The binomial theorem is a quick way of expanding a binomial expression that is raised to some power. When the binomial expressions are expanded, is there any type of pattern developing which might help us expand more quickly? It is fun to investigate how it works out in a pattern.
(a) Expand for .
(b) What observations can we make about the terms in the expansion of ?
(c) Write down the general formula for the binomial expansion of .
(d) Obtain the general formula for binomial expansion of where is a rational number, and state the condition for the validity of the binomial expansion with a negative integral exponent or a rational exponent.
Deduce the value of
The number may be defined by . Determine the approximate value for .
Question 1 (a)
Question 1 (b)
- There are terms in the expansion.
- The first term’s powers start at and go down.
- The second term’s powers start at 0 and go up.
- The coefficients are from Pascal’s triangle or .
Question 1 (c)
Question 1 (d)
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