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Q2(iv):
Write the coefficients of $x^2$ in each of the following:
(iv) $\sqrt{2}x - 1$
Solution :
Initial Setup & Theoretical Foundation
We are given the algebraic expression:
$P(x) = \sqrt{2}x - 1$
Our objective is to determine the coefficient of the $x^2$ term within this polynomial. [By definition, a coefficient is the numerical multiplier of a variable in a specific term of an algebraic expression].
Step 1: Analyzing the Degree and Terms of the Polynomial
The given polynomial, $P(x) = \sqrt{2}x - 1$, consists of two explicitly written terms:
- The linear term: $\sqrt{2}x$, where the variable $x$ is raised to the power of $1$. The coefficient here is $\sqrt{2}$.
- The constant term: $-1$, which can be mathematically expressed as $-1x^0$.
Because the highest power of the variable $x$ present in the expression is $1$, this is classified as a linear polynomial (degree $1$).
Step 2: Expressing the Polynomial in Standard Form
To rigorously identify the coefficient of a term that does not explicitly appear in the expression (in this case, the $x^2$ term), we must rewrite the polynomial in its complete standard descending form. [Per the axioms of polynomial representation, any missing term of degree $k$ can be introduced into the expression by multiplying it by zero ($0 \cdot x^k$), because adding the additive identity ($0$) does not alter the intrinsic value of the polynomial].
We expand $P(x)$ to explicitly include the quadratic ($x^2$) term:
$P(x) = 0 \cdot x^2 + \sqrt{2}x - 1$
Step 3: Visualizing the Polynomial Structure
The diagram below illustrates the structural breakdown of the polynomial when expanded to include the quadratic term, isolating the coefficient for each degree of $x$.
Step 4: Extracting the Target Coefficient
By comparing our expanded polynomial $P(x) = 0x^2 + \sqrt{2}x - 1$ with the general quadratic form $ax^2 + bx + c$, we can map the coefficients directly:
- $a$ (coefficient of $x^2$) = $0$
- $b$ (coefficient of $x$) = $\sqrt{2}$
- $c$ (constant term) = $-1$
Since the $x^2$ term is entirely absent from the original expression, its numerical multiplier must be zero.
Final Solution: The coefficient of $x^2$ in the polynomial $\sqrt{2}x - 1$ is $0$.
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(iii): Write the coefficients of $x^2$ in each of the following: (iii) $\frac{\pi}{2}x^2 + x$
- 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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