JAMB 1984 · UME · Q24

Find a factor which is common to all three expressions $4a^2 - 9b^2$ , $a^3 + 27b^3$ and $(4a + 6b)^2$ .

Worked solution (try it first)

Factorise the first and third: $4a^2 - 9b^2 = (2a - 3b)(2a + 3b)$ and $(4a + 6b)^2 = 4(2a + 3b)^2$ .
They share $2a + 3b$ .
The second is a sum of cubes: $a^3 + 27b^3 = (a + 3b)(a^2 - 3ab + 9b^2)$ .
It has no factor $2a + 3b$ : putting $a = -\frac32 b$ gives $-\frac{27}{8}b^3 + 27b^3$ , which is not zero.
So no factor is common to all three, option E.