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Q5(iii):
Classify the following as linear, quadratic and cubic polynomials:
(iii) $y + y^2 + 4$
Solution :
Initial Setup & Given Expression
We are tasked with classifying the following algebraic expression based on its degree:
$P(y) = y + y^2 + 4$
Step 1: Rearranging into Standard Form
By mathematical convention, polynomials are typically written in standard form, where the terms are ordered from the highest power of the variable to the lowest power [Descending Order of Exponents]. This makes it easier to identify the leading term and the degree.
Rearranging the terms of $P(y)$ according to the powers of $y$:
$P(y) = y^2 + y + 4$
Step 2: Analyzing the Exponents of Each Term
To classify the polynomial, we must determine the exponent of the variable $y$ in each distinct term. We apply the fundamental laws of exponents to reveal hidden powers:
- First term ($y^2$): The exponent of the variable $y$ is explicitly $2$.
- Second term ($y$): Any variable written without an explicit exponent has an implied exponent of $1$. Thus, $y = y^1$. The exponent is $1$.
- Third term ($4$): This is a constant term. By the zero exponent rule [where $y^0 = 1$ for $y \neq 0$], a constant $c$ can be written as $c \cdot y^0$. Therefore, $4 = 4y^0$. The exponent of $y$ here is $0$.
Step 3: Determining the Degree of the Polynomial
The degree of a polynomial in one variable is defined as the highest power (maximum exponent) of the variable present in the expression, provided its coefficient is non-zero.
Comparing the set of exponents we identified $\{2, 1, 0\}$, the maximum value is $2$.
Therefore, the degree of the polynomial $P(y)$ is $2$.
Step 4: Classification Based on Degree
In algebraic nomenclature, polynomials are classified strictly according to their degree. The standard classifications are outlined in the table below:
| Degree | Polynomial Classification | General Form (in variable $x$) |
|---|---|---|
| 1 | Linear | $ax + b \quad (a \neq 0)$ |
| 2 | Quadratic | $ax^2 + bx + c \quad (a \neq 0)$ |
| 3 | Cubic | $ax^3 + bx^2 + cx + d \quad (a \neq 0)$ |
Because the highest exponent of the variable $y$ in the expression $y^2 + y + 4$ is exactly $2$, it perfectly matches the definition of a quadratic polynomial.
Final Solution
Final Solution: The given polynomial $y + y^2 + 4$ has a degree of 2, and is therefore 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(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(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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