We Distribute, Yet Things Multiply

Chapter at a glance

• Distributivity relates multiplication and addition: \(a(b + c) = ab + ac\).
• This property is used to expand products such as \((a + m)(b + n)\) and to find the change in a product when numbers are increased or decreased by given amounts.
• The general expansion \((a + m)(b + n) = ab + mb + an + mn\) (Identity 1) is visualised by area diagrams and holds for any integers (positive or negative).
• Three special identities are obtained as cases: \((a + b)^2 = a^2 + 2ab + b^2\), \((a - b)^2 = a^2 - 2ab + b^2\) and \((a + b)(a - b) = a^2 - b^2\).
• The same identities enable fast multiplication of numbers involving 11, 101, 1001, etc., by breaking the multiplier and adding partial products in one line.
• Algebraic patterns (sequences of figures, calendar squares, etc.) can be expressed by more than one expression; all correct expressions simplify to the same polynomial.
• Like terms (identical letter-numbers) are combined after expansion; terms with different letter-numbers cannot be combined.

Definitions, theorems and results

• Distributive property of multiplication over addition (NCERT): If \(a\), \(b\) and \(c\) are any numbers, then \(a(b + c) = ab + ac\). The commutative form \((a + b)c = ac + bc\) follows immediately.
• Identity 1 (NCERT): \((a + m)(b + n) = ab + mb + an + mn\).
• Identity 1A (NCERT): \((a + b)^2 = a^2 + 2ab + b^2\).
• Identity 1B (NCERT): \((a - b)^2 = a^2 - 2ab + b^2\).
• Identity 1C (NCERT): \((a + b)(a - b) = a^2 - b^2\).
• An algebraic statement that remains true for all replacements of the letters by numbers is called an identity.
• All identities above are derived directly from the distributive property; no separate proof is required in the exam beyond showing the expansion steps.

Formula sheet

Solved-example patterns

• Expanding a product of two binomials — Apply distributivity term-by-term: multiply each term of the first factor by each term of the second, then combine like terms.
• Finding change in product when numbers are altered by \(m\) and \(n\) — Substitute directly into Identity 1 (or its signed variant) and simplify; verify with numerical substitution when required.
• Fast multiplication by 11, 101, 1001, … — Write the multiplier as \(10^k + 1\), perform the two partial multiplications and add digit-wise, recording carries in one line.
• Evaluating squares using identities — Rewrite the number as a sum or difference of convenient parts (e.g., \(104 = 100 + 4\)) and substitute into Identity 1A or 1B.
• Deriving an algebraic expression for a visual pattern — Count the figure in two or more independent ways, obtain different-looking polynomials, then simplify each to confirm they are identical.
• Verifying an identity or pattern — Expand both sides using distributivity and compare after combining like terms; test with specific integer values (including negatives).

Common mistakes and exam pitfalls

• Sign errors when expanding expressions containing subtraction (especially \((a - b)^2\) or \((a - u)(b - v)\)).
• Forgetting to combine like terms after expansion (e.g., leaving \(ab + ba\) uncombined).
• Treating every term inside brackets with the same sign when one factor is negative.
• In fast multiplication, omitting the carry when adding adjacent digits or misaligning place values.
• Assuming a product always increases when one factor is increased; counter-examples must be checked when both factors change.
• Using different variables for the same quantity in pattern problems, leading to non-equivalent expressions.
• Neglecting to verify that two apparently different expressions for the same pattern simplify to the identical polynomial.