Watch the signs

The idea

When a bracket contains subtraction, each term still multiplies each term, with signs given by the rules of integer multiplication, so (a + u)(b – v) = ab + ub – av – uv.

A minus inside a bracket

Expand (a + u)(b − v). The first bracket has + u. The second bracket has − v.

The cell where u meets −v holds u × (−v). What sign will it have?

• + uv
• − uv
• It depends on the numbers

Switch the signs

Each cell holds the product of its two terms. Green cells are added, red cells are taken away. Switch u and v between + and −.

The sign rules decide

When a bracket contains subtraction, each term still multiplies each term. The signs come from the rules of integer multiplication, so (a + u)(b − v) = ab + ub − av − uv.

(a − u)(b + v) = ab − ub + av − uv (a − u)(b − v) = ab − ub − av + uv

Think of (b − v) as (b + (−v)) and use the rule for (a + m)(b + n).

• Product of the two terms | Sign
• (+) × (+) | +
• (+) × (−) | −
• (−) × (+) | −
• (−) × (−) | +

Notes

Each term still multiplies each term; the sign of each product follows the rules of integer multiplication: (a + u)(b − v) = ab + ub − av − uv

Check yourself

Which is the expansion of (a + 2)(b − 3)?

In (x − 3)(y − 4), what is the last term, from (−3) × (−4)?

Use ab + ub − av − uv to find (a + u)(b − v) for a = 3, u = 5, b = 2, v = 4.

Answer: -16

6 + 10 − 12 − 20 = −16. Check: (3 + 5)(2 − 4) = 8 × (−2) = −16.

Use ab − ub − av + uv to find (a − u)(b − v) for a = 9, u = 4, b = 7, v = 2.

Answer: 25

63 − 28 − 18 + 8 = 25. Check: (9 − 4)(7 − 2) = 5 × 5 = 25.

• ab + 2b − 3a + 6. The last term is 2 × (−3) = −6. A plus times a minus is minus.
• ab + 2b − 3a − 6 — correct. Yes! a × b, 2 × b, a × (−3) and 2 × (−3) = −6.
• ab + 2b + 3a − 6. The term from a × (−3) is −3a, not +3a.
• −12. A minus times a minus is a plus. (−3) × (−4) = +12.
• −7. The terms are multiplied, not added: (−3) × (−4).
• +12 — correct. Yes! A minus times a minus is a plus: (−3) × (−4) = +12.