Question
Question content area top
Part 1
Determine whether the function is a polynomial function. If it is, identify the degree.
g(x)equals6 x cubed minus pi x squared plus one seventh x
Question content area bottom
Part 1
Choose the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
It is a polynomial. The degree of the polynomial is
enter your response here.
B.
It is not a polynomial.
Part 1
Determine whether the function is a polynomial function. If it is, identify the degree.
g(x)equals6 x cubed minus pi x squared plus one seventh x
Question content area bottom
Part 1
Choose the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
It is a polynomial. The degree of the polynomial is
enter your response here.
B.
It is not a polynomial.
Answers
Answered by
GPT-4o mini
To determine whether the function \( g(x) = 6x^3 - \pi x^2 + \frac{1}{7}x \) is a polynomial function, we need to check if it meets the criteria for being a polynomial.
A polynomial function has the following characteristics:
1. It consists of terms of the form \( ax^n \) where \( a \) is a constant (coefficient) and \( n \) is a non-negative integer (the exponent).
2. The exponents must be whole numbers (0, 1, 2, 3, ...), and coefficients can be any real numbers.
Now, let's examine the function:
- The first term is \( 6x^3 \) which is of the form \( ax^n \) where \( a = 6 \) and \( n = 3 \) (a non-negative integer).
- The second term is \( -\pi x^2 \) where \( a = -\pi \) and \( n = 2 \) (also a non-negative integer).
- The third term is \( \frac{1}{7}x \) where \( a = \frac{1}{7} \) and \( n = 1 \) (again, a non-negative integer).
Since all terms conform to the standards for polynomial functions, we conclude that \( g(x) \) is indeed a polynomial function.
Next, to identify the degree of the polynomial, we look for the term with the highest exponent.
In \( g(x) = 6x^3 - \pi x^2 + \frac{1}{7}x \), the highest exponent is 3 (from the term \( 6x^3 \)).
Thus, we can conclude the following:
**A. It is a polynomial. The degree of the polynomial is 3.**
A polynomial function has the following characteristics:
1. It consists of terms of the form \( ax^n \) where \( a \) is a constant (coefficient) and \( n \) is a non-negative integer (the exponent).
2. The exponents must be whole numbers (0, 1, 2, 3, ...), and coefficients can be any real numbers.
Now, let's examine the function:
- The first term is \( 6x^3 \) which is of the form \( ax^n \) where \( a = 6 \) and \( n = 3 \) (a non-negative integer).
- The second term is \( -\pi x^2 \) where \( a = -\pi \) and \( n = 2 \) (also a non-negative integer).
- The third term is \( \frac{1}{7}x \) where \( a = \frac{1}{7} \) and \( n = 1 \) (again, a non-negative integer).
Since all terms conform to the standards for polynomial functions, we conclude that \( g(x) \) is indeed a polynomial function.
Next, to identify the degree of the polynomial, we look for the term with the highest exponent.
In \( g(x) = 6x^3 - \pi x^2 + \frac{1}{7}x \), the highest exponent is 3 (from the term \( 6x^3 \)).
Thus, we can conclude the following:
**A. It is a polynomial. The degree of the polynomial is 3.**