Determine Whether The Function Is A Polynomial Function. F ( X ) = 6 X 4 + 5 X + 4 X F(x)=6x^4+5x+\frac{4}{x} F ( X ) = 6 X 4 + 5 X + X 4 A. Yes B. No
Introduction
In mathematics, a polynomial function is a function that can be expressed as a sum of terms, where each term is a product of a constant and a variable raised to a non-negative integer power. In this article, we will determine whether the given function is a polynomial function or not.
What is a Polynomial Function?
A polynomial function is a function that can be written in the form:
f(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0
where a_n, a_(n-1), ..., a_1, a_0 are constants, and n is a non-negative integer.
The Given Function
The given function is:
f(x) = 6x^4 + 5x + \frac{4}{x}
Is the Function a Polynomial Function?
To determine whether the function is a polynomial function, we need to check if it can be written in the form of a polynomial function.
Analysis
The function f(x) = 6x^4 + 5x + \frac{4}{x} has three terms:
- 6x^4: This term is a product of a constant (6) and a variable (x) raised to a non-negative integer power (4). Therefore, this term is a polynomial term.
- 5x: This term is also a product of a constant (5) and a variable (x) raised to a non-negative integer power (1). Therefore, this term is also a polynomial term.
- \frac{4}{x}: This term is not a product of a constant and a variable raised to a non-negative integer power. Instead, it is a fraction with a variable (x) in the denominator. Therefore, this term is not a polynomial term.
Conclusion
Since the function f(x) = 6x^4 + 5x + \frac{4}{x} has a term that is not a polynomial term (\frac{4}{x}), it cannot be written in the form of a polynomial function. Therefore, the answer is:
B. No
Why is the Function Not a Polynomial Function?
The function f(x) = 6x^4 + 5x + \frac{4}{x} is not a polynomial function because it has a term that is not a polynomial term (\frac{4}{x). This term is a fraction with a variable (x) in the denominator, which means it cannot be written in the form of a polynomial function.
What are the Characteristics of a Polynomial Function?
A polynomial function has the following characteristics:
- It can be written in the form of a sum of terms, where each term is a product of a constant and a variable raised to a non-negative integer power.
- It has a finite number of terms.
- Each term has a non-negative integer power.
- The function is defined for all real numbers.
Examples of Polynomial Functions
Some examples of polynomial functions are:
- f(x) = 2x^3 + 3x^2 - 4x + 1
- f(x) = x^2 + 2x - 3
- f(x) = 4x^4 - 2x^2 + 1
Conclusion
Introduction
In our previous article, we determined that the function f(x) = 6x^4 + 5x + \frac{4}{x} is not a polynomial function. In this article, we will answer some frequently asked questions related to polynomial functions.
Q: What is a polynomial function?
A polynomial function is a function that can be expressed as a sum of terms, where each term is a product of a constant and a variable raised to a non-negative integer power.
Q: What are the characteristics of a polynomial function?
A polynomial function has the following characteristics:
- It can be written in the form of a sum of terms, where each term is a product of a constant and a variable raised to a non-negative integer power.
- It has a finite number of terms.
- Each term has a non-negative integer power.
- The function is defined for all real numbers.
Q: What are some examples of polynomial functions?
Some examples of polynomial functions are:
- f(x) = 2x^3 + 3x^2 - 4x + 1
- f(x) = x^2 + 2x - 3
- f(x) = 4x^4 - 2x^2 + 1
Q: Can a polynomial function have a variable in the denominator?
No, a polynomial function cannot have a variable in the denominator. The variable must be raised to a non-negative integer power, and the denominator must be a constant.
Q: Can a polynomial function have a fraction as a term?
No, a polynomial function cannot have a fraction as a term. The terms of a polynomial function must be products of constants and variables raised to non-negative integer powers.
Q: Can a polynomial function have a negative exponent?
No, a polynomial function cannot have a negative exponent. The exponents of a polynomial function must be non-negative integers.
Q: Can a polynomial function have a variable with a fractional exponent?
No, a polynomial function cannot have a variable with a fractional exponent. The exponents of a polynomial function must be non-negative integers.
Q: Can a polynomial function have a constant term?
Yes, a polynomial function can have a constant term. For example, f(x) = 2x^3 + 3x^2 - 4x + 1 has a constant term of 1.
Q: Can a polynomial function have a term with a coefficient of zero?
Yes, a polynomial function can have a term with a coefficient of zero. For example, f(x) = 2x^3 + 3x^2 - 4x + 0 has a term with a coefficient of zero.
Q: Can a polynomial function have a term with a variable raised to a negative power?
No, a polynomial function cannot have a term with a variable raised to a negative power. The exponents of a polynomial function must be non-negative integers.
Conclusion
In conclusion, a polynomial function is a function that can be expressed as a sum of terms, where each term is a product of a constant and a variable raised to a non-negative integer power. A polynomial function cannot have a variable in the denominator, a fraction as a term, a negative exponent, or a variable with a fractional exponent.