Chapter 6: We Distribute, Yet Things Multiply

If a shop sells $7$ packs of pens and $7$ packs of pencils, you could count $7$ pens and $7$ pencils separately, or you could count $7$ "packs of (pen + pencil)". Either way you get the same total , and the second way is much faster. This common-sense idea, written in algebra, is the distributive law:

$a(b + c) = ab + ac.$

Despite looking modest, this law is the engine behind nearly all algebraic manipulation. From "open the brackets" in primary school to multiplying polynomials in high school, from squaring binomials to factoring expressions, every step traces back to distribution.

This chapter takes the distributive law as its hero and rides it through:

• Multiplying a monomial by a polynomial.
• Multiplying two binomials (the FOIL pattern).
• Multiplying any two polynomials, term by term.
• Famous identities like $(a+b)^2$ and $(a-b)(a+b)$.
• Factorising as the reverse of multiplication.

By the end you should be able to expand and simplify any expression you meet, and you should also be able to see the same expression written multiple ways and judge which form is most useful.

What's inside

1. Recap of algebraic vocabulary , terms, coefficients, monomials, binomials.
2. Multiplying a monomial by a polynomial , the simplest distribution.
3. Multiplying two binomials and beyond , FOIL and the general rule.
4. Algebraic identities , $(a+b)^2$, $(a-b)^2$, $(a+b)(a-b)$, and more.
5. Factorising , multiplication in reverse , common factors, identities, splitting the middle term.

Key results

How to read this chapter

Always check an identity with simple numbers. If you suspect $(a+b)^2 = a^2 + b^2$, plug in $a = 1, b = 2$: left side $= 9$, right side $= 5$. The identity is wrong, and the correct one is $(a+b)^2 = a^2 + 2ab + b^2$. This number-check should become a reflex.