Zeroes and coefficients of a polynomial
- CBSE · NCERT Class 10, Ch 2 Polynomials
For a quadratic polynomial ax² + bx + c with zeroes α and β, the sum α + β equals -b/a and the product αβ equals c/a.
The zeroes of a quadratic polynomial are tied to its coefficients, the numbers in front of each term. You can find the sum and the product of the zeroes without finding the zeroes themselves. This saves time when a question asks about an expression in α and β.
A small example
Take 2x² - 8x + 5. Here a = 2, b = -8 and c = 5.
- Sum: α + β = -(-8)/2 = 4.
- Product: αβ = 5/2.
Now find (α + 1/β) × (β + 1/α). Write each bracket over one denominator. The expression becomes (αβ + 1)² ÷ αβ. Put in αβ = 5/2: (7/2)² ÷ 5/2 = 49/4 × 2/5 = 49/10.
Where marks go
This is Question 27 of CBSE's 2026-27 Maths Standard sample paper, worth 3 marks. The board's Maths Standard marking scheme for 2026-27 gives 1 mark for writing α + β = 4 and αβ = 5/2. It gives half a mark for rewriting the expression and 1 mark for putting in the values. The last half mark is for 49/10.
So the first mark rests on the two rules alone. A sign slip in -b/a changes the sum, and every later step is built on it. How later steps are then treated is set by the rule called error carried forward.
Sources
Facts last checked against these sources on 30 September 2026.
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