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Q3:
Give one example each of a binomial of degree 35, and of a monomial of degree 100.
Solution :
Theoretical Foundation & Definitions
To construct the required algebraic expressions, we must first establish the rigorous definitions of the polynomial classifications based on their terms and degrees [Per the Fundamental Definitions of Polynomial Algebra]:
- Polynomial: An algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
- Term: A single mathematical expression separated by addition ($+$) or subtraction ($-$) signs.
- Monomial: A polynomial consisting of exactly one non-zero term.
- Binomial: A polynomial consisting of exactly two non-zero terms.
- Degree of a Polynomial in One Variable: The highest exponent (power) of the variable present in the polynomial with a non-zero coefficient.
Step 1: Constructing a Binomial of Degree 35
Based on the definitions, a binomial of degree $35$ must satisfy two independent conditions:
- It must contain exactly two terms.
- The highest power of the variable must be exactly $35$.
The general form of such a binomial in variable $x$ can be written as:
$P(x) = ax^{35} + bx^k$
Where:
- $a$ and $b$ are non-zero real numbers ($a \neq 0, b \neq 0$).
- $k$ is a non-negative integer strictly less than $35$ ($0 \le k < 35$).
By selecting $a = 3$, $b = -4$, and $k = 2$, we generate a valid example:
$3x^{35} - 4x^2$
[Justification: The expression has two terms ($3x^{35}$ and $-4x^2$), making it a binomial. The highest exponent on the variable $x$ is $35$, making its degree $35$.]
Step 2: Constructing a Monomial of Degree 100
A monomial of degree $100$ must satisfy the following conditions:
- It must contain exactly one term.
- The power of the variable in that single term must be exactly $100$.
The general form of such a monomial in variable $y$ can be written as:
$Q(y) = cy^{100}$
Where:
- $c$ is any non-zero real number ($c \neq 0$).
By selecting $c = \sqrt{5}$, we generate a valid example:
$\sqrt{5}y^{100}$
[Justification: The expression consists of a single term ($\sqrt{5}y^{100}$), making it a monomial. The exponent on the variable $y$ is $100$, making its degree $100$.]
Visual Analysis of Polynomial Anatomy
The following diagram illustrates the structural components of the constructed polynomials, verifying their terms and degrees.
Final Solution:
An example of a binomial of degree 35 is $3x^{35} - 4x^2$.
An example of a monomial of degree 100 is $\sqrt{5}y^{100}$.
(Note: Infinite valid examples exist by altering the non-zero coefficients or the lower-degree terms in the binomial).
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) $\sqrt{2}x - 1$
- 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) $r^2$
- 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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