What will happen if we want to find the square of the sum of three numbers a, b and c, that is, (a + b + c)2?
Answer: (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca: the three squares plus twice each of the three products of pairs.
Step-by-step solution
Given: (a + d)2 = a2 + 2ad + d2 (the identity for two terms)
To find: An expansion of (a + b + c)2
To find: An expansion of (a + b + c)2
Idea: Turn three terms into two. Treat (b + c) as a single number d, use the two-term identity, then put b + c back in place of d.
- Let d = b + c. Then (a + b + c)2 = (a + d)2.½ mark
- We already know (a + d)2 = a2 + 2ad + d2.½ mark
- Replace d by (b + c):
a2 + 2a(b + c) + (b + c)2½ mark - Open the brackets: 2a(b + c) = 2ab + 2ac, and (b + c)2 = b2 + 2bc + c2.1 mark
- So (a + b + c)2 = a2 + 2ab + 2ac + b2 + 2bc + c2, which is easier to remember as
a2 + b2 + c2 + 2ab + 2bc + 2ca.½ mark
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.
Check: a = 1, b = 2, c = 3: (1 + 2 + 3)2 = 36 and 1 + 4 + 9 + 4 + 12 + 6 = 36 ✓.
Answer to write in the exam
Let d = b + c
(a + b + c)2 = (a + d)2 = a2 + 2ad + d2 [(x + y)2 = x2 + 2xy + y2]
= a2 + 2a(b + c) + (b + c)2
= a2 + 2ab + 2ac + b2 + 2bc + c2
∴ (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
Common mistakes that cost marks
- Writing (a + b + c)2 = a2 + b2 + c2. All three cross terms 2ab, 2bc, 2ca are missing.
- Including only 2ab + 2bc and forgetting 2ca. Every pair (a with b, b with c, c with a) appears once.
- Writing ab + bc + ca without the factor 2.
How this can come in the exam
MCQ (1 mark)
The number of terms in the expansion of (x + y + z)2 (after simplifying) is
- 3
- 4
- 6
- 9
Show answer
(C) 6
x2, y2, z2, 2xy, 2yz, 2zx: six terms.
Short answer (2 marks)
If a + b + c = 9 and ab + bc + ca = 26, find a2 + b2 + c2.
Show answer
(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca), so 81 = a2 + b2 + c2 + 52 (1 mark). Hence a2 + b2 + c2 = 29 (1 mark).Try one yourself
Expand (x + 2y + 3)2.
Show answer
x2 + 4y2 + 9 + 2(x)(2y) + 2(2y)(3) + 2(3)(x) = x2 + 4y2 + 9 + 4xy + 12y + 6x.
More questions like this
- Label the squares and rectangles in the figure so that it represents the identity (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca.
- Let us use this identity to find the square of a number, say 119:
- Find the following squares using one of the above identities. Determine which of these identities will make these calculations easier.
- Factor using suitable identities:
- Expand the following using the identity
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca: