1. Binomial Expansion Fractional and Negative Indices
A2 Maths with Liz
Core 4

2. Starter

3. Solutions worked out

5. Aims Previous knowledge
By the end of the lesson you should be able to…
Use the binomial expansion for fractional and negative powers.
Previous knowledge
Binomial expansion with positive whole indices (Core 2)
Factorial notation n! = n(n-1)(n-2)(n-3)…(2)(1)

6. Recall binomials from Core 2
Using the binomial theorem, expand the following: (a) (b) (c)
This yields the general formula!

7. Fractional & Negative Powers
What if we have a fractional or negative power?
E.g.
E.g.
To expand these, you’ll eventually have to work out calculations such as…
and
Try these in your calculator….
It doesn’t work!
When expanding fractional and negative indices, we need to really think through our formula for
instead of simply typing it in our calculator.

8. Example 1 – Negative Indices
Expand the following as a series of ascending powers of x as far as the term in x3. State the values of x for which the expansion is valid.
Solution: First rewrite with a negative index.
Next, expand using the binomial theorem.
Hints:
Keep going until you get to the x3 term!
We know our calculators won’t work for these, so we need to work them out ourselves…
Reduce the fractions and keep simplifying!
If the original is in the form , then this expansion is valid when
In this case,

9. Negative Index - You try…
Expand the following as a series of ascending powers of x as far as the term in x3. State the values of x for which the expansion is valid.
Solution:
This expansion is valid when

10. Shortcut…
The shortcut for these type of expansions simply involves expanding and simplifying our general formula.
From our starter, we know this can be expanded as:
Hints:
All in all, this simplifies down to….
This shortcut is valid when n is negative or a fraction between 0 and 1.
Memorise it if you like, or simply work each problem out and reduce the fractions as you go!

11. Example 2 – using our shortcut
Expand the following as a series of ascending powers of x as far as the term in x3. State the values of x for which the expansion is valid.
Solution: First rewrite with a negative index.
Next, expand using our shortcut.
This expansion is valid when

12. Example 3 – Fractional Powers
Expand the following as a series of ascending powers of x as far as the term in x3. State the values of x for which the expansion is valid.
Solution: First rewrite with a fractional index.
Next, expand using our shortcut.
This expansion is valid when

13. Example 4 – What if the bracket doesn’t start with 1?
Expand the following as a series of ascending powers of x as far as the term in x3. State the values of x for which the expansion is valid.
Solution: First rewrite with a negative power and factor out the 2.
BE VERY CAREFUL WITH THE BRACKETS HERE!
Expand this part using binomial theorem, but don’t lose the factor out front!
This expansion is valid when

14. DUE NEXT LESSON, WEDNESDAY MARCH 11TH
Independent Study
Core 4 Textbook, Pg. 51, Exercise B
DUE NEXT LESSON, WEDNESDAY MARCH 11TH
Optional Practise:
Core 4 Textbook, Pg. 46, Exercise A
Mymaths – Binomial Expansion (Core 2)
Mymaths – Binomial Expansion (Core 4)