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Q1(v):
Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(v) $x^{10} + y^3 + t^{50}$
Solution :
Given Expression & Initial Setup
We are tasked with analyzing the following algebraic expression to determine if it is a polynomial in one variable:
$P(x, y, t) = x^{10} + y^3 + t^{50}$
To rigorously evaluate this, we must test the expression against two distinct mathematical criteria:
- Criterion 1 (Polynomial Status): An algebraic expression is classified as a polynomial if and only if the exponents of all its variables are non-negative integers (whole numbers).
- Criterion 2 (Single Variable Status): A polynomial is "in one variable" if it is constructed using exactly one distinct variable symbol throughout the entire expression.
Step 1: Analyzing the Exponents (Checking for Polynomial Status)
We decompose the expression into its individual terms and inspect the exponent of each variable:
| Term | Variable | Exponent | Is the Exponent a Whole Number? |
|---|---|---|---|
| $x^{10}$ | $x$ | $10$ | Yes ($10 \in \mathbb{W}$) |
| $y^3$ | $y$ | $3$ | Yes ($3 \in \mathbb{W}$) |
| $t^{50}$ | $t$ | $50$ | Yes ($50 \in \mathbb{W}$) |
[Per the fundamental definition of polynomials, since all exponents ($10$, $3$, and $50$) belong to the set of whole numbers $\mathbb{W} = \{0, 1, 2, 3, \dots\}$, the given expression is mathematically verified to be a valid polynomial.]
Step 2: Analyzing the Variables (Checking for "One Variable" Status)
Next, we must determine the dimensionality of the polynomial by counting the number of unique variables present.
- The first term contains the variable $x$.
- The second term contains the variable $y$.
- The third term contains the variable $t$.
[By the axioms of algebraic notation, $x$, $y$, and $t$ represent three distinct, independent variables. Therefore, the expression is a multivariate polynomial, specifically a polynomial in three variables.]
Step 3: Synthesizing the Final Conclusion
While the expression successfully passes the test for being a polynomial (all exponents are non-negative integers), it fails the condition of being in "one variable." The presence of $x$, $y$, and $t$ makes it a polynomial in three variables.
Final Solution: The expression $x^{10} + y^3 + t^{50}$ is NOT a polynomial in one variable. While it is a valid polynomial (because all exponents are whole numbers), it is a polynomial in THREE variables ($x$, $y$, and $t$).
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(iv): Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. (iv) $y + \frac{2}{y}$
- 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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