Polynomials — Quick Notes & Formula Sheet
Class 10 CBSE Mathematics · Chapter 2 · Complete Revision Guide (2026-27)
📌 2026-27 Syllabus Update — Read This First
The Division Algorithm for Polynomials (long division of one polynomial by another) has been officially deleted from the CBSE Class 10 board exam for 2026-27. Don't spend exam-prep time on it — focus entirely on the relationship between zeroes and coefficients, covered below. If your school still teaches it as classwork, that's fine for concept-building, but it will not be tested in boards.
1. What is a Polynomial?
A polynomial in one variable x is an expression of the form:
p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
where aₙ, aₙ₋₁, ..., a₀ are real numbers (coefficients), aₙ ≠ 0, and n is a non-negative integer called the degree of the polynomial.
| Type | Degree | General Form | Example |
|---|---|---|---|
| Constant | 0 | a₀ | 5 |
| Linear | 1 | ax + b | 2x + 3 |
| Quadratic | 2 | ax² + bx + c | x² − 5x + 6 |
| Cubic | 3 | ax³ + bx² + cx + d | x³ − 6x² + 11x − 6 |
⚠ Trap: Degree vs. number of terms
Degree is the highest power of x, not how many terms the expression has. x⁵ + 1 is degree 5, even though it only has 2 terms.
2. Zeroes of a Polynomial & Graphical Meaning
A real number k is called a zero of polynomial p(x) if p(k) = 0. Graphically, the zeroes of p(x) are the x-coordinates of the points where the graph y = p(x) intersects the x-axis.
| Polynomial Type | Maximum Possible Zeroes | Graph Shape |
|---|---|---|
| Linear | 1 | Straight line |
| Quadratic | 2 | Parabola |
| Cubic | 3 | S-shaped curve |
⚠ Trap: "Degree n always means exactly n zeroes"
A polynomial of degree n has at most n real zeroes — not always exactly n. A parabola that doesn't touch the x-axis at all has zero real zeroes, even though it's degree 2. Always read the graph carefully rather than assuming.
3. Relationship Between Zeroes and Coefficients (Most Important — High Weightage)
For a quadratic polynomial ax² + bx + c, with zeroes α and β:
Sum of zeroes: α + β = −b/a
Product of zeroes: α × β = c/a
For a cubic polynomial ax³ + bx² + cx + d, with zeroes α, β, γ:
α + β + γ = −b/a
αβ + βγ + γα = c/a
αβγ = −d/a
⚠ Trap: The sign of the sum formula
Students very commonly write sum of zeroes = b/a (forgetting the negative sign). It's always −b/a for the sum, regardless of quadratic or cubic. Only the product's sign changes based on degree (positive for quadratic, negative for cubic).
✓ Memory tip
The sign of each relationship alternates: sum is negative, sum-of-products-two-at-a-time is positive, product of all three is negative — following the pattern (−1)¹, (−1)², (−1)³ applied to each coefficient ratio in order.
4. Forming a Quadratic Polynomial from Sum & Product of Zeroes
If the sum of zeroes is S and the product of zeroes is P, the required quadratic polynomial is:
x² − Sx + P (or any constant multiple, k(x² − Sx + P))
⚠ Trap: Writing +Sx instead of −Sx
A very common slip is writing x² + Sx + P. Remember it's minus the sum, because the formula comes directly from rearranging α+β = −b/a and αβ = c/a with a = 1.
5. HOTS Pattern: "Zeroes Related to Another Polynomial"
A very common board-exam question type: "If the zeroes of a new polynomial are double (or reciprocal, or shifted by a constant) the zeroes of a given polynomial, find its coefficients."
Approach: First find the sum and product of zeroes of the given polynomial. Then apply the transformation to both values — e.g., if new zeroes are double the old ones, the new sum doubles, but the new product becomes 4 times the old product (since product scales by the square of the multiplier).
⚠ Trap: Scaling the product the same way as the sum
If zeroes are scaled by a factor k, the sum scales by k, but the product scales by k² (for two zeroes) — students often forget to square the factor for the product.
6. Full Recap — Every Trap in One Place
| Trap | Correct Understanding |
|---|---|
| Confusing degree with number of terms | Degree = highest power of x, regardless of term count |
| Assuming degree n = exactly n real zeroes | It means at most n real zeroes |
| Sum of zeroes = b/a | It's always −b/a (negative sign) |
| Forming polynomial as x² + Sx + P | Correct form is x² − Sx + P |
| Scaling product same as sum in transformation problems | Product scales by the square of the scaling factor |
| Studying the Division Algorithm for boards | Deleted from the 2026-27 CBSE board exam — skip it for exam prep |
Practice What You've Just Revised
Test yourself with the Polynomials quiz series once published.
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