How can you factorise expressions with power 3 or higher?
The key to factoring big and complicated expressions into simpler forms is to remember the factor theorem. The factor theorem says that if we plug a number k into our expression and it comes out as 0, then k must be a root (so we can factor out (x-k)). For example, say we want to factorise:x4+ 3x3+ x2- 3x - 2We'll start by plugging in some vaules for x:x=1 leads to 14+313+12-31-2 = 0 so 1 must be a root of our expression.Now we should try to take out a factor of (x-1) from the above expression. The key here is to compare coefficients to see what we need:x4+ 3x3+ x2- 3x - 2 = (x-1)(ax3+bx2+cx+d) Expanding this gives:= ax4+bx3+cx2+dx - ax3-bx2-cx-d =ax4+x3(b-a) + x2(c-b) + x(d-c) - dWe can compare the coefficients of our x4, x3, x2, x, and constant terms to get a series of simaltaneous equations:-d=-2, d-c = -3, c-b = 1, b-a = 3, a=1Solving these gives a=1, b=4, c=5, d=2, so:x4+ 3x3+ x2- 3x - 2 = (x-1)(x3+4x2+5x+2)We can repeat the process to factorise x3+4x2+5x+2, and carry on until we end up with the form:x4+ 3x3+ x2- 3x - 2 = (x-1)(x+2)(x+1)2
