Question

Is the function f left parenthesis x right parenthesis equals 5 x squared minus 3 x superscript 4 baseline plus Start Fraction 3 over 4 End Fraction x minus 7a polynomial function? If so, state its degree and the leading coefficient.
(1 point)
Responses

The function is not a polynomial.
The function is not a polynomial.

The function is a polynomial. The degree is 2 and the leading coefficient is 5.
The function is a polynomial. The degree is 2 and the leading coefficient is 5.

The function is a polynomial. The degree is 4 and the leading coefficient is negative 3.
The function is a polynomial. The degree is 4 and the leading coefficient is Image with alt text: negative 3 .

The function is a polynomial. The degree is 4 and the leading coefficient is 5.
The function is a polynomial. The degree is 4 and the leading coefficient is 5.
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Answer 1

To determine if the function \( f(x) = 5x^2 - 3x^4 + \frac{3}{4}x - 7 \) is a polynomial, we need to check its form. A polynomial function is defined as a function that can be expressed in the form \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \), where \( n \) is a non-negative integer and \( a_i \) are constants.

1. **Identify the terms of the function**:
- \( 5x^2 \) (degree 2)
- \( -3x^4 \) (degree 4)
- \( \frac{3}{4}x \) (degree 1)
- \( -7 \) (constant term, degree 0)

2. **Identify the degree**:
- The degree of a polynomial is the highest power of \( x \). In this case, the term with the highest power is \( -3x^4 \), which has degree 4.

3. **Identify the leading coefficient**:
- The leading coefficient is the coefficient of the term with the highest degree. Here, the leading coefficient is \( -3 \), which is the coefficient of \( -3x^4 \).

Based on this analysis:

- The function is a polynomial.
- The degree is 4.
- The leading coefficient is \(-3\).

So the correct response is:
**The function is a polynomial. The degree is 4 and the leading coefficient is \(-3\).**