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Q1(i):
Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(i) $4x^2 – 3x + 7$
Solution :
Initial Setup & Algebraic Definition
Given the algebraic expression: $P(x) = 4x^2 - 3x + 7$
[Per the Fundamental Theorem of Algebra and Polynomial Definitions], an algebraic expression is classified as a polynomial in one variable if and only if it satisfies two strict mathematical conditions:
- Condition 1 (Single Variable): The expression must contain only one distinct variable symbol (e.g., only $x$, only $y$, or only $z$).
- Condition 2 (Non-Negative Integer Exponents): The exponent of the variable in every single term must be a whole number ($\mathbb{W} = \{0, 1, 2, 3, \dots\}$). Fractional, negative, or irrational exponents disqualify the expression from being a polynomial.
Step 1: Evaluating the Variable Count
By inspecting the expression $4x^2 - 3x + 7$, we observe that the only alphabetical symbol representing an unknown quantity is $x$. There are no secondary variables (such as $y$ or $t$) present in any of the terms.
Conclusion for Condition 1: The expression is strictly in one variable.
Step 2: Analyzing the Exponents of Each Term
To rigorously verify Condition 2, we must decompose the expression into its constituent terms and isolate the exponent of $x$ for each. Note that constants can be expressed as coefficients of $x^0$ [By the Zero Exponent Rule: $x^0 = 1$ for $x \neq 0$].
| Term | Algebraic Expansion | Variable | Exponent | Is Exponent a Whole Number ($\in \mathbb{W}$)? |
|---|---|---|---|---|
| $4x^2$ | $4 \cdot x^2$ | $x$ | $2$ | Yes ($2 \in \mathbb{W}$) |
| $-3x$ | $-3 \cdot x^1$ | $x$ | $1$ | Yes ($1 \in \mathbb{W}$) |
| $7$ | $7 \cdot x^0$ | $x$ | $0$ | Yes ($0 \in \mathbb{W}$) |
Step 3: Visualizing the Polynomial Structure
The following diagram maps the anatomical structure of the given expression, proving that all exponents belong to the set of whole numbers and only a single variable is utilized.
Step 4: Synthesizing the Findings
Because the expression contains exactly one variable ($x$) and every exponent of $x$ across all terms is a non-negative integer ($2, 1, \text{ and } 0$), the expression perfectly satisfies all algebraic axioms required to be classified as a polynomial in one variable.
Final Solution: The expression $4x^2 - 3x + 7$ is a polynomial in one variable. The reason is that it contains only a single variable ($x$), and the exponents of the variable in all terms ($2, 1, \text{ and } 0$) are whole numbers.
More Questions from Class 9 Mathematics Polynomials EXERCISE 2.1
- 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(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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