The Binomial Expansion After completing this chapter you
The Binomial Expansion
After completing this chapter you should be able to: • Expand (1 + x)n for any constant n • Expand (a + bx)n for any constants a, b and n • Determine the range of values of x for which the expansion is valid • Use partial fractions to expand more complex fractional expressions
3. 1 The binomial expansion is When n is a positive integer, this expansion is finite and exact. This is not generally the case when n is negative or a fraction. You have already used the binomial expansion in Core 2 to expand (1 – x)6, and (4 -3 x)7 Expand (1+x)4 and (1 -2 x)3 to remind yourselves Now we will look at negative and fractional powers
Using Gives
And now a fractional power
You can use these expansions to find approximations
2. 2 You can use the binomial expansion of (1+x)n to expand (a + bx)n for any constants a and b by simply taking out a as a factor
Example 1 Find the expansion in ascending powers x as far as x³ of
Example 2 And now a fractional one This is something we can expand!
Example 3
Example 5 When (1 + ax)n is expanded as a series in ascending powers of x the coefficients of x and x² are -8 and 48 respectively. (a) Find the values of a and n (b) Find the coefficient of x³ (c) State the values of x for which the expansion is valid
substitute n = -2 back into ① -2 x a = -8 a=4
Find the coefficient of x³ State the values of x for which the expansion is valid Exercise 3 B page 31 now please
3. 3 You can use partial fractions to simplify the expansions of many more difficult expressions Step 1 – split into partial fractions
Expand each expression up to and including x³
now combine these two expansions to actually answer the question!
Exercise 3 C page 33
- Slides: 20