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Class 10 Mathematics Chapter 2 Polynomials — Formulas & Key Points

Chapter 2 Polynomials in Class 10 Mathematics builds the algebraic toolkit you will use in every higher-level problem — from quadratic equations to coordinate geometry. This formula sheet organises every identity, theorem, and factorisation rule into quick-reference tables, memory aids, and worked examples. Use it as your last-minute revision companion before term tests and the board exam, or as a weekly checklist to ensure you have internalised each formula.

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Key takeaways

✓Polynomials have variables with non-negative integer exponents only; expressions like x⁻¹ or √x are not polynomials.
✓Degree of a polynomial is the highest power of the variable; it determines the maximum number of zeroes the polynomial can have.
✓Factor Theorem: (x – a) is a factor of p(x) if and only if p(a) = 0; use this to factorise cubics quickly.
✓Remainder Theorem: dividing p(x) by (x – a) leaves remainder p(a), no long division needed.
✓Seven core algebraic identities (squares, cubes, sums of cubes) save time in expansions and factorisations in board exams.
✓Splitting the middle term is the workhorse method for factorising quadratics; find two numbers that add to b and multiply to ac.
✓Always verify your factorisation by expanding; common sign errors occur when handling (x – y)² and (x – y)³.

Core Definitions and Terminology

Understanding the precise language of polynomials is non-negotiable in board exams. A polynomial in one variable x is an algebraic expression of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ +... + a₁x + a₀, where all exponents are non-negative whole numbers and all coefficients are real. The degree is the highest power of x with a non-zero coefficient. A zero (or root) of p(x) is any real number c such that p(c) = 0. The constant term is the term with x⁰, and the leading coefficient is the coefficient of the highest-degree term. Linear polynomials have degree 1, quadratics have degree 2, and cubics have degree 3. The zero polynomial (0) has no defined degree. Memorising these definitions verbatim helps you answer MCQs and write precise step-by-step solutions.

Polynomial: aₙxⁿ + aₙ₋₁xⁿ⁻¹ +... + a₁x + a₀, where n ∈ ℕ ∪ {0} and all aᵢ ∈ ℝ
Degree: Highest exponent of x with non-zero coefficient (zero polynomial has undefined degree)
Zero of p(x): A real number c such that p(c) = 0
Linear polynomial: ax + b, a ≠ 0 (degree 1, exactly one zero)
Quadratic polynomial: ax² + bx + c, a ≠ 0 (degree 2, up to 2 zeroes)
Cubic polynomial: ax³ + bx² + cx + d, a ≠ 0 (degree 3, up to 3 zeroes)
Factor: (x – a) is a factor of p(x) if p(x) = (x – a)·q(x) for some polynomial q(x)

All Polynomial Formulas in One Table

This table lists every formula and theorem you need from NCERT Class 10 Mathematics Chapter 2. Use the 'When to use' column to recognise patterns in exam questions. The CBSE board often asks you to expand (a + 2b)³ or factorise x² – 9, and spotting the correct identity saves minutes. Practice reading a problem, identifying which row applies, then writing the formula in your answer book before substituting numbers. This habit prevents careless mistakes and earns you full method marks even if the final answer has a minor error.

Seven Algebraic Identities — Expanded Forms

These seven identities are the backbone of polynomial algebra in Class 10. The CBSE board exam will test your ability to recognise which identity applies, substitute correctly, and simplify. Identity I and II handle squares, III is the difference of squares (the fastest factorisation trick), IV extends to three variables, and V, VI, VII handle cubes. Many students confuse (x – y)² and (x + y)² or forget the sign in front of 3xy in the cube formulas. Write each identity on a flashcard, work through five practice problems per identity, and you will never hesitate in the exam hall. The alternative expanded forms (like (x + y)³ = x³ + 3x²y + 3xy² + y³) are useful when the question asks you to show all intermediate steps.

(x + y)² = x² + 2xy + y² — remember the middle term is +2xy
(x – y)² = x² – 2xy + y² — middle term is –2xy, NOT +2xy
x² – y² = (x + y)(x – y) — instant factorisation of difference of squares
(x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx — three squares plus three cross-products
(x + y)³ = x³ + y³ + 3xy(x + y) = x³ + 3x²y + 3xy² + y³
(x – y)³ = x³ – y³ – 3xy(x – y) = x³ – 3x²y + 3xy² – y³
x³ + y³ + z³ – 3xyz = (x + y + z)(x² + y² + z² – xy – yz – zx) — key for symmetric sums

Factorisation Techniques — Step-by-Step

Factorisation is the inverse of expansion, and CBSE loves to ask you to factorise quadratics and cubics. For quadratics ax² + bx + c, split the middle term: find two numbers that add to b and multiply to ac, rewrite bx as their sum, then factor by grouping. For cubics, use the Factor Theorem: test small integer divisors of the constant term (±1, ±2,...) until you find a zero, say x = a, then (x – a) is a factor. Divide p(x) by (x – a) using long division or synthetic division to get a quadratic, then factorise that quadratic. Always verify by expanding your factors back to the original polynomial. This two-step check catches sign errors and ensures full marks.

Quadratic splitting: For ax² + bx + c, find m, n such that m + n = b and m·n = ac; rewrite as ax² + mx + nx + c; factor by grouping
Difference of squares: If you see x² – k², write (x + √k)(x – √k) or (x + k)(x – k) if k is a perfect square
Perfect square trinomial: If x² + 2kx + k², recognise (x + k)²; if x² – 2kx + k², recognise (x – k)²
Cubic factorisation: Find one zero using Factor Theorem, divide to reduce to quadratic, factorise quadratic
Sum/difference of cubes: a³ + b³ = (a + b)(a² – ab + b²), a³ – b³ = (a – b)(a² + ab + b²)

Remainder Theorem and Factor Theorem — Quick Reference

The Remainder Theorem states that when you divide polynomial p(x) by (x – a), the remainder is simply p(a). No need for long division: substitute x = a into p(x) and evaluate. The Factor Theorem is the special case where the remainder is zero: (x – a) is a factor of p(x) if and only if p(a) = 0. These two theorems are your shortcuts for checking divisibility and finding factors. In a 3-mark question, if you are asked whether (x – 2) is a factor of p(x), compute p(2); if it equals zero, write 'By Factor Theorem, (x – 2) is a factor' and earn full marks. If the question asks for the remainder, write 'By Remainder Theorem, remainder = p(a)' and show the substitution clearly.

Remainder Theorem: R = p(a) when p(x) is divided by (x – a)
Factor Theorem: (x – a) is a factor ⟺ p(a) = 0
To find remainder when dividing by (ax – b), rewrite as a(x – b/a), so remainder = p(b/a)
If p(x) has (x – a) as a factor, you can write p(x) = (x – a)·q(x) for some polynomial q(x)

Common Mistakes and How to Avoid Them

CBSE marking schemes deduct heavily for sign errors, incorrect application of identities, and incomplete factorisation. The most frequent mistake is confusing (x – y)² = x² – 2xy + y² with (x + y)² or writing (x – y)³ = x³ – y³ + 3xy(x – y) (wrong sign). Another pitfall: forgetting to check whether all factors are fully factorised (e.g. stopping at (x² – 4) instead of (x + 2)(x – 2)). Always expand your final answer to verify. When using the Factor Theorem, students often test random values instead of testing divisors of the constant term. Finally, in division problems, remember that remainder when dividing by (2x – 1) is p(1/2), not p(–1/2). Write each step clearly and label theorems by name to maximise method marks.

Sign error: (x – y)² ≠ x² + 2xy + y². The middle term must be –2xy.
Incomplete factorisation: Always check if factors can be factorised further (e.g. x² – 9 = (x+3)(x–3), not left as x²–9)
Wrong test values: For Factor Theorem, test divisors of the constant term, not arbitrary numbers
Remainder with (ax – b): Rewrite as a(x – b/a); remainder is p(b/a), not p(–b/a)
Forgetting zero polynomial: The zero polynomial has no defined degree; do not write 'degree 0'
Notation: Write p(x), not p·x or p(x) without parentheses around the argument

Memory Tricks and Mnemonics

Remembering seven identities under exam pressure is easier with mnemonics and patterns. For (x + y)², think 'Square – Double – Square': x², 2xy, y². For (x – y)², same pattern but the middle term is negative. For cubes, recall that (x + y)³ includes the term +3xy(x + y), which you can also write as +3x²y + 3xy². The symmetry helps: (x + y)³ has all positive signs when expanded fully, while (x – y)³ alternates signs. For Identity VII (sum of cubes), remember 'if x + y + z = 0, then x³ + y³ + z³ = 3xyz' as a special case. Use the mnemonic SSP for quadratic factorisation: Split, Separate, Pair (split the middle term, separate into groups, pair and factor). Write these on the first page of your exam answer book as a quick reference before attempting polynomial questions.

Square of sum: 'First² + 2·First·Second + Second²' → (x+y)² = x² + 2xy + y²
Square of difference: Same, but middle term is minus → (x–y)² = x² – 2xy + y²
Difference of squares: 'Sum times Difference' → x²–y² = (x+y)(x–y)
Cube of sum: 'Cube + Cube + 3·Product·Sum' → (x+y)³ = x³ + y³ + 3xy(x+y)
Cube of difference: 'Cube minus Cube minus 3·Product·Difference' → (x–y)³ = x³ – y³ – 3xy(x–y)
SSP for quadratics: Split middle term, Separate into pairs, Pull out common factors

Three Solved Mini-Examples Applying the Formulas

Working through short, focused examples cements your understanding and builds exam speed. Each example below demonstrates one key technique: verifying a factor using the Factor Theorem, expanding a trinomial square, and factorising a cubic by finding one zero. Practice these patterns until you can reproduce them in under two minutes each. In the board exam, a 3-mark factorisation question should take you 3–4 minutes if you have these methods internalised. Always write the theorem name, show substitution, and box your final answer.

One-Glance Last-Minute Revision Box

The night before your exam, review this box to ensure you have not missed any formula. Each line is a must-know fact. If any line is unclear, revisit that section above. Print this box, stick it in your notebook, and read it once before you sleep and once in the morning. In the exam hall, if you blank on an identity, close your eyes and visualise this box. Muscle memory from repeated revision will bring the formula back. This box is your safety net for those high-pressure moments.

Polynomial: aₙxⁿ +... + a₀, exponents ∈ ℕ∪{0}, coefficients ∈ ℝ
Degree = highest power with non-zero coefficient; zero polynomial has no degree
Zero of p(x): c such that p(c) = 0; linear has 1, quadratic up to 2, cubic up to 3
Remainder Theorem: remainder when p(x)÷(x–a) is p(a)
Factor Theorem: (x–a) factor ⟺ p(a)=0
(x+y)² = x²+2xy+y²; (x–y)² = x²–2xy+y²
x²–y² = (x+y)(x–y)
(x+y+z)² = x²+y²+z²+2xy+2yz+2zx
(x+y)³ = x³+y³+3xy(x+y); (x–y)³ = x³–y³–3xy(x–y)
x³+y³+z³–3xyz = (x+y+z)(x²+y²+z²–xy–yz–zx)
Quadratic factorisation: split middle term, factor by grouping
Cubic factorisation: find one zero, divide, factorise quotient

How CBSETUTOR.ai Helps You Master Polynomials in Minutes

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Quick Notation and Symbol Guide

Using correct mathematical notation earns you clarity marks and prevents misinterpretation by the examiner. Write p(x) for 'polynomial in x', not p·x or p(x without closing parenthesis. Use the symbol ≡ (identical to) for identities when you want to emphasise the equation is true for all values, though = is acceptable in CBSE exams. Write '⟺' (if and only if) when stating the Factor Theorem to show the two-way relationship. For division, write p(x) = (x – a)·q(x) + R, where R is the remainder. Always label your final answer with 'Hence,...' or box it. When expanding, show at least one intermediate step; jumping straight to the answer loses method marks even if correct. For factorisation, write each factor in lowest terms: (2x + 4) should be simplified to 2(x + 2) if possible. These small habits distinguish a 90 percenter from a 95 percenter.

p(x) denotes polynomial in variable x; p(a) denotes value when x = a
Degree notation: deg(p) = n or 'degree of p(x) is n'
Factorisation: p(x) = (x – a)(x – b)..., each factor in simplest form
Remainder/quotient: p(x) = (x – a)·q(x) + R
Identity symbol: (x+y)² ≡ x² + 2xy + y² (optional, = is also fine)
If and only if: (x–a) is a factor ⟺ p(a) = 0

Frequently asked questions

What is the difference between a zero and a root of a polynomial?+

In CBSE Class 10 Mathematics, the terms 'zero' and 'root' are used interchangeably. Both refer to a value c such that p(c) = 0. Technically, c is a zero of the polynomial p(x) and a root of the equation p(x) = 0. NCERT uses 'zero' more often in Chapter 2.

How do I know which algebraic identity to use in an expansion question?+

Look at the structure: if you see something squared like (a + b)², use Identity I or II; if cubed, use Identity V or VI. If it is a difference of squares x² – y², use Identity III. For three terms squared like (x+y+z)², use Identity IV. Practice 10 examples of each type and pattern recognition becomes automatic within a week.

Can a polynomial have a negative degree?+

No. By definition, a polynomial has only non-negative integer exponents (0, 1, 2, 3,...). Expressions with negative exponents like x⁻¹ or fractional exponents like x^(1/2) are not polynomials. This is a common MCQ trap in CBSE exams.

What is the fastest way to factorise a quadratic in the board exam?+

For ax² + bx + c, use splitting the middle term: find two numbers m and n such that m + n = b and m·n = ac. Rewrite bx as mx + nx, then factor by grouping. With practice, this takes under 90 seconds. If ac is small, trial and error works; for large ac, this method is foolproof and earns full marks.

Why does the zero polynomial have no defined degree?+

The zero polynomial is 0, which can be written as 0·xⁿ for any n. Since there is no highest non-zero term, the degree is undefined. CBSE marking schemes penalise writing 'degree 0' for the zero polynomial; always write 'degree is not defined' or 'undefined'.

How do I verify my factorisation is correct?+

Expand your factors and check whether you get the original polynomial. For example, if you factorised x² – 5x + 6 as (x – 2)(x – 3), multiply (x–2)(x–3) = x² – 3x – 2x + 6 = x² – 5x + 6 ✓. This one-step check prevents silly errors and is worth the 10 seconds in the exam.

What is the remainder when p(x) is divided by (2x – 3)?+

Rewrite (2x – 3) as 2(x – 3/2). By the Remainder Theorem, the remainder is p(3/2). A common mistake is computing p(–3/2) or p(2/3); always solve 2x – 3 = 0 to get x = 3/2, then substitute.

Is (x + y)² the same as x² + y²?+

No! (x + y)² = x² + 2xy + y², which includes the cross-term 2xy. This is one of the most frequent errors in Class 10. Never skip the middle term. Memorise the identity: 'square of sum = sum of squares plus twice the product'.

How many zeroes can a cubic polynomial have?+

A cubic polynomial has degree 3, so it can have at most 3 real zeroes. It may have 1, 2, or 3 distinct zeroes depending on its factorisation. For example, x³ – x = x(x–1)(x+1) has zeroes 0, 1, –1 (three zeroes).

Can I use the Factor Theorem to find all factors of a polynomial?+

The Factor Theorem helps you find linear factors of the form (x – a). For a cubic, you typically find one linear factor, divide, and then factorise the resulting quadratic. So it is a starting point, not a complete method. Combine it with splitting or other techniques for full factorisation.

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