Question

How do you use the binomial series to expand `(x - sqrt 2)^5`?

Answer 1

Binomial Expansion: `(a+b)^n=\sum_{k=0}^{n} C(n,k)a^{n-k}b^k`, where `C(n,k)` is the the binomial coefficient which is defined as:

`C(n,k) \equiv \frac{n!}{k!(n-k)!}`.

`(a-b)^n=\sum_{k=0}^{n}C(n,k)a^{n-k}(-1.b)^k`
`\qquad \qquad \qquad \quad =\sum_{k=0}^{n}(-1)^kC(n,k)a^kb^{n-k}`.

Solution: `a=x; \qquad b=\sqrt{2}; \qquad n=5`
`(x-\sqrt{2})^5 = (-1)^{0}C(5,0)x^5+(-1)^1C(5,1)x^4(\sqrt{2})^3 +`
`(-1)^2C(5,2)x^3(\sqrt{2})^2+(-1)^3C(5,3)x^2(\sqrt{2})^3+`
`(-1)^4C(5,4)x(\sqrt{2})^4+(-1)^5C(5,5)(\sqrt{2})^5`

Binomial Coefficients: `C(n,k)=C(n, n-k)`
`C(5,0)=C(5,5)=1; \qquad C(5,1)=C(5,4)=5;`
`C(5,2)=C(5,3)=10`

`(x-\sqrt{2})^5=x^5-5\sqrt{2}x^4+20x^3-20\sqrt{2}x^2+20x-4\sqrt{2}`