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Q2(ii):
Write the coefficients of $x^2$ in each of the following:
(ii) $2 – x^2 + x^3$
Solution :
Given Polynomial & Initial Setup
We are given the algebraic expression:
$P(x) = 2 - x^2 + x^3$
Our objective is to determine the numerical coefficient of the $x^2$ term within this polynomial.
Step 1: Understanding Polynomial Terms and Standard Form
[Per the algebraic definition of a polynomial], an expression is composed of distinct terms separated by addition ($+$) or subtraction ($-$) operators. To analyze the polynomial systematically, it is best practice to rewrite it in standard form, where the terms are ordered by descending powers of the variable $x$.
Rearranging $P(x)$ yields:
$P(x) = x^3 - x^2 + 2$
Step 2: Isolating the Target Term
We must identify the specific term that contains the variable $x^2$. By dissecting the polynomial into its constituent terms, we get:
- Cubic term: $+x^3$
- Quadratic term: $-x^2$
- Constant term: $+2$
The target term containing $x^2$ is explicitly $-x^2$.
Step 3: Extracting the Numerical Coefficient
[By the fundamental axioms of algebra], the coefficient is the constant multiplicative factor attached to a variable in a specific term. When a variable appears without an explicit numerical multiplier but carries a negative sign, the multiplicative identity property dictates that the hidden multiplier is $-1$.
We can expand the target term algebraically to reveal its coefficient:
$-x^2 = (-1) \cdot x^2$
Thus, the numerical factor multiplying $x^2$ is $-1$.
Visual Breakdown of the Polynomial Terms
Final Solution: The coefficient of $x^2$ in the polynomial $2 - x^2 + x^3$ is $-1$.
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(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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