Let's determine the degrees of each polynomial expression given in the table. The degree of a polynomial is the highest power of the variable [tex]\(x\)[/tex] in the polynomial.
1. Polynomial: [tex]\(x-9\)[/tex]
- The highest power of [tex]\(x\)[/tex] in this polynomial is 1 ([tex]\(x^1\)[/tex]).
- Degree: 1
2. Polynomial: [tex]\(-4x^2 - 6x + 9\)[/tex]
- The highest power of [tex]\(x\)[/tex] in this polynomial is 2 ([tex]\(x^2\)[/tex]).
- Degree: 2
3. Polynomial: [tex]\(x^2 - 2x + 9\)[/tex]
- The highest power of [tex]\(x\)[/tex] in this polynomial is 2 ([tex]\(x^2\)[/tex]).
- Degree: 2
4. Polynomial: [tex]\(-3\)[/tex]
- This is a constant polynomial. The degree of a non-zero constant polynomial is 0.
- Degree: 0
5. Polynomial: [tex]\(3x - 2\)[/tex]
- The highest power of [tex]\(x\)[/tex] in this polynomial is 1 ([tex]\(x^1\)[/tex]).
- Degree: 1
6. Polynomial: [tex]\(6x + 2\)[/tex]
- The highest power of [tex]\(x\)[/tex] in this polynomial is 1 ([tex]\(x^1\)[/tex]).
- Degree: 1
Now, let's add the missing degrees to the table:
[tex]\[
\begin{tabular}{|c|c|}
\hline
\text{Polynomial Expression} & \text{Degree} \\
\hline
x-9 & 1 \\
\hline
-4 x^2-6 x+9 & 2 \\
\hline
x^2-2 x+9 & 2 \\
\hline
-3 & 0 \\
\hline
3 x-2 & 1 \\
\hline
6 x+2 & 1 \\
\hline
5 & 0 \\
\hline
\end{tabular}
\][/tex]
To summarize, the degrees are as follows:
- [tex]\(x-9: 1\)[/tex]
- [tex]\(-4x^2 - 6x + 9: 2\)[/tex]
- [tex]\(x^2 - 2x + 9: 2\)[/tex]
- [tex]\(-3: 0\)[/tex]
- [tex]\(3x - 2: 1\)[/tex]
- [tex]\(6x + 2: 1\)[/tex]
- [tex]\(5: 0\)[/tex]
This completes our table and confirms the degrees of each polynomial.
