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Q1(iv):
Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(iv) $y + \frac{2}{y}$
Solution :
Step 1: Initial Setup & Algebraic Transformation
The given algebraic expression is:
$y + \frac{2}{y}$
To rigorously analyze whether this expression is a polynomial, we must first express all terms such that the variable appears in the numerator. [Per the Laws of Exponents, specifically the negative exponent rule $\frac{1}{a^n} = a^{-n}$], we rewrite the second term of the expression:
$y^1 + 2y^{-1}$
Step 2: Variable Analysis
The problem requires us to verify two distinct conditions: whether the expression is in one variable, and whether it is a polynomial.
Looking at the transformed expression $y^1 + 2y^{-1}$, the only alphabetical symbol representing an unknown quantity is $y$. There are no other variables (such as $x$, $z$, etc.) present. Therefore, the expression strictly satisfies the condition of being in one variable.
Step 3: Theoretical Definition of a Polynomial
By mathematical definition, an algebraic expression is classified as a polynomial if and only if it can be written in the form:
$P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$
Where:
- $a_n, a_{n-1}, \dots, a_0$ are real number coefficients.
- The exponents $n, n-1, \dots$ must belong strictly to the set of non-negative integers (i.e., the set of whole numbers, $\mathbb{W} = \{0, 1, 2, 3, \dots\}$).
Step 4: Exponent Analysis
We now evaluate the exponents of the variable $y$ in our transformed expression $y^1 + 2y^{-1}$ against the theoretical definition:
- First Term ($y^1$): The exponent of $y$ is $1$. Because $1 \in \mathbb{W}$, this specific term satisfies the polynomial condition.
- Second Term ($2y^{-1}$): The exponent of $y$ is $-1$. Because $-1 \notin \mathbb{W}$ (it is a negative integer, not a whole number), this term violates the fundamental definition of a polynomial.
Step 5: Logical Conclusion
For an entire expression to be classified as a polynomial, every single term must possess a whole number exponent. The presence of even one term with a negative or fractional exponent immediately disqualifies the entire expression.
Final Solution: The expression $y + \frac{2}{y}$ is NOT a polynomial. While it is an expression in exactly one variable ($y$), the exponent of the variable in the second term is $-1$, 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(iii): Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. (iii) $3\sqrt{t} + t\sqrt{2}$
- 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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