Certainly! Let's analyze each polynomial to determine its degree and the number of variables involved.
### Step-by-Step Solution
#### (a) [tex]\(2 x^2 + 5 x + 12\)[/tex]
1. Degree: The degree of a polynomial is the highest power of the variable with a non-zero coefficient. Here, the terms are [tex]\(2x^2\)[/tex], [tex]\(5x\)[/tex], and [tex]\(12\)[/tex]. The highest power is [tex]\(x^2\)[/tex], so the degree is [tex]\(2\)[/tex].
2. Number of Variables: This polynomial only has one variable, [tex]\(x\)[/tex].
Therefore, for (a):
- Degree: [tex]\(2\)[/tex]
- Number of Variables: [tex]\(1\)[/tex]
#### (b) [tex]\(4 x^2 y + 5 x y + 6\)[/tex]
1. Degree: We need to consider the total power of each term. The terms are:
- [tex]\(4x^2y\)[/tex]: The power is [tex]\(2 + 1 = 3\)[/tex].
- [tex]\(5xy\)[/tex]: The power is [tex]\(1 + 1 = 2\)[/tex].
- [tex]\(6\)[/tex]: This is a constant term with degree [tex]\(0\)[/tex].
The highest combined power is [tex]\(3\)[/tex], so the degree is [tex]\(3\)[/tex].
2. Number of Variables: This polynomial involves two variables, [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
Therefore, for (b):
- Degree: [tex]\(3\)[/tex]
- Number of Variables: [tex]\(2\)[/tex]
#### (c) [tex]\(4 x + 9\)[/tex]
1. Degree: The terms are [tex]\(4x\)[/tex] and [tex]\(9\)[/tex]. The highest power is [tex]\(x\)[/tex], so the degree is [tex]\(1\)[/tex].
2. Number of Variables: This polynomial only has one variable, [tex]\(x\)[/tex].
Therefore, for (c):
- Degree: [tex]\(1\)[/tex]
- Number of Variables: [tex]\(1\)[/tex]
#### (d) [tex]\(4 x^2 + 2 x + 10\)[/tex]
1. Degree: The terms are [tex]\(4x^2\)[/tex], [tex]\(2x\)[/tex], and [tex]\(10\)[/tex]. The highest power is [tex]\(x^2\)[/tex], so the degree is [tex]\(2\)[/tex].
2. Number of Variables: This polynomial only has one variable, [tex]\(x\)[/tex].
Therefore, for (d):
- Degree: [tex]\(2\)[/tex]
- Number of Variables: [tex]\(1\)[/tex]
#### (e) [tex]\(2 x^3 - 18 x^2 + 2 x - 2\)[/tex]
1. Degree: The terms are [tex]\(2x^3\)[/tex], [tex]\(-18x^2\)[/tex], [tex]\(2x\)[/tex], and [tex]\(-2\)[/tex]. The highest power is [tex]\(x^3\)[/tex], so the degree is [tex]\(3\)[/tex].
2. Number of Variables: This polynomial only has one variable, [tex]\(x\)[/tex].
Therefore, for (e):
- Degree: [tex]\(3\)[/tex]
- Number of Variables: [tex]\(1\)[/tex]
### Final Table
[tex]\[
\begin{tabular}{|l|c|c|c|}
\hline
\text{S.N.} & \text{Polynomial} & \text{Degree} & \text{No of Variables} \\
\hline
(a) & 2x^2 + 5x + 12 & 2 & 1 \\
(b) & 4x^2y + 5xy + 6 & 3 & 2 \\
(c) & 4x + 9 & 1 & 1 \\
(d) & 4x^2 + 2x + 10 & 2 & 1 \\
(e) & 2x^3 - 18x^2 + 2x - 2 & 3 & 1 \\
\hline
\end{tabular}
\][/tex]
I hope this explanation clarifies the degree and the number of variables in each polynomial!
