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Q5(iv):
Classify the following as linear, quadratic and cubic polynomials:
(iv) $1 + x$
Solution :
Initial Setup & Given Expression
We are given the algebraic expression:
$P(x) = 1 + x$
Our objective is to classify this polynomial into one of three categories: linear, quadratic, or cubic. This classification is strictly determined by the degree of the polynomial.
Step 1: Standardizing the Polynomial
First, we rewrite the polynomial in its standard form, where the terms are ordered in descending powers of the variable $x$.
$P(x) = x + 1$
We can explicitly write the exponents for every term to ensure absolute clarity:
- The term $x$ is equivalent to $x^1$.
- The constant term $1$ is equivalent to $1 \cdot x^0$ [Per the Zero Exponent Rule, where $x^0 = 1$ for $x \neq 0$].
Thus, the fully expanded standard form is:
$P(x) = 1 \cdot x^1 + 1 \cdot x^0$
Step 2: Determining the Degree of the Polynomial
The degree of a polynomial in one variable is defined as the highest exponent (power) of the variable present in the expression with a non-zero coefficient.
Analyzing the exponents in $P(x) = x^1 + 1 \cdot x^0$:
- The exponent of the first term is $1$.
- The exponent of the second term is $0$.
Comparing these values, the maximum exponent is $1$. Therefore, the degree of the polynomial $P(x) = 1 + x$ is exactly $1$.
Step 3: Applying the Classification Theorem
Polynomials are classified by their degree according to the following universally accepted algebraic definitions:
| Degree | Classification Name | Standard Form |
|---|---|---|
| $1$ | Linear | $ax + b$ (where $a \neq 0$) |
| $2$ | Quadratic | $ax^2 + bx + c$ (where $a \neq 0$) |
| $3$ | Cubic | $ax^3 + bx^2 + cx + d$ (where $a \neq 0$) |
Since we have rigorously determined that the degree of $1 + x$ is $1$, it maps directly to the definition of a linear polynomial.
Visual Verification: Geometric Interpretation
A linear polynomial, when graphed on a Cartesian coordinate system as a function $y = P(x)$, will always produce a perfectly straight line [Per the geometric definition of first-degree equations]. Below is the precise graphical representation of $y = x + 1$, demonstrating its constant slope ($m = 1$) and y-intercept ($c = 1$).
Final Solution: The polynomial $1 + x$ has a highest degree of $1$. Therefore, it is classified as a linear 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(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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