Q5(vi):

Classify the following as linear, quadratic and cubic polynomials: (vi) $r^2$

Initial Setup & Given Expression

We are given the following algebraic expression:

$P(r) = r^2$

Our objective is to classify this polynomial as linear, quadratic, or cubic based on its mathematical properties.

Step 1: Theoretical Foundation of Polynomial Classification

In algebra, polynomials are classified according to their degree. The degree of a polynomial in one variable is defined as the highest power (exponent) of the variable present in the expression with a non-zero coefficient. [Per the Fundamental Theorem of Algebra and standard polynomial definitions].

The standard classifications based on degree are as follows:

• Linear Polynomial: A polynomial of degree $1$. General form: $ax + b$ (where $a \neq 0$).
• Quadratic Polynomial: A polynomial of degree $2$. General form: $ax^2 + bx + c$ (where $a \neq 0$).
• Cubic Polynomial: A polynomial of degree $3$. General form: $ax^3 + bx^2 + cx + d$ (where $a \neq 0$).

Step 2: Analyzing the Degree of the Given Polynomial

Let us examine the given polynomial:

$P(r) = r^2$

This expression is a monomial (a polynomial consisting of a single term). We analyze the components of this term:

• The variable is $r$.
• The coefficient is $1$ (since $r^2$ is equivalent to $1 \cdot r^2$).
• The exponent of the variable $r$ is $2$.

Because there are no other terms in the polynomial, the highest power of the variable $r$ is exactly $2$. Therefore, the degree of the polynomial $P(r) = r^2$ is $2$.

Step 3: Visualizing the Quadratic Nature

To rigorously confirm the nature of this polynomial, we can observe its graphical representation. A polynomial of degree $2$ forms a parabola when plotted on a Cartesian coordinate system. The precise geometric mapping of $P(r) = r^2$ demonstrates a non-linear, symmetric curve with a single vertex at the origin $(0,0)$.

Step 4: Final Classification

Because the highest exponent of the variable $r$ is $2$, the polynomial satisfies the strict definition of a quadratic polynomial. It does not possess a degree of $1$ (which would make it linear) nor a degree of $3$ (which would make it cubic).

Final Solution: The polynomial $r^2$ is a quadratic polynomial.