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Q1(iii):
Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. (iii) $3\sqrt{t} + t\sqrt{2}$

Solution :

Initial Setup & Theoretical Foundation

We are given the algebraic expression:

$P(t) = 3\sqrt{t} + t\sqrt{2}$

To determine whether this expression is a polynomial in one variable, we must evaluate it against the fundamental definition of a polynomial. [Per the algebraic definition of polynomials], an expression is classified as a polynomial in one variable if and only if:

  • It contains exactly one variable.
  • The exponent of the variable in every term is a non-negative integer (i.e., a whole number belonging to the set $\mathbb{W} = \{0, 1, 2, 3, \dots\}$).
  • The coefficients of the variables are real numbers ($\mathbb{R}$).

Step 1: Algebraic Transformation of Radical Terms

To properly analyze the exponents, we must convert all radical notations into fractional exponents using the laws of indices. [By the definition of rational exponents, $\sqrt[n]{x^m} = x^{m/n}$].

Applying this to the given expression:

  • The first term is $3\sqrt{t}$. The square root of $t$ can be rewritten as $t^{1/2}$. Thus, the term becomes $3t^{1/2}$.
  • The second term is $t\sqrt{2}$. The variable $t$ is raised to the first power, and the coefficient is $\sqrt{2}$. This can be rewritten as $\sqrt{2}t^1$.

The transformed expression is:

$P(t) = 3t^{1/2} + \sqrt{2}t^1$

Step 2: Exponent Analysis & Verification

We now isolate and examine the exponent of the variable $t$ in each term of the transformed expression.

Term Variable Coefficient Exponent Is Exponent a Whole Number ($\in \mathbb{W}$)?
$3t^{1/2}$ $t$ $3$ $1/2$ No ($1/2 \notin \mathbb{W}$)
$\sqrt{2}t^1$ $t$ $\sqrt{2}$ $1$ Yes ($1 \in \mathbb{W}$)
Exponent Analysis of 3√t + t√2 3t1/2 + √2 t1 Term 1: 3t1/2 Exponent = 1/2 Not a Whole Number Term 2: √2 t1 Exponent = 1 Whole Number Conclusion: Fails Polynomial Condition

Step 3: Logical Deduction

While the expression contains only one variable ($t$), the exponent of $t$ in the first term is $\frac{1}{2}$. Because $\frac{1}{2}$ is a rational number but not a non-negative integer (whole number), the expression violates the strict algebraic criteria required for polynomials.

Final Solution: The expression $3\sqrt{t} + t\sqrt{2}$ is NOT a polynomial in one variable. Reason: The exponent of the variable $t$ in the first term is $\frac{1}{2}$, which is not a whole number.


More Questions from Class 9 Mathematics Polynomials EXERCISE 2.1


CBSE Solutions for Class 9 Mathematics Polynomials


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