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Polynomials is a short Class 10 chapter, but it sets up quadratic equations, the next chapter, and many board questions mix the two. The ideas are simple once you see them on a graph: a zero is where the curve meets the x-axis, and the coefficients tell you the sum and product of those zeros. This post covers what is in the current syllabus, works through four examples, and lists the slips that cost marks.
What does the Class 10 syllabus include?
CBSE's Class X Mathematics syllabus for 2026-27 lists "zeros of a polynomial" and "relationship between zeros and coefficients of quadratic polynomials". It expects you to find zeros "graphically and algebraically" and to verify the relationship. Algebra as a whole carries 20 of the 80 theory marks.
The NCERT chapter (rationalised, reprinted for 2026-27) has a footnote worth knowing: plotting graphs of quadratic or cubic polynomials "is not meant to be done by the students, nor is to be evaluated". You must read zeros off a given graph, but you will not be asked to draw one. The chapter's cubic example is starred as not from the examination point of view, and the syllabus names quadratics only.
What is a zero, and how do you read it from a graph?
A zero of p(x) is a value of x for which p(x) = 0. On the graph of y = p(x), it is the x-coordinate of a point where the curve meets the x-axis.
A linear polynomial ax + b (a ≠ 0) has exactly one zero, −b/a.
A quadratic has at most two zeros. Its graph, a parabola, can cut the x-axis twice, touch it once, or miss it.
In general, a polynomial of degree n has at most n zeros.
So when a question shows a graph and asks "how many zeros?", count the points on the x-axis, not the turns of the curve and not where it crosses the y-axis.
How do the zeros relate to the coefficients?
If α and β are the zeros of ax² + bx + c, then:
α + β = −b/a
αβ = c/a
Example 1. Find the zeros of 2x² − 7x + 3 and verify the relationship.
Split the middle term: 2x² − 7x + 3 = 2x² − 6x − x + 3 = 2x(x − 3) − 1(x − 3) = (2x − 1)(x − 3).
Zeros: x = 1/2 and x = 3.
Sum: 1/2 + 3 = 7/2. And −b/a = −(−7)/2 = 7/2. They match.
Product: 1/2 × 3 = 3/2. And c/a = 3/2. They match.
Example 2. Find a quadratic polynomial whose zeros have sum 3/2 and product −1.
A quadratic with sum S and product P is k(x² − Sx + P).
So k(x² − (3/2)x − 1). Take k = 2 to clear the fraction: 2x² − 3x − 2.
Check: 2x² − 3x − 2 = (2x + 1)(x − 2), zeros −1/2 and 2. Sum 3/2, product −1.
Example 3. For 2x² − 7x + 3, find α² + β² without finding the zeros.
α² + β² = (α + β)² − 2αβ.
= (7/2)² − 2 × 3/2 = 49/4 − 3 = 37/4.
Check with the zeros from Example 1: (1/2)² + 3² = 1/4 + 9 = 37/4.
Example 4. If one zero of x² + kx − 12 is 3, find k and the other zero.
Substitute x = 3: 9 + 3k − 12 = 0, so k = 1.
Product of zeros = c/a = −12, so the other zero is −12 ÷ 3 = −4.
Check the sum: 3 + (−4) = −1 = −b/a = −1. It works.
Which mistakes lose marks?
Dropping the minus in −b/a. For 2x² − 7x + 3, the sum is +7/2, not −7/2. Write −b/a first, then substitute b in brackets: −(−7)/2.
Forgetting to divide by a. For 2x² − 7x + 3, the product is 3/2, not 3.
Writing x² + Sx + P. The middle sign is minus: x² − Sx + P.
Not verifying. Questions often say "verify the relationship". Show both sides.
Counting zeros wrongly from a graph. Count x-axis meetings only, including a point where the curve just touches.
How should you practise?
Mixing problem types helps more than doing one type twelve times. In a study of 126 seventh-grade students over three months, Rohrer, Dedrick and Stershic (2015) found that interleaved practice, where different kinds of problems were mixed, produced higher test scores than the usual blocked practice, both one day and 30 days later. Mixing forces you to choose the method from the question itself, as the board paper does.
A routine for one week:
Day 1: Five quadratics: factorise, find zeros, verify sum and product.
Day 2: Five "find the polynomial" questions from given sum and product.
Day 3: Five graph questions: count zeros from sketches in the NCERT exercise.
Day 4: Five expression questions such as α² + β², 1/α + 1/β and α − β.
Day 5: A mixed set of eight, shuffled, including two from quadratic equations.
When a question goes wrong and you cannot see where, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning, so you can find the exact line where your method went off. The next chapter you will need is quadratic equations in Class 10.
Tonight, pick three quadratics from the NCERT exercise and check every answer by substituting the zeros back in.
