1. Chapter at a Glance
In Class 9 you met polynomials and their zeros. Now we look at zeros graphically — as the points where a graph meets the x-axis — and discover a beautiful link: for a quadratic polynomial, the sum and product of its zeros can be read straight from its coefficients, without solving anything. This lets us check answers quickly and build a polynomial from given zeros. The chapter belongs to the Algebra unit (about 20 marks in the CBSE board paper, together with linear equations, quadratics and APs).
- Degree and types of polynomials (recap).
- Geometrical meaning of zeros: x-intercepts of y = p(x).
- Maximum number of zeros of a polynomial of degree n.
- Shape of graphs: line (linear), parabola (quadratic).
- Relationship between zeros and coefficients of a quadratic polynomial.
- Forming a quadratic polynomial from the sum and product of its zeros.
2. Polynomials — Recap
| Degree | Name | General form | Max zeros |
|---|---|---|---|
| 1 | Linear | ax + b, a ≠ 0 | 1 |
| 2 | Quadratic | ax² + bx + c, a ≠ 0 | 2 |
| 3 | Cubic | ax³ + bx² + cx + d, a ≠ 0 | 3 |
A real number k is a zero of p(x) if p(k) = 0. A polynomial of degree n has at most n zeros.
3. Geometrical Meaning of Zeros
The zeros of p(x) are exactly the x-coordinates of the points where the graph of y = p(x) meets the x-axis.
- Linear: the graph is a straight line meeting the x-axis at exactly one point (−b/a, 0).
- Quadratic: the graph is a parabola — opening upward if a > 0, downward if a < 0.
The graph of y = p(x) crosses the x-axis at x = −2, 1 and 4. How many zeros does p(x) have, and what is its least possible degree?
Solution: 3 zeros; degree at least 3.
4. Zeros and Coefficients of a Quadratic
If α and β are the zeros of p(x) = ax² + bx + c, then p(x) = a(x − α)(x − β) = a[x² − (α + β)x + αβ]. Comparing coefficients:
α + β = −b/a = −(coefficient of x)/(coefficient of x²)
αβ = c/a = (constant term)/(coefficient of x²)
Find the zeros of x² + 7x + 10 and verify the relationship.
Solution: x² + 7x + 10 = (x + 2)(x + 5) → zeros −2, −5. Sum = −7 = −b/a = −7/1 ✓. Product = 10 = c/a ✓.
Find the zeros of 6x² − 3 − 7x and verify.
Solution: 6x² − 7x − 3 = 6x² − 9x + 2x − 3 = 3x(2x − 3) + 1(2x − 3) = (3x + 1)(2x − 3). Zeros −1/3 and 3/2. Sum = 7/6 = −(−7)/6 ✓. Product = −1/2 = −3/6 ✓.
Find the zeros of x² − 3.
Solution: x² − 3 = (x − √3)(x + √3) → zeros ±√3. Sum 0 = −0/1 ✓; product −3 = −3/1 ✓.
5. Forming a Quadratic from Its Zeros
A quadratic polynomial with sum of zeros S and product P is k[x² − Sx + P], for any non-zero constant k.
Find a quadratic polynomial whose zeros have sum −1/4 and product 1/4.
Solution: x² + ¼x + ¼. Multiply by 4 for whole coefficients: 4x² + x + 1.
If α, β are zeros of x² − 5x + 6, find α² + β² and 1/α + 1/β.
Solution: α + β = 5, αβ = 6. α² + β² = (α + β)² − 2αβ = 25 − 12 = 13. 1/α + 1/β = (α + β)/αβ = 5/6.
6. Enrichment: Cubic Polynomials
For ax³ + bx² + cx + d with zeros α, β, γ:
- α + β + γ = −b/a
- αβ + βγ + γα = c/a
- αβγ = −d/a
Example: 2x³ − 5x² − 14x + 8 has zeros 4, −2, ½. Sum = 2.5 = 5/2 ✓; product = −4 = −8/2 ✓.
7. Solved Problems (Board Pattern)
If one zero of x² − 4x + k is 1, find k and the other zero.
Solution: 1 − 4 + k = 0 → k = 3. Sum of zeros = 4 → other zero = 3.
If the zeros of 5x² + 13x + k are reciprocals of each other, find k.
Solution: αβ = 1 → k/5 = 1 → k = 5.
If the zeros of x² + (a + 1)x + b are equal in magnitude but opposite in sign, find a.
Solution: Sum = 0 → −(a + 1) = 0 → a = −1.
8. Master Formula Table
| Result | Formula |
|---|---|
| Zero of ax + b | −b/a |
| Sum of zeros (quadratic) | α + β = −b/a |
| Product of zeros (quadratic) | αβ = c/a |
| Quadratic from zeros | k[x² − (α + β)x + αβ] |
| α² + β² | (α + β)² − 2αβ |
| (α − β)² | (α + β)² − 4αβ |
| 1/α + 1/β | (α + β)/αβ |
| Cubic | Σα = −b/a, Σαβ = c/a, αβγ = −d/a |
9. Master Summary Table
| Topic | Key idea | Exam focus |
|---|---|---|
| Zeros on a graph | x-intercepts | Count zeros from a graph |
| Parabola | Opens up (a > 0) / down (a < 0) | Sign of a from graph |
| Sum and product | −b/a and c/a | Find zeros and verify |
| Form polynomial | x² − Sx + P | Given sum/product |
| Symmetric expressions | Use identities | α² + β², 1/α + 1/β |
10. Mind Map
- Linear, quadratic, cubic
- ≤ n zeros
- Zeros = x-intercepts
- Parabola
- α + β = −b/a
- αβ = c/a
- x² − Sx + P
- Symmetric forms
11. Tips and Tricks
- Rearrange in standard form (descending powers) before reading a, b, c.
- Signs: sum = −b/a (note the minus), product = +c/a.
- When fractions appear in S or P, multiply by the LCM to get integer coefficients.
- Reading b and c from an unsorted expression like 6x² − 3 − 7x.
- Saying a graph with 2 x-intercepts must be a quadratic — it could be any degree ≥ 2.
- Forgetting that x-intercepts are zeros, not y-intercepts.
12. Important One-liners
- The zeros of p(x) are where y = p(x) meets the x-axis.
- A quadratic polynomial has at most 2 zeros.
- The graph of a quadratic polynomial is a parabola.
- Sum of zeros = −b/a; product = c/a.
- If the product of zeros is negative, the zeros have opposite signs.
- If zeros are reciprocals, a = c.
13. Exam-pattern Questions with Detailed Answers
The number of zeros of a polynomial whose graph does not meet the x-axis at all is:
- 0
- 1
- 2
- Cannot say
Show answer
No intersection → no real zeros.
Answer: (A)
Find the sum and product of zeros of 3x² − 6x + 4.
Show answer
Sum = 6/3 = 2; product = 4/3.
Answer: 2 and 4/3
Find a quadratic polynomial with zeros 3 and −5.
Show answer
S = −2, P = −15 → x² + 2x − 15.
Answer: x² + 2x − 15
Find the zeros of 4u² + 8u and verify the relationship.
Show answer
4u(u + 2) → zeros 0, −2. Sum −2 = −8/4 ✓; product 0 = 0/4 ✓.
Answer: 0, −2
Find the zeros of 3x² − x − 4 and verify.
Show answer
3x² − 4x + 3x − 4 = (3x − 4)(x + 1) → 4/3, −1. Sum 1/3 ✓; product −4/3 ✓.
Answer: 4/3, −1
If α, β are zeros of x² − 6x + 8, find α² + β².
Show answer
36 − 16 = 20.
Answer: 20
Find a quadratic polynomial with sum of zeros √2 and product 1/3.
Show answer
x² − √2x + 1/3 → multiply by 3: 3x² − 3√2x + 1.
Answer: 3x² − 3√2x + 1
If one zero of 2x² + 3x + k is −2, find k.
Show answer
8 − 6 + k = 0 → k = −2.
Answer: k = −2
If α and β are zeros of x² − 7x + 10, find a polynomial whose zeros are α + 2 and β + 2.
Show answer
α + β = 7, αβ = 10. New sum = 11; new product = αβ + 2(α + β) + 4 = 28. Polynomial: x² − 11x + 28.
Answer: x² − 11x + 28
If the difference of the zeros of x² − 5x + k is 1, find k.
Show answer
(α − β)² = (α + β)² − 4αβ → 1 = 25 − 4k → k = 6.
Answer: k = 6
If α, β are zeros of 2x² − 5x + 2, find 1/α² + 1/β² without finding the zeros. Then check by finding them.
Show answer
α + β = 5/2, αβ = 1. α² + β² = 25/4 − 2 = 17/4. 1/α² + 1/β² = (α² + β²)/(αβ)² = 17/4. Check: zeros are 2 and ½ → ¼ + 4 = 17/4 ✓.
Answer: 17/4
For p(x) = x² + (k − 1)x + 2k, the sum of zeros equals the product. Find k.
Show answer
Sum = −(k − 1) = 1 − k; product = 2k. 1 − k = 2k → k = 1/3.
Answer: k = 1/3
14. Quick Revision Summary
- Degree n → at most n zeros; zeros = x-intercepts of the graph.
- Quadratic graph: parabola (up if a > 0).
- α + β = −b/a, αβ = c/a.
- Polynomial from zeros: k[x² − Sx + P].
- Use identities for α² + β², (α − β)², 1/α + 1/β.