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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 ($1 \in \mathbb{W}$) |
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
- Q1(i): Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. (i) $4x^2 – 3x + 7$
- Q1(ii): Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. (ii) $y^2 + \sqrt{2}$
- Q1(iv): Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. (iv) $y + \frac{2}{y}$
- Q1(v): Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. (v) $x^{10} + y^3 + t^{50}$
- Q2(i): Write the coefficients of $x^2$ in each of the following: (i) $2 + x^2 + x$
- Q2(ii): Write the coefficients of $x^2$ in each of the following: (ii) $2 – x^2 + x^3$
- Q2(iii): Write the coefficients of $x^2$ in each of the following: (iii) $\frac{\pi}{2}x^2 + x$
- Q2(iv): Write the coefficients of $x^2$ in each of the following: (iv) $\sqrt{2}x - 1$
- Q3: Give one example each of a binomial of degree 35, and of a monomial of degree 100.
- Q4(i): Write the degree of each of the following polynomials: (i) $5x^3 + 4x^2 + 7x$
- Q4(ii): Write the degree of each of the following polynomials: (ii) $4 – y^2$
- Q4(iii): Write the degree of each of the following polynomials: (iii) $5t – \sqrt{7}$
- Q4(iv): Write the degree of each of the following polynomials: (iv) $3$
- Q5(i): Classify the following as linear, quadratic and cubic polynomials: (i) $x^2 + x$
- Q5(ii): Classify the following as linear, quadratic and cubic polynomials: (ii) $x – x^3$
- Q5(iii): Classify the following as linear, quadratic and cubic polynomials: (iii) $y + y^2 + 4$
- Q5(iv): Classify the following as linear, quadratic and cubic polynomials: (iv) $1 + x$
- Q5(v): Classify the following as linear, quadratic and cubic polynomials: (v) $3t$
- Q5(vi): Classify the following as linear, quadratic and cubic polynomials: (vi) $r^2$
- Q5(vii): Classify the following as linear, quadratic and cubic polynomials: (vii) $7x^3$
CBSE Solutions for Class 9 Mathematics Polynomials
Chapters in CBSE - Class 9 Mathematics
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