Q1.
We have seen that (x + 3)(x + 4) = x² + 7x + 12. Also (x + 6)(x + 7) = x² + 13x + 42. Generalise the pattern to get an expression for (x + a)(x + b).
Answer
Look at where the numbers 7 and 12 came from, and where 13 and 42 came from:
| Product | Coefficient of x | Constant term |
|---|---|---|
| (x + 3)(x + 4) | 7 = 3 + 4 | 12 = 3 × 4 |
| (x + 6)(x + 7) | 13 = 6 + 7 | 42 = 6 × 7 |
| (x + a)(x + b) | a + b | ab |
(x + a)(x + b) = x² + (a + b)x + ab
Proof by the distributive property:
(x + a)(x + b) = x(x + b) + a(x + b)
= x² + bx + ax + ab
= x² + (a + b)x + ab
= x² + bx + ax + ab
= x² + (a + b)x + ab
Why the sum and the product both appear: multiplying out produces four terms. The two ‘mixed’ terms bx and ax are the only ones carrying a single x, so they merge into (a + b)x; the term with no x at all is ab. In the tile picture these are the two arms of x-tiles and the corner block of unit tiles. The identity is the same statement as “a rectangle’s area is the sum of its parts”.
Tip: Read the identity right to left and it becomes a factorising rule: to factor x² + px + q, hunt for two numbers whose sum is p and whose product is q. That is exactly what Section 4.6 does.