Q1. Find the zeroes of x² − 5x + 6 and verify the relationship.
x² − 5x + 6 = (x − 2)(x − 3) ⇒ zeroes 2 and 3
Sum = 2 + 3 = 5 = −(−5)/1 ✓
Product = 2 × 3 = 6 = 6/1 ✓
Polynomials — Class 10
The zeroes of a polynomial are where its graph cuts the x-axis. Find them instantly and learn how they link to the coefficients.
Polynomial & degreeAn expression like ax² + bx + c. The highest power is the degree: linear (1), quadratic (2), cubic (3).
Zero of a polynomialA value of x for which p(x) = 0. A quadratic has at most 2 zeroes, a cubic at most 3.
Zeroes ↔ coefficients (quadratic)For ax² + bx + c: sum of zeroes = −b/a, product of zeroes = c/a.
Division algorithmp(x) = g(x)·q(x) + r(x), where the remainder r(x) has a smaller degree than the divisor g(x).
Quadratic zeroes finder Interactive
Enter a, b, c to get the zeroes of ax² + bx + c, plus the sum and product check.
x² + x + = 0
Discriminant D—
Nature of roots—
Root x₁—
Root x₂—
Numericals practice Interactive
Sum and product of zeroes with steps.
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Solved examples
Formula sheet
| Quantity | Formula |
|---|---|
| Sum of zeroes | α + β = −b/a |
| Product of zeroes | α·β = c/a |
| Quadratic from zeroes | x² − (α+β)x + αβ |
Common mistakes & exam wins
- The number of zeroes = the number of times the graph meets the x-axis.
- For ax² + bx + c: remember "sum = −b/a, product = c/a".
- A quadratic with zeroes α, β is k[x² − (α+β)x + αβ] for any non-zero k.
Frequently asked questions
What is a zero of a polynomial?
A value of the variable that makes the polynomial equal to zero; graphically, where the curve crosses the x-axis.
What is the relationship between zeroes and coefficients?
For a quadratic ax² + bx + c, the sum of the zeroes is −b/a and the product is c/a.
How many zeroes can a polynomial have?
At most as many as its degree — a quadratic has up to 2, a cubic up to 3.