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Q5(ii):
Classify the following as linear, quadratic and cubic polynomials:
(ii) $x – x^3$
Solution :
Given Variables & Initial Setup
We are given the algebraic expression:
$P(x) = x - x^3$
Our objective is to classify this polynomial into one of three categories: linear, quadratic, or cubic. To do this, we must determine the degree of the polynomial.
Step 1: Expressing the Polynomial in Standard Form
The standard form of a polynomial in one variable requires arranging the terms in descending order of their exponents [Per the conventions of algebraic notation].
The given polynomial has two terms:
- The term $x$, which can be written as $x^1$.
- The term $-x^3$.
Rearranging these terms in descending order of their powers, we get:
$P(x) = -x^3 + x^1$
Step 2: Determining the Degree of the Polynomial
The degree of a polynomial in one variable is defined as the highest power (exponent) of the variable that has a non-zero coefficient.
Analyzing the exponents in our standard form polynomial $P(x) = -x^3 + x^1$:
- The exponent of the first term ($-x^3$) is $3$.
- The exponent of the second term ($x^1$) is $1$.
Comparing the exponents, $3 > 1$. Therefore, the highest power of the variable $x$ is $3$.
$ \text{Degree of } P(x) = 3 $
Step 3: Classifying the Polynomial
Polynomials are classified based on their degree according to the following definitions:
| Degree | Classification | 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 | $ax^3 + bx^2 + cx + d \quad (a \neq 0)$ |
Since the degree of $P(x) = -x^3 + x$ is exactly $3$, it satisfies the definition of a cubic polynomial [Where $a = -1, b = 0, c = 1, d = 0$].
Visual Verification: Graphical Representation of a Cubic Polynomial
A cubic polynomial typically exhibits a characteristic "S-shaped" curve and can intersect the x-axis up to three times [Per the Fundamental Theorem of Algebra]. Below is the precise geometric plot of $y = x - x^3$, clearly showing its three real roots at $x = -1$, $x = 0$, and $x = 1$.
Final Solution: The highest power of the variable $x$ in the polynomial $x - x^3$ is $3$. Therefore, it 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(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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