Alright, let's analyze the given function step by step.
The function is:
[tex]\[ d(x) = 8x^2 + 7x - 5x^2 - 1 \][/tex]
### Step 1: Combine like terms
We see that there are terms involving [tex]\(x^2\)[/tex] and terms involving [tex]\(x\)[/tex], as well as a constant term. Let's combine the like terms:
- Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[ 8x^2 - 5x^2 = 3x^2 \][/tex]
- The [tex]\(x\)[/tex] term remains as it is:
[tex]\[ + 7x \][/tex]
- The constant term remains as it is:
[tex]\[ - 1 \][/tex]
So, after combining like terms, the function simplifies to:
[tex]\[ d(x) = 3x^2 + 7x - 1 \][/tex]
### Step 2: Determine the degree of the polynomial
The degree of a polynomial is the highest power of [tex]\(x\)[/tex] that appears with a non-zero coefficient.
In the simplified expression [tex]\(d(x) = 3x^2 + 7x - 1\)[/tex]:
- The highest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex], which indicates that the degree of the polynomial is [tex]\(2\)[/tex].
### Conclusion
Since the degree of the polynomial is [tex]\(2\)[/tex], the function [tex]\(d(x) = 3x^2 + 7x - 1\)[/tex] is a quadratic function.
So the correct classification of the function [tex]\(d(x)\)[/tex] is:
Quadratic.
