Q3:

Give one example each of a binomial of degree 35, and of a monomial of degree 100.

Theoretical Foundation & Definitions

To construct the required algebraic expressions, we must first establish the rigorous definitions of the polynomial classifications based on their terms and degrees [Per the Fundamental Definitions of Polynomial Algebra]:

• Polynomial: An algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
• Term: A single mathematical expression separated by addition ($+$) or subtraction ($-$) signs.
• Monomial: A polynomial consisting of exactly one non-zero term.
• Binomial: A polynomial consisting of exactly two non-zero terms.
• Degree of a Polynomial in One Variable: The highest exponent (power) of the variable present in the polynomial with a non-zero coefficient.

Step 1: Constructing a Binomial of Degree 35

Based on the definitions, a binomial of degree $35$ must satisfy two independent conditions:

1. It must contain exactly two terms.
2. The highest power of the variable must be exactly $35$.

The general form of such a binomial in variable $x$ can be written as:

$P(x) = ax^{35} + bx^k$

Where:

• $a$ and $b$ are non-zero real numbers ($a \neq 0, b \neq 0$).
• $k$ is a non-negative integer strictly less than $35$ ($0 \le k < 35$).

By selecting $a = 3$, $b = -4$, and $k = 2$, we generate a valid example:

$3x^{35} - 4x^2$

[Justification: The expression has two terms ($3x^{35}$ and $-4x^2$), making it a binomial. The highest exponent on the variable $x$ is $35$, making its degree $35$.]

Step 2: Constructing a Monomial of Degree 100

A monomial of degree $100$ must satisfy the following conditions:

1. It must contain exactly one term.
2. The power of the variable in that single term must be exactly $100$.

The general form of such a monomial in variable $y$ can be written as:

$Q(y) = cy^{100}$

Where:

• $c$ is any non-zero real number ($c \neq 0$).

By selecting $c = \sqrt{5}$, we generate a valid example:

$\sqrt{5}y^{100}$

[Justification: The expression consists of a single term ($\sqrt{5}y^{100}$), making it a monomial. The exponent on the variable $y$ is $100$, making its degree $100$.]

Visual Analysis of Polynomial Anatomy

The following diagram illustrates the structural components of the constructed polynomials, verifying their terms and degrees.

Final Solution:
An example of a binomial of degree 35 is $3x^{35} - 4x^2$.
An example of a monomial of degree 100 is $\sqrt{5}y^{100}$.
(Note: Infinite valid examples exist by altering the non-zero coefficients or the lower-degree terms in the binomial).