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Q5(vii):
Classify the following as linear, quadratic and cubic polynomials:
(vii) $7x^3$
Solution :
Step 1: Initial Setup and Definition of the Polynomial
We are given the algebraic expression:
$P(x) = 7x^3$
To classify this mathematical expression, we must first verify that it satisfies the formal definition of a polynomial in one variable. A polynomial in a single variable $x$ is an expression consisting of variables and coefficients, where the exponents of the variables are non-negative integers.
- Coefficient: $7$ (a real number)
- Variable: $x$
- Exponent: $3$ (a non-negative integer)
Since the exponent is a non-negative integer, $P(x) = 7x^3$ is strictly a polynomial.
Step 2: Determining the Degree of the Polynomial
The classification of a polynomial is fundamentally determined by its degree. [By definition, the degree of a polynomial in one variable is the highest power (exponent) of the variable present in the expression with a non-zero coefficient].
Analyzing our given polynomial:
$P(x) = 7x^3$
This is a monomial (a polynomial with a single term). The only variable present is $x$, and its exponent is $3$. Therefore, the highest power of $x$ in this polynomial is $3$.
$ \text{Degree of } P(x) = 3 $
Step 3: Classifying the Polynomial
Polynomials are classified by their degree according to the following standard algebraic nomenclature:
| Degree | Classification Name | Standard Form |
|---|---|---|
| $1$ | Linear Polynomial | $ax + b \quad (a \neq 0)$ |
| $2$ | Quadratic Polynomial | $ax^2 + bx + c \quad (a \neq 0)$ |
| $3$ | Cubic Polynomial | $ax^3 + bx^2 + cx + d \quad (a \neq 0)$ |
Because the degree of $7x^3$ is exactly $3$, it falls into the category of a cubic polynomial.
Step 4: Geometric Interpretation (Visualizing the Cubic Polynomial)
A cubic polynomial of the form $P(x) = ax^3$ (where $a > 0$) produces a characteristic curve that passes through the origin $(0,0)$ and exhibits point symmetry about the origin. This is known as an inflection point, where the concavity of the graph changes. Below is the precise geometric representation of $y = 7x^3$.
Final Solution: The highest power of the variable $x$ in the expression $7x^3$ is $3$. Therefore, $7x^3$ is classified as a cubic 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(vi): Classify the following as linear, quadratic and cubic polynomials: (vi) $r^2$
CBSE Solutions for Class 9 Mathematics Polynomials
Chapters in CBSE - Class 9 Mathematics
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