• Fully factorise \(x^4-1\)

•   Define \(u = x^2\). Then \(u^2=x^4\).

•   \(x^4-1 \equiv u^2-1 \equiv (u+1)(u-1) \equiv (x^2+1)(x^2-1) \equiv (x^2+1)(x+1)(x-1)\)

1. Expand \((x+\frac{1}{2})(x+6)\). Is it possible factorise this expanded expression so that the factors only contain integer coefficients and constants?
2. Expand \((x+\frac{1}{2})(2x+6)\). Is it possible factorise this expanded expression so that the factors only contain integer coefficients and constants?

1.   \((x+\frac{1}{2})(x+6)\equiv x^2+\frac{13}{2}x+3\); no
2.   \((x+\frac{1}{2})(2x+6)\equiv 2x^2+7x+3 \equiv (2x+1)(x+3)\)

Stan and Ollie are trying to factorise \(x^2-5x+6\). Stan writes \((x-2)(x-3)\). Ollie writes \((2-x)(3-x)\). Are either of them correct? If so, who?

Hover for answers:

•   They’re both correct, but Stan’s factorisation is more conventional.