# Maths - Binomial Theorem

> Source: https://ollybritton.com/notes/a-level/maths/topics/binomial-theorem/ · Updated: 2021-03-01 · Tags: school, maths, year-2

##### $$(a + b)^12$$ What would you use to expand this??
The binomial theorem.

##### What is the $r$-th term in the binomial expansion of $(a + b)^n$??
$$
 \left(\begin{matrix} n \\\\ r \end{matrix}\right) a^n b^{n - r}
$$

##### $$1 \quad 5 \quad 10 \quad 10 \quad 5 \quad 1$$ What is the next row of Pascal's triangle??
$$
1 \quad 6 \quad 15 \quad 20 \quad 15 \quad 6 \quad 1
$$

##### What does the $n$-th row of Pascal's triangle start with (ignoring the top)??
$$
1 \quad n
$$

##### What are the coefficients for $(a + b)^3$??
$$
1 \quad 3 \quad 3 \quad 1
$$

##### What is $(a + b)^3$??
$$
a^3 + 3a^2b + 3ab^2 + b^3
$$

##### $$ \left(\begin{matrix} 8 \\ 3 \end{matrix}\right) $$ Because of the symmetry property, what is this equal to??
$$
\left(\begin{matrix} 8 \\\\ 5 \end{matrix}\right)
$$

##### $$ \left(\begin{matrix} n \\ r \end{matrix}\right) $$ Because of the symmetry property, what is this equal to??
$$
\left(\begin{matrix} n \\\\ n-r \end{matrix}\right)
$$

##### What would be the expression for working out the $x^3$ term of $(2x + 6)^7$??
$$
\left(\begin{matrix} 7 \\\\ 3 \end{matrix}\right) (2x)^3 (6)^4
$$

### 2021-10-14
##### What is the formula for $(1 + x)^n$ where $|x| < 1$??
$$
1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + ... + \frac{n(n-1)(n-2)...(n-(r-1))}{r!}x^r
$$

##### Why does the result $$(1 + x)^n \equiv 1 + nx + \frac{n(n-1)}{2!}x^2 + ...$$ hold when $n > 1$ even though the sequence is infinite??
Because you get $0$ in the numerator for later terms and so they disappear.

##### What is the $x^2$ term in the formula for $(1 + x)^n$??
$$
\frac{n(n-1)}{2!} x^2
$$

##### What is the $x^7$ term in the formula for $(1 + x)^n$??
$$
\frac{n(n-1)(n-2)(n-3)(n-4)(n-5)(n-6)}{7!} x^7
$$
##### When is the expansion for $$(1 + x)^n$$ valid??
$$
|x| < 1
$$

##### When is the expansion for $$(1 + 4x)^n$$ valid??
$$
|x| < \frac{1}{4}
$$

##### When is the approximation for $$(1 + x)^n$$ the best??
When the values of $x$ are small.

### 2021-10-22
##### What is $$(a + bx)^n$$ equivalent too??
$$
a^n\left( 1 + \frac{b}{a}x \right)^n
$$

##### How would you tackle finding the binomial expansion for $$\frac{4 - 5x}{(1 + x)(2 - x)}$$??
Use partial fractions.

### 2022-01-11
##### How would you tackle finding the binomial expansion for $$\sqrt{\frac{1-x}{1+4x}}$$??
Treat it as $(1-x)^{\frac{1}{2}}(1+4x)^{-\frac{1}{2}}$ and multiply the expansions together.

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