More than two terms
Every term meets every term, then add the like terms.
Three terms in a bracket
You can expand a bracket with two terms times a bracket with two terms. Now try (a + b) × (a² + 2ab + b²). The second bracket has three terms.
How many separate products will you get?
What this lesson covers
The idea
The distributive property is not restricted to two terms in a bracket: each term multiplies every term inside, products such as a × a are written a², and like terms in the result (ba = ab) are added.
Three terms in a bracket
You can expand a bracket with two terms times a bracket with two terms. Now try (a + b) × (a² + 2ab + b²). The second bracket has three terms.
How many separate products will you get?
- 2
- 3
- 6
Tap every cell, then join
Tap a cell: its two terms multiply. a × a² becomes a³. Collect all six products. Then tap two like terms to join them.
Each term times every term
The distributive property is not restricted to two terms in a bracket: each term multiplies every term inside. Products such as a × a are written a². Like terms (same letter-numbers, even if ba = ab) are added.
(a + b)(a² + 2ab + b²) = a³ + a²b + 2a²b + 2ab² + ab² + b³ = a³ + 3a²b + 3ab² + b³
2 terms × 3 terms = 6 products. The like terms a²b + 2a²b join into (1 + 2)a²b = 3a²b. But a²b and ab² have different letter-numbers: they are not like terms.
Notes
Each term of one bracket multiplies every term of the other. Then add like terms: terms with exactly the same letter-numbers.
Check yourself
How many separate products are made when (x + y + z)(p + q) is expanded?
Answer: 6
3 terms × 2 terms = 6 products: xp, xq, yp, yq, zp, zq.
Which two terms are like terms?
Which is the expansion of a(a + 2b + 3)?
Expand (a + 1)(a² + a + 1) and join like terms. What is the number in front of a²?
Answer: 2
The products are a³, a², a, a², a, 1. The two a² terms join into 2a², the two a terms into 2a: a³ + 2a² + 2a + 1.
- a²b and ab². The letters are the same, but the powers are different (a twice and b once, against a once and b twice).
- 2ab and 5ba — correct. Yes! ba = ab, so both terms have exactly the same letter-numbers.
- 3a and 3a². a and a² are different letter-numbers. Only the number in front is the same.
- a² + 2b + 3. The a must multiply every term: 2b and 3 as well.
- a² + 2ab + 3. The a must multiply the last term too: a × 3 = 3a.
- a² + 2ab + 3a — correct. Yes! a × a = a², a × 2b = 2ab and a × 3 = 3a.