Binomial Theorem

The Binomial Theorem states that, where n is a positive integer:

• (a + b)ⁿ = aⁿ + (ⁿC₁)a^n-1b + (ⁿC₂)a^n-2b² + … + (ⁿC_n-1)ab^n-1 + bⁿ

Example

Expand (4 + 2x)⁶ in ascending powers of x up to the term in x³

This means use the Binomial theorem to expand the terms in the brackets, but only go as high as x³.

So to find the answer we substitute 4 for a in the Binomial theorem and 2x for b:

4⁶ + (⁶C₁)(4⁵)(2x) + (⁶C₂)(4⁴)(2x)² + (⁶C₃)(4³)(2x)³ + …
= 4096 + (6 ×1024 ×2x) + (15 ×256 ×4x²) + (20 ×64 ×8x³) + …
= 4096 + 12288x + 15360x² + 10240x³ + …

It is, of course, often impractical to write out Pascal"s triangle every time, when all that we need to know are the entries on the nth line. Clearly, the first number on the nth line is 1. The second number is n. The third number is:
n(n - 1)   .
 1 × 2

In general, the rth number in the nth line is:
    n!        (which is ⁿC_r on your calculator)
r! (n - r)!

where n! means ‘n factorial’ and is equal to n × (n-1) × … × 2 × 1

ⁿC_r   is pronounced “n choose r”.

Find (3 + x)³

The power that we are expanding the bracket to is 3, so we look at the third line of Pascal’s triangle, which is 1 3 3 1.

So the answer is: 3³ + 3 × (3² × x) + 3 × (x² × 3) + x³(we are replacing a by 3 and b by x in the expansion of (a + b)³ above)

You should know that (a + b)² = a² + 2ab + b² and you should be able to work out that (a + b)³ = a³ + 3a²b + 3b²a + b³ .
It should also be obvious to you that (a + b)¹ = a + b .

so (a + b)¹  =        a + b
(a + b)² =       a² + 2ab + b²
(a + b)³ =  a³ + 3a²b + 3b²a + b³