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Q5(i):
Classify the following as linear, quadratic and cubic polynomials:
(i) $x^2 + x$
Solution :
Given Polynomial & Initial Setup
We are tasked with classifying the following algebraic expression based on its degree:
$P(x) = x^2 + x$
Step 1: Theoretical Foundation of Polynomial Classification
In algebra, polynomials in a single variable are classified according to their degree. [Per the Fundamental Theorem of Algebra and polynomial definitions], the degree of a polynomial is defined as the highest exponent (power) of the variable present in the expression with a non-zero coefficient.
- Linear Polynomial: A polynomial of degree $1$. Standard form: $ax + b$ (where $a \neq 0$).
- Quadratic Polynomial: A polynomial of degree $2$. Standard form: $ax^2 + bx + c$ (where $a \neq 0$).
- Cubic Polynomial: A polynomial of degree $3$. Standard form: $ax^3 + bx^2 + cx + d$ (where $a \neq 0$).
Step 2: Term-by-Term Analysis of $P(x)$
We decompose the given polynomial $P(x) = x^2 + x$ into its constituent terms to identify the exponents of the variable $x$:
- First Term ($x^2$): The variable $x$ is raised to the power of $2$.
- Second Term ($x$): The variable $x$ is implicitly raised to the power of $1$ (since $x = x^1$).
Comparing the exponents $\{2, 1\}$, the maximum value is $2$. Therefore, the highest power of the variable $x$ in the polynomial is $2$.
Step 3: Classification and Geometric Justification
Since the highest exponent is $2$, the degree of the polynomial $P(x) = x^2 + x$ is exactly $2$. By definition, a polynomial of degree $2$ is classified as a quadratic polynomial.
[Geometrically, a quadratic polynomial represents a parabola when graphed on a Cartesian plane. The presence of the $x^2$ term dictates this parabolic curvature, distinguishing it from the straight line of a linear polynomial or the inflection curve of a cubic polynomial.]
Figure 1: The parabolic graph of $y = x^2 + x$ confirms its quadratic nature.
Final Solution: The highest power of the variable $x$ in the expression $x^2 + x$ is $2$. Therefore, it is classified as a quadratic polynomial.
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) $\frac{\pi}{2}x^2 + x$
- 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(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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