Class 9 Mathematics
Chapter 2: Polynomials
Exercise 2.1 – Complete Stepwise Solutions
Important Formula / Rules Used
1. Polynomial in one variable:
An expression is called a polynomial in one variable if:
- it has only one variable,
- the powers of the variable are non-negative whole numbers (0, 1, 2, 3, …),
- the variable does not appear in the denominator,
- coefficients may be any real numbers.
2. Degree of a polynomial:
The highest power of the variable in the polynomial is called its degree.
3. Types of polynomials based on number of terms:
- Monomial: 1 term
- Binomial: 2 terms
- Trinomial: 3 terms
4. Types of polynomials based on degree:
- Linear polynomial → degree 1
- Quadratic polynomial → degree 2
- Cubic polynomial → degree 3
Q1. Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
Given expressions:
(i) 4x2 − 3x + 7
(ii) y2 + √2
(iii) 3√t + t√2
(iv) y + 2/y
(v) x10 + y3 + t50
(i) 4x2 − 3x + 7
Check:
- Only one variable: x
- Powers of x are 2 and 1, both non-negative whole numbers
Conclusion: It is a polynomial in one variable.
(ii) y2 + √2
Check:
- Only one variable: y
- Power of y is 2, a non-negative whole number
- √2 is a real number, so it can be a constant coefficient
Conclusion: It is a polynomial in one variable.
(iii) 3√t + t√2
Rewrite:
3√t = 3t1/2
Check:
- Variable t has power 1/2 in the first term
- 1/2 is not a non-negative whole number
Conclusion: It is not a polynomial.
(iv) y + 2/y
Rewrite:
2/y = 2y−1
Check:
- The variable y has power −1
- Negative powers are not allowed in a polynomial
Conclusion: It is not a polynomial.
(v) x10 + y3 + t50
Check:
- It has three variables: x, y and t
- A polynomial in one variable must contain only one variable
Conclusion: It is not a polynomial in one variable.
Q2. Write the coefficients of x2 in each of the following:
Given:
(i) 2 + x2 + x
(ii) 2 − x2 + x3
(iii) (π/2)x2 + x
(iv) √2x − 1
Rule used:
The coefficient of x2 is the number multiplied by x2.
(i) 2 + x2 + x
The coefficient of x2 is 1.
(ii) 2 − x2 + x3
The coefficient of x2 is −1.
(iii) (π/2)x2 + x
The coefficient of x2 is π/2.
(iv) √2x − 1
There is no x2 term.
So, the coefficient of x2 is 0.
Q3. Give one example each of a binomial of degree 35, and of a monomial of degree 100.
Rule used:
- Binomial has 2 terms
- Monomial has 1 term
- Degree is the highest power of the variable
Example of a binomial of degree 35:
x35 + 2
This has two terms and highest power is 35.
Example of a monomial of degree 100:
5x100
This has one term and degree 100.
Q4. Write the degree of each of the following polynomials:
Given:
(i) 5x3 + 4x2 + 7x
(ii) 4 − y2
(iii) 5t − √7
(iv) 3
Rule used:
The degree of a polynomial is the highest power of the variable.
(i) 5x3 + 4x2 + 7x
Highest power of x = 3
Degree = 3
(ii) 4 − y2
Highest power of y = 2
Degree = 2
(iii) 5t − √7
Highest power of t = 1
Degree = 1
(iv) 3
3 is a non-zero constant polynomial.
The degree of a non-zero constant polynomial is 0.
Q5. Classify the following as linear, quadratic and cubic polynomials:
Given:
(i) x2 + x
(ii) x − x3
(iii) y + y2 + 4
(iv) 1 + x
(v) 3t
(vi) r2
(vii) 7x3
Rule used:
- Degree 1 → Linear
- Degree 2 → Quadratic
- Degree 3 → Cubic
(i) x2 + x
Highest power = 2
Quadratic polynomial
(ii) x − x3
Highest power = 3
Cubic polynomial
(iii) y + y2 + 4
Highest power = 2
Quadratic polynomial
(iv) 1 + x
Highest power = 1
Linear polynomial
(v) 3t
Highest power = 1
Linear polynomial
(vi) r2
Highest power = 2
Quadratic polynomial
(vii) 7x3
Highest power = 3
Cubic polynomial
Final Answers Summary
Q1.
- (i) Polynomial in one variable
- (ii) Polynomial in one variable
- (iii) Not a polynomial
- (iv) Not a polynomial
- (v) Not a polynomial in one variable
Q2. Coefficients of x2:
- (i) 1
- (ii) −1
- (iii) π/2
- (iv) 0
Q3.
- Binomial of degree 35: x35 + 2
- Monomial of degree 100: 5x100
Q4. Degrees:
- (i) 3
- (ii) 2
- (iii) 1
- (iv) 0
Q5.
- Linear: (iv), (v)
- Quadratic: (i), (iii), (vi)
- Cubic: (ii), (vii)