Q1.
Try to multiply (x + y)(x² − xy + y²) using the distributive property. Predict what (x + y)(x² − xy + y²) will be.
Answer
Prediction: x³ + y³. The chapter has just shown (x − y)(x² + xy + y²) = x³ − y³; replacing y by −y turns the left side into (x + y)(x² − xy + y²) and the right side into x³ + y³. Now confirm it by multiplying out.
(x + y)(x² − xy + y²)
= x(x² − xy + y²) + y(x² − xy + y²)
= (x³ − x²y + xy²) + (x²y − xy² + y³)
= x³ − x²y + x²y + xy² − xy² + y³
= x³ + y³
= x(x² − xy + y²) + y(x² − xy + y²)
= (x³ − x²y + xy²) + (x²y − xy² + y³)
= x³ − x²y + x²y + xy² − xy² + y³
= x³ + y³
Why the middle terms cancel: the second bracket is arranged so that each of its terms, multiplied by x, produces exactly what the next term produces when multiplied by y — but with the opposite sign. The six terms therefore fall into two cancelling pairs, leaving only the first and the last. That is why x² − xy + y² is the right partner for x + y, while x² + xy + y² is the right partner for x − y.
Check it yourself: x = 2, y = 1. Left: (3)(4 − 2 + 1) = 3 × 3 = 9. Right: 8 + 1 = 9. ✓