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