Q4(iv):

Write the degree of each of the following polynomials: (iv) $3$

Initial Setup & Theoretical Foundation

We are given the mathematical expression:

$P(x) = 3$

To determine the degree of this polynomial, we must first establish the formal definition of a polynomial's degree. [Per the Fundamental Theorem of Algebra and standard polynomial theory], the degree of a polynomial in a single variable is defined as the highest exponent (power) of the variable that possesses a non-zero coefficient.

Step 1: Algebraic Manipulation of the Constant

The given expression is a constant number, $3$. In algebra, any non-zero constant can be expressed as a product of that constant and a variable raised to the power of zero.

Let us introduce a variable, $x$. [By the Zero Exponent Rule of indices, $x^0 = 1$ for all $x \neq 0$]. Therefore, we can rewrite the constant polynomial as follows:

$P(x) = 3 \cdot 1$

$P(x) = 3 \cdot x^0$

Step 2: Identification of the Highest Power

A polynomial in one variable $x$ is generally written in the standard descending form:

$P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x^1 + a_0 x^0$

By mapping our expression $P(x) = 3x^0$ to the standard form, we observe:

• The coefficient $a_0 = 3$.
• All other coefficients ($a_1, a_2, \dots, a_n$) are exactly $0$.

Since the only term with a non-zero coefficient is $3x^0$, the highest power of the variable $x$ present in the polynomial is $0$.

Step 3: Graphical Verification (Geometric Interpretation)

Geometrically, the degree of a polynomial $P(x)$ correlates with the maximum number of times its graph can intersect the x-axis (its roots). The graph of $P(x) = 3$ is a horizontal line parallel to the x-axis, maintaining a constant y-value of $3$. Because it is parallel to the x-axis and not coincident with it, it never intersects the x-axis. Zero intersections imply zero roots, which perfectly aligns with a polynomial of degree $0$.

Theoretical Exception Note

It is critical to distinguish between a non-zero constant polynomial (like $3$) and the zero polynomial ($P(x) = 0$). While the degree of any non-zero constant polynomial is strictly $0$, the degree of the zero polynomial is mathematically undefined because $0$ can be written as $0x^0$, $0x^1$, $0x^{100}$, etc., making it impossible to define a unique highest power.

Final Solution: The degree of the polynomial $3$ is $0$.