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BINOMIAL THEOREM

Helen Scanlon

Reflections, May 1990 - Journal of the Mathematical Association of NSW

HISTORICAL

Euclid (c. 300 BC) discovered

Omar Khayyam (c.AD 1100) developed expansions for

Pascal (1623 - 1662) is associated with the array of coefficients, but this array was used by Chinese writers in AD 1300 and by a German Apianus (AD 1527). Sir Isaac Newton extended this idea, working with coefficients. Today applied mathematicians make frequent use of approximations based on the

first few terms of

POSSIBLE TEACHING SEQUENCE

1. Have students expand and tabulate the final line of each.

expansion

By observing only the coefficients, we have Pascal's triangle:

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1

Points to draw out:

the leading and final coefficients are 1;

the coefficient of the second term is the index of the power;

symmetry;

there is one more term in each expansion than the index of the power;

powers of a decrease by one; powers of b increase by one;

the sum of the powers of a and b in each term is constant and equals the power of the binomial expression;

any coefficient in the Pascal triangle array is the sum of the two written just above it in the line before, for example:

new line in Pascal's triangle

2. Add more lines (2 or 3) to Pascal's triangle and use these to expand .

3. Replace a by 1, b by x and observe (1 + x)ⁿ for n = 0,1,2...,5.

4. Examine (1 - x)ⁿ by putting (-x) for b, 1 for a,
for example, in (a + b)⁴, put a = 1, b = -x.

expansion

Observe:

powers of x increase by 1, and form a polynomial in xⁿ

in (1 - x)ⁿ, signs alternate + and - ;

for terms in odd powers of x, the term is -ve.

5. Extend to (1 + 3x)⁴
(1 - 2a)³, etc.

6. Introduce a shorthand notation. C stands for coefficient, so ⁿC_k stands for the coefficient of the term in the binomial expansion.

Hence, we could write (1 + x)⁵

Now, put x = 0 in this expansion:
x=0 in the expansion

7. Looking again at how a new line of Pascal's triangle is obtained:

obtain new line of Pascal's triangle

using coefficients:
use coefficients

After more examples, we generalise to
.

8. This formula has to be proved. The method in the syllabus is fine.
Write
Then multiply it by x and add the two to give The result follows.

9. A good way to derive the coefficients in the expansion of (1 + x)ⁿ is to use calculus.

Differentiate

Differentiate again:

10. Use symmetry and then show .

For example:
Calculation

Practise evaluation of and then use calculator to confirm.

11. Proof by induction for ⁿC_k. (Not examinable and quite difficult. Syllabus proof is quite good and full.)

12. Introduce factorial notation and show that
by giving first arithmetical examples, for example:

factoral notation

13. Introduce "choose" notation. ⁿC_k is written
and can be read "n choose k".

If (like me) you have not done permutations and combinations before the binomial theorem, you will favour ⁿC_k, but use both notations.

14. The syllabus requires use of "sigma" notation. Perhaps a careful listing of all notations and expansions helps:

Sigma expansion

Each of these can be written using "sigma" as follows:

Sigma expansion

Deciding which of these to use in any one instance is difficult for students. It is notation overload for many, so they need help in translation to something which has meaning for them. Only very good 3 Unit (and 4 Unit) students are comfortable using "sigma". [Also, explain the use of (depends on context).]