To classify the polynomial [tex]\(3x^4 + 9x^2 - 1\)[/tex] by its degree, let's follow these steps:
### Step 1: Identify the Degree of the Polynomial
The degree of a polynomial is the highest power of the variable [tex]\( x \)[/tex] that appears with a non-zero coefficient.
The given polynomial is:
[tex]\[ 3x^4 + 9x^2 - 1 \][/tex]
Here, we can see the terms:
- [tex]\(3x^4\)[/tex]
- [tex]\(9x^2\)[/tex]
- [tex]\(-1\)[/tex]
The degrees of these terms are:
- [tex]\( x^4 \)[/tex] has a degree of [tex]\( 4 \)[/tex]
- [tex]\( x^2 \)[/tex] has a degree of [tex]\( 2 \)[/tex]
- The constant term [tex]\( -1 \)[/tex] has a degree of [tex]\( 0 \)[/tex]
The highest degree among these is [tex]\( 4 \)[/tex]. Therefore, the degree of the polynomial is [tex]\( 4 \)[/tex].
### Step 2: Know the Classification by Degree
Polynomials are classified based on their degrees as follows:
- A polynomial of degree [tex]\( 2 \)[/tex] is called quadratic.
- A polynomial of degree [tex]\( 3 \)[/tex] is called cubic.
- A polynomial of degree [tex]\( 4 \)[/tex] is called quartic.
- A polynomial of degree [tex]\( 5 \)[/tex] is called quintic.
- Polynomials of degree higher than [tex]\( 5 \)[/tex] or other than specified have different classifications or are simply referred to by their degree.
### Step 3: Classify the Polynomial
Since the polynomial [tex]\( 3x^4 + 9x^2 - 1 \)[/tex] has a degree of [tex]\( 4 \)[/tex], it is classified as a quartic polynomial.
### Conclusion
The polynomial [tex]\( 3x^4 + 9x^2 - 1 \)[/tex] is classified as quartic based on its degree.
So the correct classification is:
quartic
