Polynomials Class 10 Notes: Zeros & Worked Examples
Understand polynomial zeros, the graph connection and the sum–product relations for quadratics, with worked examples and practice answers.
A polynomial in x is built from constant coefficients and non-negative integer powers of x. Its degree is the largest power with a non-zero coefficient. A zero is an input that makes the value of the polynomial equal to zero. This lesson focuses on finding and checking zeros of quadratic polynomials.
From a polynomial to its zeros
Find the zeros of x² − 7x + 12
- 1Look for two numbers whose product is 12 and whose sum is −7: −3 and −4.
- 2x² − 7x + 12 = (x − 3)(x − 4).
- 3The product is zero when x = 3 or x = 4. Substitution confirms 9 − 21 + 12 = 0 and 16 − 28 + 12 = 0.
On the graph y = p(x), real zeros occur at intersections with the x-axis. A quadratic can intersect the x-axis twice, touch it once, or fail to intersect it. Its degree is two in all three cases; degree is not the same as the number of distinct real zeros.
Sum and product of zeros
| For ax² + bx + c, a ≠ 0 | Relation |
|---|---|
| Sum of zeros α + β | −b/a |
| Product of zeros αβ | c/a |
Check 2x² − 9x + 4
- 1Factorise as (2x − 1)(x − 4), giving zeros 1/2 and 4.
- 2Sum = 1/2 + 4 = 9/2 = −(−9)/2. Product = 2 = 4/2.
- 3Keep the minus sign in −b/a: here b itself is negative.
Construct a polynomial
Zeros −2 and 5
- 1Use (x − (−2))(x − 5) = (x + 2)(x − 5).
- 2A suitable polynomial is x² − 3x − 10. Its non-zero constant multiples have the same zeros.
- 3The answer is therefore not unique unless a leading coefficient is specified.
Practice and answers
- 1. Find the zeros of x² + x − 12. Answer: 3 and −4, because (x − 3)(x + 4) expands to the polynomial.
- 2. For 3x² + 6x − 9, find the sum and product of zeros. Answer: −2 and −3.
- 3. Form a monic quadratic with zeros 2 and −6. Answer: x² + 4x − 12. “Monic” means leading coefficient 1.
- 4. Is 2 a zero of x² − 5x + 4? No: its value at 2 is −2, not zero.
A quick self-check
After factorising, expand the factors and substitute each proposed zero. If the factors multiply to the wrong middle term, revisit the signs. Keep “zero of a polynomial” separate from “constant term”: the constant term is a coefficient, while a zero is an input.