Binomial Theorem

[Maximum mark: 4]

Expand $(2x + 1)^4$ in descending powers of $x$ and simplify your answer.

[Maximum mark: 5]

Consider the expansion of $(x-3)^8$.

1. Write down the number of terms in this expansion. [1]
   

2. Find the coefficient of the term in $x^6$. [4]
   

[Maximum mark: 6]

Consider the expansion of $(2x-1)^9$.

1. Write down the number of terms in this expansion. [1]
   

2. Find the coefficient of the term in $x^2$. [5]
   

[Maximum mark: 4]

Expand $(2x - 3)^4$ in descending powers of $x$ and simplify your answer.

[Maximum mark: 6]

1. Show that $(2n-1)^3 + (2n+1)^3 = 16n^3+12n$ for $n \in \mathbb{Z}$. [3]
   

2. Hence, or otherwise, prove that the sum of the cubes of any two consecutive odd integers is divisible by four. [3]
   

[Maximum mark: 5]

The third term, in descending powers of $x$, in the expansion of $(x+p)^8$ is $252x^6$. Find the possible values of $p$.

[Maximum mark: 5]

Consider the expansion of $\left(\dfrac{x^2}{2} + \dfrac{a}{x}\right)^6$. The constant term is $960$.

Find the possible values of $a$.

[Maximum mark: 6]

Consider the expansion of $\bigg(x^3+\dfrac{2}{x}\bigg)^8$.

1. Write down the number of terms in this expansion. [1]
   

2. Find the coefficient of the term in $x^4$. [5]
   

[Maximum mark: 7]

Let $f(x) = (x^2 + a)^5$.

In the expansion of the derivative, $f'(x)$, the coefficient of the term in $x^5$ is $960$. Find the possible values of $a$.

[Maximum mark: 6]

In the expansion of $px^2(5 + px)^8$, the coefficient of the term in $x^6$ is $3402$. Find the value of $p$.

[Maximum mark: 6]

Consider the expansion of $\bigg(3x + \dfrac{p}{x}\bigg)^8$, where $p > 0$. The coefficient of the term

in $x^4$ is equal to the coefficient of the term in $x^6$. Find $p$.

[Maximum mark: 6]

Consider the expansion of $\dfrac{(x+a)^7}{bx}$, where $a > 0$. The coefficient of the term in $x^5$ is $2$, and the coefficient of the term in $x^3$ is $1690$.

Find the value of $a$ and the value of $b$.

[Maximum mark: 7]

Consider the expansion of $\bigg(2x^6+\dfrac{x^2}{q}\bigg)^{10}$,  $q \neq 0$. The coefficient of the term

in $x^{40}$ is twelve times the coefficient of the term in $x^{36}$. Find $q$.

[Maximum mark: 6]

1. Write down and simplify the expansion of $(3-x)^5$ in descending order of powers of $x$. [3]
   

2. Hence find the exact value of $(2.9)^5$. [3]
   

[Maximum mark: 5]

In the expansion of $x(2x + 1)^n$, the coefficient of the term in $x^3$ is $20n$, where $n \in \mathbb{Z}^+$. Find $n$.

[Maximum mark: 5]

In the expansion of $(2x + 1)^n$, the coefficient of the term in $x^2$ is $40n$, where $n \in \mathbb{Z}^+$. Find $n$.

[Maximum mark: 5]

Use the extension of the binomial theorem for $n \in \mathbb{Q}$ to show that $\dfrac{x}{(1+x)^2} \approx x - 2x^2 + 3x^3$, $|x| < 1$.

[Maximum mark: 7]

Given that $(5+nx)^2\bigg(1+\dfrac{3}{5}x\bigg)^n\hspace{-0.25em}=\hspace{0.05em}25+100x+\cdots$, find the value of $n$.

[Maximum mark: 8]

Use the extension of the binomial theorem for $n \in \mathbb{Q}$ to show that $\sqrt{\dfrac{1+x}{1-x}} \approx 1 + x + \dfrac{x^2}{2}$, $|x| < 1$.

[Maximum mark: 7]

Given that $(1 + x)^3(1 + px)^4 = 1 + qx + 93x^2 + \dots + p^4x^7$, find the possible values of $p$ and $q$.

[Maximum mark: 7]

1. Write down the quadratic expression $3x^2 + 5x - 2$ in the form $(ax-b)(x+c)$.[2]
   

2. Hence, or otherwise, find the coefficient of the term in $x^9$ in the expansion
   of $(3x^2+5x-2)^5$. [5]
   

[Maximum mark: 30]

This question will investigate power series, as an extension to the
Binomial Theorem for negative and fractional indices.

A power series in $x$ is defined as a function of the form

where the $a_i \in \mathbb{R}$.

1. Expand $(1-x)^5$ using the Binomial Theorem.[3]
   

Consider the geometric series

2. 1. State for which values the geometric series is convergent.

   2. Show that, for this set of values, the sum of the series is $(1+x)^{-1}$.[4]
      

3. By differentiating the series $S$, show that

   $(1+x)^{-2} = 1 -2x +3x^2-4x^3+5x^4-\cdots$ [2]
   

4. By differentiating the equation obtained in part (c), show that

   $(1+x)^{-3} = 1 - 3x + 6x^2 - 10x^3 + \cdots$ [2]
   

5. Hence by recognising the pattern, deduce that for $n\in \mathbb{Z}^+$,

   $(1+x)^{-n} = 1 - nx + \dfrac{n(n+1)}{2!}x^2 - \dfrac{n(n+1)(n+2)}{3!}x^3 + \cdots$ [4]
   

Now, we will determine how to generalize the expansion of $(1+x)^q$ for $q\in \mathbb{Q}$.

Suppose $(1+x)^q$ with $q\in \mathbb{Q}$ can be written as the power series

6. By substituting $x=0$, find the value of $a_0$.[1]
   

7. By differentiating the series of $(1+x)^q$ and evaluating at $x=0$ find the value of $a_1$.[2]
   

8. By repeating the procedure of part (g) find the value of $a_2$ and $a_3$.[4]
   

9. Hence, write down the first four terms of the series expansion for $(1+x)^q$ called the Extended Binomial Theorem.[1]
   

10. Write down the power series for $\dfrac{1}{\sqrt{1-x^2}}$, including the first four terms.[3]
    

11. Hence, integrating the series found in part (j), find the power series for $\arcsin x$, including the first four terms.[4]