Expanding and simplifying expressions

The building blocks

• A term is a number, a variable, or numbers and variables multiplied together (e.g. $4x^2$). A coefficient is the number in front of a variable.
• Like terms have exactly the same variable part ($3x$ and $5x$; but $3x$ and $3x^2$ are not like terms). Only like terms can be added or subtracted.
• Expanding means removing brackets by multiplying out; it uses the distributive law — the term outside multiplies every term inside.

Expanding a single bracket

• Multiply every term inside the bracket by the term outside: $a(b + c) = ab + ac$.
• A negative term outside flips the sign of every term inside.
• Collect like terms whenever expanding leaves two terms of the same kind.

Expanding two brackets

• Every term in the first bracket must multiply every term in the second. A handy order is FOIL — First, Outer, Inner, Last:
  • First: $x \times x = x^2$
  • Outer: $x \times (-5) = -5x$
  • Inner: $3 \times x = 3x$
  • Last: $3 \times (-5) = -15$
• Then collect the two middle (like) terms:

$(x + 3)(x - 5) = x^2 - 5x + 3x - 15 = x^2 - 2x - 15$

Perfect squares and difference of two squares

• These are two patterns worth memorising, because they appear constantly:
• Perfect square — a bracket multiplied by itself:

$(a + b)^2 = a^2 + 2ab + b^2 \qquad (a - b)^2 = a^2 - 2ab + b^2$

• The middle term $2ab$ comes from the Outer and Inner products, which are equal, so they add rather than cancel.
• Difference of two squares — a sum times a difference:

$(a + b)(a - b) = a^2 - b^2$

• Here the Outer ($-ab$) and Inner ($+ab$) products cancel, leaving no middle term.