Question 5
Consider the expression

a)
Write out the binomial expansion up to and including the term.

The binomial expansion of can be expressed using the binomial series formula. The first few terms are:

Calculating the terms, we get:

Thus, the expansion up to is:

Did you know that the binomial expansion was first systematically studied by Sir Isaac Newton? He introduced it in the 17th century, extending Pascal’s triangle concepts to non-integer and even fractional exponents. It’s fascinating how his discoveries birthed a fundamental tool that we still use today in fields ranging from algebra to statistics!

When using binomial expansion, one common mistake is forgetting the factorial in the denominator for higher-order terms. It’s crucial to apply properly, especially in complex expansions. A good tip is to write out the first few terms of the series to catch errors—nothing like double-checking our math to keep those mistakes at bay!