Polynomials
A polynomial is an algebraic expression made of terms with whole-number powers of a variable. In CBSE Class 10, the focus is on the zeroes of a polynomial — where its graph meets the x-axis — and the neat relationship between those zeroes and the polynomial's coefficients. The chapter is short and closely linked to Quadratic Equations.
Learning objectives
- Identify the degree of a polynomial and classify it as linear, quadratic or cubic.
- Find the zeroes of a quadratic polynomial.
- Use the relationship between the zeroes and coefficients.
- Form a quadratic polynomial when its zeroes are given.
- Interpret zeroes graphically as x-intercepts.
Key concepts
Degree and types of polynomials
The degree is the highest power of the variable. A polynomial of degree 1 is linear (ax + b), degree 2 is quadratic (ax² + bx + c), and degree 3 is cubic. A polynomial of degree n has at most n zeroes.
Zeroes of a polynomial
A zero of a polynomial p(x) is a value of x for which p(x) = 0. Geometrically, the zeroes are the x-coordinates of the points where the graph of the polynomial meets the x-axis.
Zeroes and coefficients (quadratic)
If α and β are the zeroes of ax² + bx + c (a ≠ 0), then the sum α + β = −b/a and the product αβ = c/a. These let you check answers and form polynomials quickly.
Forming a quadratic from its zeroes
If the zeroes are known, the quadratic polynomial is x² − (sum of zeroes)x + (product of zeroes), i.e. x² − (α + β)x + αβ.
Important formulas
Sum of zeroes (quadratic)
α + β = −b ÷ a
Product of zeroes (quadratic)
α × β = c ÷ a
Quadratic from zeroes
x² − (α + β)x + αβ
Key definitions
- Polynomial
- An expression of the form aₙxⁿ + … + a₁x + a₀ where the powers of x are whole numbers.
- Degree
- The highest power of the variable in the polynomial.
- Zero of a polynomial
- A value of the variable that makes the polynomial equal to zero.
Solved examples
Q1. Find the zeroes of x² − 2x − 8 and verify the relationship with the coefficients.
Solution: Factorise: x² − 4x + 2x − 8 = (x − 4)(x + 2) = 0, so the zeroes are 4 and −2. Sum = 4 + (−2) = 2 = −(−2)/1 = −b/a. Product = 4 × (−2) = −8 = c/a. Both relations hold.
Q2. Find a quadratic polynomial whose zeroes are 3 and −2.
Solution: Sum = 3 + (−2) = 1, product = 3 × (−2) = −6. Polynomial = x² − (sum)x + product = x² − x − 6.
Q3. If α and β are the zeroes of x² − 5x + 6, find α + β and αβ without solving.
Solution: α + β = −b/a = 5 and αβ = c/a = 6.
Common mistakes to avoid
- Forgetting the minus sign: the sum of zeroes is −b/a, not b/a.
- Confusing the degree of a polynomial with the number of terms.
- Writing the polynomial as x² + (sum)x + product instead of x² − (sum)x + product.
- Assuming every polynomial is quadratic — always check the degree first.
Polynomials — MCQ Quiz
12 questions with instant feedback. Use number keys 1–4 to answer.
The degree of the polynomial 7x³ − 2x + 1 is:
Practice questions
Short answer
State the degree of 4x² − 3x⁵ + 7.
The highest power is 5, so the degree is 5.
Find the sum and product of the zeroes of x² + 4x + 3.
Sum = −4, product = 3.
Write a quadratic polynomial whose zeroes are 0 and 5.
x² − (0 + 5)x + 0 = x² − 5x.
Long answer
Find the zeroes of 6x² − 7x − 3 and verify the relationship between zeroes and coefficients.
Split −7x into −9x + 2x (product 6 × −3 = −18, sum −7): 6x² − 9x + 2x − 3 = 3x(2x − 3) + 1(2x − 3) = (2x − 3)(3x + 1). Zeroes: x = 3/2 and x = −1/3. Sum = 3/2 − 1/3 = 7/6 = −b/a = 7/6. Product = (3/2)(−1/3) = −1/2 = c/a = −3/6. Verified.
If one zero of x² − 4x + k is 3, find k and the other zero.
Since 3 is a zero: 9 − 12 + k = 0 ⇒ k = 3. Sum of zeroes = 4, so the other zero = 4 − 3 = 1.
HOTS (Higher Order Thinking)
If the sum of the zeroes of kx² + 2x + 3k equals their product, find k.
Sum = −2/k, product = 3k/k = 3. Setting them equal: −2/k = 3 ⇒ k = −2/3.
The zeroes of a quadratic polynomial are 5 and −3. Can the polynomial be 2x² − 4x − 30? Justify.
Yes. x² − (5 + (−3))x + (5)(−3) = x² − 2x − 15. Multiplying by 2 gives 2x² − 4x − 30, which has the same zeroes.
Quick revision
Revision notes
- Degree = highest power; linear (1), quadratic (2), cubic (3).
- Zeroes are the x-intercepts of the graph.
- Quadratic: sum of zeroes = −b/a, product = c/a.
- Quadratic from zeroes: x² − (sum)x + product.
Key takeaways
- Use sum = −b/a and product = c/a to check or build quadratics fast.
- A degree-n polynomial has at most n zeroes.
- This chapter feeds directly into Quadratic Equations.
Frequently asked questions
What is the difference between a zero and a root?
They mean the same value — 'zero' is used for a polynomial p(x), and 'root' is used for the equation p(x) = 0.
Is the division algorithm for polynomials in the Class 10 syllabus?
The current rationalised CBSE syllabus focuses on zeroes and their relationship with coefficients; check your edition, but most of the marks come from that relationship.
How do I form a quadratic polynomial from its zeroes?
Use x² − (sum of zeroes)x + (product of zeroes).