Exploring further, let us multiply (x + y + z) and (x2 + y2 + z2 − xy − xz − yz).
Step-by-step solution
To find: Their product
Idea: Multiply the long bracket by x, then by y, then by z, writing the three results in rows. Then cross out every pair of equal terms with opposite signs.
- × x: x3 + xy2 + xz2 − x2y − x2z − xyz½ mark
- × y: x2y + y3 + yz2 − xy2 − xyz − y2z½ mark
- × z: x2z + y2z + z3 − xyz − xz2 − yz2½ mark
- Cancel pairs: xy2 with −xy2; xz2 with −xz2; −x2y with x2y; −x2z with x2z; yz2 with −yz2; −y2z with y2z.1 mark
- What is left: x3 + y3 + z3 and three copies of −xyz, i.e. x3 + y3 + z3 − 3xyz. This is another identity.½ mark
Check: x = 1, y = 2, z = 3: left = 6 × (14 − 2 − 3 − 6) = 6 × 3 = 18; right = 1 + 8 + 27 − 18 = 18 ✓. Useful special case: if x + y + z = 0, then x3 + y3 + z3 = 3xyz.
Answer to write in the exam
(x + y + z)(x2 + y2 + z2 − xy − xz − yz)
= x3 + xy2 + xz2 − x2y − x2z − xyz
+ x2y + y3 + yz2 − xy2 − xyz − y2z
+ x2z + y2z + z3 − xyz − xz2 − yz2
= x3 + y3 + z3 − 3xyz
∴ (x + y + z)(x2 + y2 + z2 − xy − xz − yz) = x3 + y3 + z3 − 3xyz
Common mistakes that cost marks
- Writing −xyz only once. It comes from each of the three rows, so it is −3xyz.
- Cancelling terms that only look alike, e.g. xy2 with x2y. They are different terms.
- Missing a term when multiplying, since each row must have 6 terms (18 in all).
How this can come in the exam
If x + y + z = 0, then x3 + y3 + z3 equals
- 0
- xyz
- 3xyz
- −3xyz
Show answer
(C) 3xyz
The left factor is 0, so x3 + y3 + z3 − 3xyz = 0.
Without cubing, find (−12)3 + 73 + 53.
Show answer
−12 + 7 + 5 = 0 (½ mark), so the sum = 3 × (−12) × 7 × 5 (1 mark) = −1260 (½ mark).Try one yourself
Factorise a3 + 8b3 + 27c3 − 18abc.
Show answer
With x = a, y = 2b, z = 3c (3xyz = 18abc): (a + 2b + 3c)(a2 + 4b2 + 9c2 − 2ab − 6bc − 3ca).
More questions like this
- The sum of three numbers is 10 and their product is 25. The sum of their squares is 38. Try to use the previous identity to find the sum of the cubes of these three numbers.
- Simplify the rational expression x2 − 7x + 125x2 + 5x − 100, assuming that 5x2 + 5x − 100 ≠ 0.
- Try to simplify the following rational expression:
36s2 − 12st + t2t2 + 2ts − 48s2 = (6s − t)2(___ + ___)(___ + ___) .
(Hint: Factor t2 + 2ts − 48s2 and simplify the rational expressions assuming that t2 + 2ts − 48s2 ≠ 0). - Simplify the following rational expressions assuming that the expressions in the denominators are not equal to zero:
- Saira has arranged a square of side x units, 8 rectangular strips of sides x units and width 1 unit, and 15 squares of side 1 unit to form a bigger rectangle. Find the length and breadth of the rectangle in terms of x.