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Q5(vi):
Classify the following as linear, quadratic and cubic polynomials:
(vi) $r^2$
Solution :
Initial Setup & Given Expression
We are given the following algebraic expression:
$P(r) = r^2$
Our objective is to classify this polynomial as linear, quadratic, or cubic based on its mathematical properties.
Step 1: Theoretical Foundation of Polynomial Classification
In algebra, polynomials are classified according to their degree. The degree of a polynomial in one variable is defined as the highest power (exponent) of the variable present in the expression with a non-zero coefficient. [Per the Fundamental Theorem of Algebra and standard polynomial definitions].
The standard classifications based on degree are as follows:
- Linear Polynomial: A polynomial of degree $1$. General form: $ax + b$ (where $a \neq 0$).
- Quadratic Polynomial: A polynomial of degree $2$. General form: $ax^2 + bx + c$ (where $a \neq 0$).
- Cubic Polynomial: A polynomial of degree $3$. General form: $ax^3 + bx^2 + cx + d$ (where $a \neq 0$).
Step 2: Analyzing the Degree of the Given Polynomial
Let us examine the given polynomial:
$P(r) = r^2$
This expression is a monomial (a polynomial consisting of a single term). We analyze the components of this term:
- The variable is $r$.
- The coefficient is $1$ (since $r^2$ is equivalent to $1 \cdot r^2$).
- The exponent of the variable $r$ is $2$.
Because there are no other terms in the polynomial, the highest power of the variable $r$ is exactly $2$. Therefore, the degree of the polynomial $P(r) = r^2$ is $2$.
Step 3: Visualizing the Quadratic Nature
To rigorously confirm the nature of this polynomial, we can observe its graphical representation. A polynomial of degree $2$ forms a parabola when plotted on a Cartesian coordinate system. The precise geometric mapping of $P(r) = r^2$ demonstrates a non-linear, symmetric curve with a single vertex at the origin $(0,0)$.
Step 4: Final Classification
Because the highest exponent of the variable $r$ is $2$, the polynomial satisfies the strict definition of a quadratic polynomial. It does not possess a degree of $1$ (which would make it linear) nor a degree of $3$ (which would make it cubic).
Final Solution: The polynomial $r^2$ is 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(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(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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