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Binomial Expansion (Edexcel A Level Maths: Pure): Flashcards

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FrontBinomial Expansion
Define the binomial coefficient.

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Define the binomial coefficient.
The number multiplying each term when $(a + b)^{n}$ is expanded with $n$ a positive integer. It is written in two ways, both of which appear in the formula booklet and in exam papers:
${}^{n}C_{r} = (\binom{n}{r}) = \frac{n!}{r! (n - r)!}$
The $!$ symbol means factorial, so $4! = 4 \times 3 \times 2 \times 1$ .
How many terms does the expansion of $(a + b)^{n}$ have when $n$ is a positive integer?
It has $n + 1$ terms.
The coefficients run from ${}^{n}C_{0}$ up to ${}^{n}C_{n}$ , which is $n + 1$ values, so the expansion is finite and stops on its own.
For example $(a + b)^{4}$ has 5 terms.
Fill in the missing values, where $n$ is a positive integer:
${}^{n}C_{0} = {}^{n}C_{n} =______$
${}^{n}C_{1} = {}^{n}C_{n - 1} =______$
The completed results are:
${}^{n}C_{0} = {}^{n}C_{n} = 1$
${}^{n}C_{1} = {}^{n}C_{n - 1} = n$
These are worth knowing, because they let you write the first two and last two terms of any expansion without touching a calculator.
In the expansion of $(a + b)^{n}$ with $n$ a positive integer, how do the powers of $a$ and $b$ behave from term to term?
The power of $a$ starts at $n$ and decreases by 1 each term. The power of $b$ starts at 0 and increases by 1.
The two powers always add up to $n$ , which is a quick way to check a term is right.
When is Pascal's triangle a good way to get binomial coefficients, and when is it not?
It is useful for small values of $n$ , where reading a row off is quick.
For larger $n$ it is slow and prone to arithmetic slips, because you have to build every row up to the one you want. Use ${}^{n}C_{r}$ on a calculator instead.
When expanding $(3 + 2 x)^{4}$ , why must the $2 x$ be kept in brackets?
Because the power applies to the whole term, not just the $x$ .
$(2 x)^{3} = 8 x^{3}$ , not $2 x^{3}$ . Forgetting the brackets loses the factor of $2^{3}$ .
True or False?
In the expansion of $(2 - 3 x)^{6}$ , all the terms are positive.
False.
The signs alternate.
Here $b = - 3 x$ , so $(- 3 x)^{r}$ is negative whenever $r$ is odd. The expansion is $64 - 576 x + 2160 x^{2} - 4320 x^{3} + \dots$
How do you find the coefficient of a particular power of $x$ without expanding the whole binomial?
Pick out the single term whose power of $b$ gives you the power of $x$ you want, using the fact that the subscript of the coefficient matches the power of $b$ .
For the $x^{6}$ term of $(3 + 2 x)^{8}$ you need $(2 x)^{6}$ , so the term is ${}^{8}C_{6} (3)^{2} (2 x)^{6}$ , and the powers still add to 8.
What does "in ascending powers of $x$ " tell you to do?
Start with the constant term, the one in $x^{0}$ , and work upwards in powers of $x$ .
If only the first few terms are asked for, write $\approx$ rather than $=$ , since the expansion you are giving is incomplete.

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