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Q2(i):
Write the coefficients of $x^2$ in each of the following:
(i) $2 + x^2 + x$
Solution :
Given Variables & Initial Setup
Let the given algebraic expression be denoted as a polynomial function $P(x)$:
$P(x) = 2 + x^2 + x$
Our objective is to determine the coefficient of the $x^2$ term. [A coefficient is defined mathematically as the numerical or constant multiplier of a specific variable within an algebraic term].
Step 1: Standardizing the Polynomial Form
To systematically analyze the polynomial, we first rearrange its terms in descending order of their degree to express it in standard form.
$P(x) = x^2 + x + 2$
[Per the standard convention of polynomial representation, where terms are ordered from the highest exponent to the lowest exponent].
Step 2: Isolating the Target Term
We inspect the standardized polynomial to locate the specific term containing the variable $x$ raised to the power of $2$. The polynomial consists of three distinct terms separated by addition:
- First term: $x^2$ (Quadratic term, Degree 2)
- Second term: $x$ (Linear term, Degree 1)
- Third term: $2$ (Constant term, Degree 0)
The target term for our analysis is $+x^2$.
Step 3: Extracting the Coefficient
We must identify the constant numerical value that multiplies the variable $x^2$. In algebra, when a variable does not have an explicitly written numerical prefix, it is implicitly multiplied by $1$.
$x^2 = 1 \cdot x^2$
[By the Multiplicative Identity Property of Real Numbers, which states that any real number or variable multiplied by $1$ remains unchanged, i.e., $a \cdot 1 = a$].
Therefore, the numerical multiplier (coefficient) of the $x^2$ term is exactly $1$.
Final Solution: The coefficient of $x^2$ in the polynomial $2 + x^2 + x$ 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(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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