Answer :

To determine if the function [tex]\( G(x) = -4x + 3x^6 + 9 + 5x^7 \)[/tex] is a polynomial function, let's break it down using the definition of a polynomial.

A polynomial function is composed of terms in the form of [tex]\( a_n \cdot x^n \)[/tex], where:
- [tex]\( a_n \)[/tex] is a constant coefficient.
- [tex]\( n \)[/tex] is a non-negative integer.
- The terms are combined using addition or subtraction.

Let's look at each term in the given function:

1. Term: [tex]\(-4x\)[/tex]
- Coefficient: [tex]\(-4\)[/tex] (a constant)
- Power of [tex]\(x\)[/tex]: [tex]\(1\)[/tex] (which is a non-negative integer)

2. Term: [tex]\(3x^6\)[/tex]
- Coefficient: [tex]\(3\)[/tex] (a constant)
- Power of [tex]\(x\)[/tex]: [tex]\(6\)[/tex] (which is a non-negative integer)

3. Term: [tex]\(9\)[/tex]
- Coefficient: [tex]\(9\)[/tex] (a constant)
- This can be considered as [tex]\(9 \cdot x^0\)[/tex], because any non-zero number raised to the power of [tex]\(0\)[/tex] is [tex]\(1\)[/tex].

4. Term: [tex]\(5x^7\)[/tex]
- Coefficient: [tex]\(5\)[/tex] (a constant)
- Power of [tex]\(x\)[/tex]: [tex]\(7\)[/tex] (which is a non-negative integer)

Each term in [tex]\( G(x) \)[/tex] fits the criteria for a polynomial term. Therefore, the entire function [tex]\( G(x) = -4x + 3x^6 + 9 + 5x^7 \)[/tex] is a polynomial function.

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Rewritten by : Barada