Binomial Theorem

Binomial Threom xplained

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The Binomial Theorem

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

• (a + b)n = an + (nC1)an-1b + (nC2)an-2b2 + … + (nCn-1)abn-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 + …

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Generally

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 nCr on your calculator)
r! (n - r)!

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

nCr   is pronounced “n choose r”.

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Example

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)

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Pascal’s Triangle

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

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