What is the general binomial expansion?
- The general binomial expansion, as given in the formula booklet, is
- If n ∈ ℕ then the expansion is finite (see Binomial Expansion)
- Otherwise the expansion is infinitely long
- It is only valid for |x| < 1 (-1 < x < 1)
- Only the first few terms of an expansion are usually needed
How do I use a binomial expansion to approximate a value?
- Ignoring higher powers of x leads to an approximation
- The more terms the closer the approximation is to the true value
- For most purposes, squared or cubed terms are accurate enough
STEP 1 Compare the value you are approximating to (a + bx)n
STEP 2 Solve the appropriate equation to find the value of x
STEP 3 Substitute this value of x into the expansion to find the approximation
- You can get a good idea if your approximation is correct by working out the “real” answer using your calculator.
- Sometimes it helps to factorise out a number before approximating