Polynomials in One Variable Questions
Chapter – Polynomials
Topic – Polynomials in One Variable
Question 1. What is a polynomial in one variable? Give three examples.
Solution:
A polynomial in one variable x is an algebraic expression of the form:
p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
Here, the exponents of x are non-negative integers and the coefficients are real numbers.
Examples:
(i) 3x² + 5x − 7
(ii) 2x³ − 4x + 1
(iii) x⁴ + 6
All three expressions are polynomials in one variable because they contain only the variable x with non-negative integer exponents.
Question 2. Determine which of the following expressions are polynomials in one variable.
(a) 4x³ − 2x + 5
(b) 3/x + 2
(c) √5x² + 7
(d) x¹ᐟ² + 4
Solution:
(a) 4x³ − 2x + 5 is a polynomial because the powers of x are non-negative integers.
(b) 3/x + 2 is not a polynomial because 3/x = 3x⁻¹ contains a negative exponent.
(c) √5x² + 7 is a polynomial because irrational coefficients are allowed.
(d) x¹ᐟ² + 4 is not a polynomial because the exponent 1/2 is not a non-negative integer.
Answer: (a) and (c) are polynomials in one variable.
Question 3. Find the degree of each polynomial.
(a) 7x⁴ − 3x² + 5
(b) 9x − 2
(c) 6x³ + 4x⁵ − 1
(d) 8
Solution:
The degree of a non-zero polynomial is the highest power of the variable with a non-zero coefficient.
(a) 7x⁴ − 3x² + 5
Highest power = 4
Degree = 4
(b) 9x − 2
Highest power = 1
Degree = 1
(c) 6x³ + 4x⁵ − 1
Highest power = 5
Degree = 5
(d) 8
A non-zero constant polynomial has degree 0.
Answer: (a) 4, (b) 1, (c) 5, (d) 0.
Question 4. Identify the coefficients and constant term of the polynomial p(x) = 6x³ − 5x² + 4x − 9.
Solution:
Given:
p(x) = 6x³ − 5x² + 4x − 9
Coefficient of x³ = 6
Coefficient of x² = −5
Coefficient of x = 4
Constant term = −9
The degree of the polynomial is 3.
Answer: Coefficients are 6, −5 and 4; constant term is −9.
Question 5. Classify the following polynomials as linear, quadratic or cubic.
(a) 5x + 3
(b) x² − 7x + 10
(c) 2x³ + 4x − 1
(d) 9 − x²
Solution:
A linear polynomial has degree 1.
A quadratic polynomial has degree 2.
A cubic polynomial has degree 3.
(a) 5x + 3 has degree 1, so it is linear.
(b) x² − 7x + 10 has degree 2, so it is quadratic.
(c) 2x³ + 4x − 1 has degree 3, so it is cubic.
(d) 9 − x² has degree 2, so it is quadratic.
Answer:
(a) Linear
(b) Quadratic
(c) Cubic
(d) Quadratic
Question 6. Write a quadratic polynomial whose coefficients of x² and x are 3 and −5, respectively, and whose constant term is 7.
Solution:
The general form of a quadratic polynomial is:
p(x) = ax² + bx + c
Given:
a = 3
b = −5
c = 7
Substituting:
p(x) = 3x² − 5x + 7
Answer: 3x² − 5x + 7.
Question 7. Find the value of p(x) = 2x³ − 3x² + 4x − 5 at x = −1.
Solution:
Given:
p(x) = 2x³ − 3x² + 4x − 5
Substitute x = −1:
p(−1) = 2(−1)³ − 3(−1)² + 4(−1) − 5
= 2(−1) − 3(1) − 4 − 5
= −2 − 3 − 4 − 5
= −14
Answer: p(−1) = −14.
Question 8. Find the number of terms and the degree of the polynomial p(x) = 4x⁵ − 3x³ + 2x − 8.
Solution:
Given:
p(x) = 4x⁵ − 3x³ + 2x − 8
The terms are:
4x⁵
−3x³
2x
−8
Therefore, the polynomial has four terms.
The highest power of x is 5.
Answer: Number of terms = 4; degree = 5.
Question 9. Find the value of k if the degree of p(x) = (k − 3)x³ + 5x² − 2x + 1 is 2.
Solution:
Given:
p(x) = (k − 3)x³ + 5x² − 2x + 1
For the polynomial to have degree 2, the coefficient of x³ must be zero.
Therefore:
k − 3 = 0
k = 3
Substituting k = 3:
p(x) = 5x² − 2x + 1
Its highest power is 2.
Answer: k = 3.
Question 10. Explain the difference between a monomial, binomial and trinomial with examples.
Solution:
A monomial is a polynomial containing exactly one non-zero term.
Example: 7x³
A binomial is a polynomial containing exactly two non-zero terms.
Example: 3x² + 5
A trinomial is a polynomial containing exactly three non-zero terms.
Example: 2x² − 4x + 7
Therefore:
Monomial – One term
Binomial – Two terms
Trinomial – Three terms
Answer: Polynomials are classified as monomials, binomials or trinomials according to the number of non-zero terms they contain.