Introduction
This chapter explores algebraic expressions and how multiplication of polynomials uses the distributive law. When we expand brackets, we distribute — yet the number of terms grows (multiplies). Mastering this process is essential for all of higher algebra.
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Key Concepts
Monomial: An expression with a single term, e.g. 3x, 5xy2.
Binomial: Two terms, e.g. (a + b), (2x - 3).
Trinomial: Three terms, e.g. (a2 + 2ab + b2).
Polynomial: Any algebraic expression with one or more terms.
Multiplication of Monomials:
3x x 4y = 12xy; 2a2 x 3a3 = 6a5.
Monomial x Polynomial (Distributive Law):
3x x (2x + 5) = 6x2 + 15x.
Binomial x Binomial (FOIL / Distributive Law twice):
(a + b)(c + d) = ac + ad + bc + bd.
- 1.Standard Identities:
- 2.(a + b)2 = a2 + 2ab + b2
- 3.(a - b)2 = a2 - 2ab + b2
- 4.(a + b)(a - b) = a2 - b2
- 5.(x + a)(x + b) = x2 + (a+b)x + ab
- Factorisation (reverse of expansion):
- Common factor: 6x2 + 9x = 3x(2x + 3).
- Using identities: x2 - 9 = (x+3)(x-3).
- By regrouping: ax + bx + ay + by = (a+b)(x+y).
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Worked Examples
Multiply 3x2 by (2x + 5y).
= 3x2 x 2x + 3x2 x 5y = 6x3 + 15x2 y.
Expand (x + 4)(x + 3) using identity 4.
= x2 + (4+3)x + 4 x 3 = x2 + 7x + 12.
Use the identity (a + b)2 to find (3x + 2y)2.
= (3x)2 + 2 x 3x x 2y + (2y)2 = 9x2 + 12xy + 4y2.
Evaluate 1022 using (a + b)2.
(100 + 2)2 = 10000 + 400 + 4 = 10404.
Evaluate 99 x 101 using (a + b)(a - b).
= (100 + 1)(100 - 1) = 1002 - 12 = 10000 - 1 = 9999.
Factorise 4a2 - 25b2.
= (2a)2 - (5b)2 = (2a + 5b)(2a - 5b).
Factorise x2 + 5x + 6.
Find two numbers whose product = 6 and sum = 5: those are 2 and 3.
= (x + 2)(x + 3).
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Common mistakes
- Writing (a + b)2 = a2 + b2 — this omits the middle term 2ab. Always include it.
- Confusing (a - b)2 with (a + b)(a - b) — they are different identities.
- When multiplying binomials, forgetting to multiply ALL pairs (missing cross terms).
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Summary
Multiplying polynomials uses the distributive law repeatedly. Standard identities provide shortcuts for squaring binomials and multiplying conjugates. Factorisation reverses expansion — it is essential for simplifying expressions and solving equations. Recognising which identity fits a given expression is a key algebraic skill.