Class 10 Polynomials Complete Guide 2027 — Zeros, Division Algorithm, Graphs + Tricks

In short: Polynomials carries 5 marks in CBSE Class 10 Maths 2027 under the Algebra unit. The chapter is more conceptual than computational — you need to understand the relationship between zeros and coefficients (sum and product of zeros), the division algorithm, and what the graph of a polynomial tells you about its zeros. Two formulas — sum of zeros = −b/a and product of zeros = c/a — are the backbone.

Polynomials is Chapter 2 in CBSE Class 10 Maths. It sits in the Algebra unit (20 marks) and carries about 5 marks. While Chapter 1 (Real Numbers) tests proof-based reasoning and Chapter 4 (Quadratic Equations) tests solving skills, Polynomials tests your understanding of the connection between a polynomial's algebraic form and its graphical behaviour.

The chapter is short and conceptual. Once you understand the zeros-coefficients relationship and the division algorithm, the problems become repetitive. For the full marks breakdown, see the CBSE Class 10 chapter-wise weightage.

Core Concepts

Degree and Types of Polynomials

Key fact: A polynomial of degree n has at most n zeros. This is tested in MCQs.

Zeros of a Polynomial

A zero (or root) of a polynomial p(x) is a value of x for which p(x) = 0.

Graphical meaning: Zeros are the x-coordinates where the graph of the polynomial crosses or touches the x-axis.

• A quadratic that crosses the x-axis at two points has 2 zeros
• A quadratic that touches the x-axis at one point has 1 zero (repeated)
• A quadratic that does not touch the x-axis has 0 real zeros

The Two Key Formulas — Zeros and Coefficients

For a quadratic polynomial ax² + bx + c with zeros α and β:

Sum of zeros: α + β = −b/a

Product of zeros: α × β = c/a

Memory trick — "Negative b over a, c over a": Sum has the negative sign, product does not. Or think "SuN" — Sum = Negative b/a. Product is straightforward: c/a.

For a cubic polynomial ax³ + bx² + cx + d with zeros α, β, γ:

• α + β + γ = −b/a
• αβ + βγ + γα = c/a
• αβγ = −d/a

Pattern: The signs alternate — negative, positive, negative for the three relationships. The denominators are always a.

Using the Formulas — Solved Examples

"Find a quadratic polynomial whose zeros are 3 and −2."

Sum = 3 + (−2) = 1 = −b/a → if a = 1, then b = −1 Product = 3 × (−2) = −6 = c/a → c = −6 Polynomial: x² − x − 6

General form: If zeros are α and β, the polynomial is: k[x² − (α+β)x + αβ] for any non-zero constant k.

"If one zero of x² − 6x + k is twice the other, find k."

Let zeros be α and 2α. Sum: α + 2α = 6 (since −b/a = 6) → 3α = 6 → α = 2 Product: α × 2α = k → 2 × 4 = k → k = 8

Division Algorithm for Polynomials

Statement: For polynomials p(x) and g(x) where degree of g(x) > 0:

p(x) = g(x) × q(x) + r(x)

Where q(x) is the quotient, r(x) is the remainder, and degree of r(x) < degree of g(x).

This is exactly like integer division: Dividend = Divisor × Quotient + Remainder.

Board exam application: If two zeros of a cubic polynomial are given, use them to form a quadratic factor, divide the cubic by this factor, and the quotient gives the third zero.

Example: "Find all zeros of 2x³ − x² − 13x − 6, given that two zeros are 3 and −1/2."

The two zeros give factors (x − 3) and (2x + 1). Their product = 2x² − 5x − 3. Divide: (2x³ − x² − 13x − 6) ÷ (2x² − 5x − 3) = (x + 2) Third zero: x = −2. All zeros: 3, −1/2, −2.

Common Mistakes That Cost Marks

1. Forgetting the negative sign in sum of zeros. Sum = −b/a, NOT b/a. The negative sign is critical. If the polynomial is x² − 7x + 12, the sum of zeros is 7 (since −(−7)/1 = 7), not −7.

2. Wrong polynomial from given zeros. If zeros are 2 and 3, the polynomial is x² − 5x + 6, NOT x² + 5x + 6. Use the formula: x² − (sum)x + (product). The minus sign before the sum is part of the formula.

3. Confusing zeros with coefficients. Zeros are the values of x where p(x) = 0. Coefficients are the numbers a, b, c in the polynomial expression. These are related by the formulas but are NOT the same thing.

4. Not verifying zeros. After finding zeros, substitute them back into the polynomial. If p(α) ≠ 0, your zero is wrong. This 30-second check catches errors.

5. Division algorithm errors. When dividing polynomials by long division, align terms by degree and don't skip missing terms. If the polynomial is x³ + 2x − 5 (missing x² term), write it as x³ + 0x² + 2x − 5 before dividing.

4-Week Practice Plan

Week 1: Zeros and Graphs

• Read NCERT Sections 2.1-2.2
• Plot graphs of 5 quadratic polynomials and identify zeros visually
• Practice: given a graph, determine the number of zeros and their approximate values
• Memorise sum and product formulas — write them 3 times daily

Week 2: Zeros-Coefficients Relationship

• Solve NCERT Exercise 2.2 completely (finding polynomials from zeros, finding zeros from relationships)
• Practice "if one zero is k times the other" type problems — solve at least 5
• Practice forming quadratic polynomials from given zeros — 10 problems
• Cross-check with the complete formulas list

Week 3: Division Algorithm

• Read NCERT Section 2.3
• Solve NCERT Exercise 2.3 completely (polynomial division)
• Practice "find all zeros of a cubic given two zeros" — this is the most common board question type
• Make sure you can do long division of polynomials cleanly

Week 4: Board Paper Practice

• Solve Polynomials questions from the last 5 years of CBSE board papers
• Focus on 3-mark questions (zeros-coefficients) and 4-mark questions (division algorithm + find all zeros)
• Time yourself: 3-mark in 4-5 minutes, 4-mark in 6-7 minutes
• Final check: can you write sum and product formulas from memory instantly?

Quick Revision Card for Exam Day

• Sum of zeros (quadratic): α + β = −b/a → "SuN = Negative b/a"
• Product of zeros (quadratic): αβ = c/a
• Polynomial from zeros: x² − (α+β)x + αβ → "minus sum, plus product"
• Max zeros = degree of polynomial
• Zeros = x-intercepts on the graph
• Division algorithm: p(x) = g(x) × q(x) + r(x), degree(r) < degree(g)
• Cubic zeros (if 2 given): form quadratic from 2 zeros → divide cubic → quotient gives 3rd zero
• Missing terms in division: write as 0x², 0x, etc.
• Always verify: p(zero) must equal 0

For interactive practice on finding zeros, verifying the zeros-coefficients relationship, and polynomial division, try Padhaao's AI tutor.

Frequently Asked Questions

How many marks does Polynomials carry in CBSE Class 10 Maths 2027?

Polynomials carries approximately 5 marks in the CBSE Class 10 board exam 2027, as part of the Algebra unit (20 marks total). You can expect one 1-2 mark MCQ on identifying zeros from graphs or basic coefficient relationships, and one 3-4 mark question on finding zeros using the division algorithm or forming polynomials from given zeros.

What is the relationship between zeros and coefficients of a quadratic polynomial?

For a quadratic polynomial ax² + bx + c with zeros α and β: the sum of zeros (α + β) equals −b/a, and the product of zeros (αβ) equals c/a. These two relationships let you find the polynomial if zeros are given, find missing zeros if one is known, or verify whether given values are actually zeros.

How do I find all zeros of a cubic polynomial when two zeros are given?

If two zeros α and β of a cubic polynomial p(x) are given, form the quadratic factor (x − α)(x − β). Divide p(x) by this quadratic using long division. The quotient will be a linear polynomial (x − γ), where γ is the third zero. This division-algorithm-based approach is the standard board exam method.

What does the graph of a polynomial tell us about its zeros?

The zeros of a polynomial are the x-coordinates of the points where its graph intersects the x-axis. A quadratic graph (parabola) can cross the x-axis at 0, 1, or 2 points — meaning 0, 1, or 2 real zeros respectively. If the parabola just touches the x-axis at one point, the polynomial has a repeated zero.

Why is there a negative sign in the sum of zeros formula?

The sum of zeros is −b/a (negative b over a), not b/a. This comes from the factor form of the quadratic: a(x − α)(x − β) = ax² − a(α+β)x + aαβ. Comparing with ax² + bx + c, we get b = −a(α+β), so α + β = −b/a. The negative sign is built into the algebra of the factorisation. Forgetting it is the most common error in this chapter.

Polynomials is a conceptual chapter that rewards understanding over brute-force calculation. Know the two zeros-coefficients formulas, practise the division algorithm, and always verify your zeros by substitution.