Terms with brackets

Four friends each order a drink for 3 euros and a pizza for 8 euros. You can calculate 4 · (3 + 8) or 4 · 3 + 4 · 8: both give 44 euros. That both paths always lead to the same result is one of the most useful rules in maths, and it's exactly what you'll explore here.

Expanding: the distributive law

The rule says: a · (b + c) = a · b + a · c. Why is that true? Picture a rectangle that is a tall and b + c wide. Its area is a · (b + c). Cut it at the border between b and c and you get two smaller rectangles with areas a · b and a · c. It's the same area, just split differently. That's why 3 · (4 + 2) is the same as 3 · 4 + 3 · 2: the factor in front of the brackets is multiplied by every part inside.

Factoring: the reverse

Factoring reads the rule backwards: if every summand contains the same factor, you may pull it in front of the brackets. So 6x + 9 becomes 3 · (2x + 3), because 6x and 9 each contain the factor 3. Expanding serves as the check: 3 · 2x + 3 · 3 = 6x + 9, it fits. And in mental maths expanding helps you too: split 7 · 98 into 7 · (100 − 2), expand, and work out 700 − 14 = 686.

Careful with the minus before brackets

If a minus stands in front of brackets, it flips every sign inside when you remove them: 10 − (4 + x) = 10 − 4 − x = 6 − x. The minus applies to everything in the brackets, not just the first part. A minus inside the brackets flips as well and turns into a plus: 10 − (4 − x) = 10 − 4 + x = 6 + x. With these tools you can simplify terms: for 2 · (x + 3) + 4x you first expand (2x + 6 + 4x) and then combine like parts: 6x + 6.