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Q1(ii):
Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(ii) $y^2 + \sqrt{2}$
Solution :
Step 1: Theoretical Foundation of Polynomials
To determine whether a given algebraic expression is a polynomial in one variable, we must evaluate it against two strict algebraic criteria:
- Single Variable Condition: The expression must contain exactly one distinct variable (e.g., only $x$, only $y$, or only $z$).
- Non-Negative Integer Exponent Condition: [Per the fundamental definition of polynomials], the exponent of the variable in every term must be a whole number (i.e., $0, 1, 2, 3, \dots$). Fractional, negative, or irrational exponents on the variable disqualify the expression from being a polynomial.
Step 2: Structural Analysis of the Given Expression
The given mathematical expression is:
$y^2 + \sqrt{2}$
We can rewrite this expression in its standard canonical form by explicitly showing the variable in the constant term. Any non-zero constant $c$ can be written as $c \cdot y^0$ because $y^0 = 1$ (for $y \neq 0$).
$y^2 + \sqrt{2} \cdot y^0$
Visual representation of the polynomial function showing a continuous, smooth parabolic curve characteristic of quadratic polynomials.
Step 3: Evaluating the Variable Condition
Observing the expression $y^2 + \sqrt{2}$, the only alphabetical symbol representing an unknown quantity is $y$. There are no other variables (such as $x$ or $z$) present in the expression. Therefore, the expression satisfies the condition of being in one variable.
Step 4: Evaluating the Exponent Condition
We must now inspect the exponent of the variable $y$ in each term:
| Term | Variable Part | Exponent | Is it a Whole Number? |
|---|---|---|---|
| First Term: $y^2$ | $y^2$ | $2$ | Yes |
| Second Term: $\sqrt{2}$ | $y^0$ | $0$ | Yes |
Note: The presence of $\sqrt{2}$ does not violate the polynomial definition. The restriction of having non-negative integer powers applies only to the variables, not to the numerical coefficients or constants. $\sqrt{2}$ is simply a real number acting as the constant term.
Step 5: Logical Conclusion
Since the expression contains exactly one variable ($y$) and all exponents of this variable ($2$ and $0$) are non-negative integers (whole numbers), it perfectly satisfies all algebraic axioms defining a polynomial in one variable.
Final Solution: Yes, the expression $y^2 + \sqrt{2}$ is a polynomial in one variable. The reason is that it contains only one variable ($y$), and the exponent of the variable in every term is 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(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(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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