = 3x + 1 are clearly polynomials. They involve either no variable or a variable raised to power 1, with appropriate real-number coefficients. f(x)=x7−32×6−πx3+4584f(x) = x^7 – 32x^6 – \pi x^3 + \frac{45}{84} is also a polynomial: despite the presence of irrational and fractional constants, all variable exponents are non-negative integers.
On the other hand, f(x)=x1/2−x+8f(x) = x^{1/2} – x + 8 includes a fractional exponent, which violates the polynomial definition. Similarly, f(x)=−4x−3+5x−1+7−18x2f(x) = -4x^{-3} + 5x^{-1} + 7 – 18x^2 uses negative exponents, which are not allowed. The function f(x)=(x+1)(x−1)+ex−exf(x) = (x + 1)(x – 1) + e^x – e^x simplifies algebraically, but the presence of the exponential function exe^x, even though it cancels, breaks the rule of only allowing algebraic operations. Finally, f(x)=x2−3x+2x−2f(x) = \frac{x^2 – 3x + 2}{x – 2} involves division by a variable, making it undefined at x=2x = 2, and hence not a true polynomial.
Thus, only four of the given functions qualify as polynomial functions.
