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Q2(iii):
Write the coefficients of $x^2$ in each of the following:
(iii) $\frac{\pi}{2}x^2 + x$
Solution :
Initial Setup & Theoretical Foundation
We are given the algebraic expression:
$P(x) = \frac{\pi}{2}x^2 + x$
In algebra, a polynomial is defined as an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. [Per the Fundamental Theorem of Algebra and standard polynomial definitions], a coefficient is the numerical or constant multiplier attached to a specific power of a variable within a term.
Step 1: Term-by-Term Analysis of the Polynomial
To rigorously identify the coefficient, we must first decompose the polynomial $P(x)$ into its constituent terms. The terms of a polynomial are separated by addition ($+$) or subtraction ($-$) operators.
- Term 1: $\frac{\pi}{2}x^2$ (This is the quadratic term, possessing a degree of 2).
- Term 2: $x$ (This is the linear term, possessing a degree of 1, which can be written as $1 \cdot x^1$).
Step 2: Isolating the Target Variable
The objective is to find the coefficient specifically for the $x^2$ variable. We isolate the term containing $x^2$, which is:
$\frac{\pi}{2}x^2$
Step 3: Extraction and Classification of the Coefficient
By observing the isolated term $\frac{\pi}{2}x^2$, we separate the variable component ($x^2$) from its constant multiplier.
The constant multiplier is $\frac{\pi}{2}$.
Analytical Note: It is crucial to recognize that polynomials over the real numbers ($\mathbb{R}$) can have irrational coefficients. The number $\pi$ is an irrational mathematical constant (approximately $3.14159...$). Therefore, the fraction $\frac{\pi}{2}$ is a perfectly valid real number and serves as the exact coefficient of the $x^2$ term.
Final Solution: The coefficient of $x^2$ in the polynomial $\frac{\pi}{2}x^2 + x$ is $\frac{\pi}{2}$.
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) $y + \frac{2}{y}$
- 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(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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