Q4(iii):

Write the degree of each of the following polynomials: (iii) $5t – \sqrt{7}$

Initial Setup & Theoretical Foundation

We are given the algebraic expression:

$P(t) = 5t - \sqrt{7}$

The degree of a polynomial in one variable is defined as the highest exponent (power) of the variable present in the polynomial with a non-zero coefficient. [Per the Fundamental Theorem of Algebra and standard polynomial definitions]. To find the degree, we must analyze the exponent of the variable $t$ in every term of the expression.

Step 1: Isolate and Analyze Each Term

A polynomial is constructed as a sum of individual terms. Let us break down the given polynomial $P(t)$ into its constituent terms:

• Term 1: $5t$
• Term 2: $-\sqrt{7}$

Step 2: Determine the Exponent of the Variable in Each Term

We rewrite each term to explicitly reveal the power of the variable $t$:

• Analyzing Term 1 ($5t$): The variable $t$ is written without an explicit exponent. By algebraic convention, any variable without a written exponent has an implicit exponent of $1$. Thus, it can be written as $5t^1$. The power of the variable in this term is $1$.
• Analyzing Term 2 ($-\sqrt{7}$): This is a constant term and contains no visible variable. By the laws of exponents, any non-zero variable raised to the power of $0$ equals $1$ (i.e., $t^0 = 1$). Thus, the constant $-\sqrt{7}$ can be mathematically expressed as $-\sqrt{7}t^0$. The power of the variable in this term is $0$.

Step 3: Identify the Highest Power (Degree)

We now compare the exponents of $t$ extracted from all terms:

• Power in Term 1: $1$
• Power in Term 2: $0$

The degree is the maximum value among these exponents: $\max(1, 0) = 1$. Therefore, the highest power of the variable $t$ is $1$.

Visual Representation: The Linear Nature of Degree 1 Polynomials

A polynomial of degree $1$ is classified geometrically as a linear polynomial. When graphed on a Cartesian coordinate system, a degree $1$ polynomial always forms a perfectly straight line. Below is the exact geometric representation of $P(t) = 5t - \sqrt{7}$, demonstrating its constant rate of change (slope = $5$) and its intercepts.

Final Solution: The degree of the polynomial $5t - \sqrt{7}$ is $1$.