Binomial Theorem SL

The following page will help you with questions about the Binomial Theorem (or Binomial Expansion). The Binomial Theorem is used for expanding brackets in the form (a + b)n . Questions on this topic are usually short ones: you usually only have to find one term in an expansion rather than find the full expansion. In this case, it is important to be comfortable using combinations, but you should also be familiar with Pascal's triangle.


Key Concepts

On this page, you should learn to

  • Expand \((a+b)^n\) using Pascal's triangle and \(^nC_r\)

Essentials

The following videos will help you understand all the concepts from this page

Binomial Expansion - Spotting the Pattern

Spotting the Pattern

At first look, the formula for the Binomial Theorem looks quite complicated

\(\left(a+b\right)^{n}=a^{n}+\left(\begin{array}{c} n\\ 1 \end{array}\right) a^{n-1}b+...+\left(\begin{array}{c} n\\ r \end{array}\right)a^{n-r}b^{r}+...+b^{n} \)

The formula is in the information booklet, so no need to try and lean it by heart! Applying it is not too difficult, but before we do that, it is important that we understand the pattern (or patterns) that lie behind it.

The following video shows you the patterns behind the Binomial Theorem

Using Pascal's Triangle

Using Pascal's Triangle

The numbers in Pascal's triangle might well be familiar to you. Each number can be calculated by adding to the numbers immediately above it. As we saw in the previous video, the numbers in Pascal's triangle make up part of the pattern in the expansion of \((a + b)^n\)

We can use Pascal's triangle to help us expand brackets in the form \((a + b)^n\) . Watch the following video to see how to expand \((1 +4x)^3\).

What are Combinations?

What are Combinations?

The numbers that appear in Pascal's triangle can also be found using combinations. HL students should know that a combination is a way of selecting items from a certain number of objects. We can write combinations in different ways \(_{n} P_{r} \) or \(\binom {n} {r}\)

The formula for combinations can be found in the information booklet \(\binom {n} {r} = \frac{n!}{r!(n-r)!}\)

The diagram below shows how each number in Pascal's triangle can be written as a combination.

Here's a video to explain in detail how we can find each number in Pascal's triangle with a combination.

Binomial Expansion using Combinations

Binomial Expansion using Combinations

Now we can use combinations to expand \((a + b)^n\).

The following video shows you how to expand \({(1+2x)^4}\) using combinations

The following video shows you how to expand \({(2-3x)^3}\) using combinations

The following video shows you how to expand \((2x+{3 \over x})^5\) using combinations

Binomial Expansion - Exam-style Questions

In the examination, it is unlikely that you are going to be asked to expand fully a set of brackets - it takes too long! Usually you are asked to find just one term. In the following video, we look at some typical exam-style questions:

  1. Consider the expansion of (1 + 2x)4 . Find the term containing x3.

  2. The third term in the expansion of (2x + p)6 is 2160x4. Find the possible values of p.

  3. Find the term independent of x in the expansion of \((x^3+ \frac{2}{x})^8\)

Binomial Expansion - Exam-style Questions Difficult

In the following video, we look at a very challenging exam-style question:

Find the term x² in the expansion of \((2-3x)^2(4- \frac{x}{2})^5\)

Summary

Print from here

Test Yourself

Here is a short quiz to test your understanding of combinations


START QUIZ!

Here is a quiz that practises the Binomial expansion (a + b)n 


START QUIZ!

Exam-style Questions

Question 1

The values of the third row of Pascal's triangle are given below

a) Write down the values in the fourth row of Pascal's triangle

b) Hence or otherwise, find the term in x² in the expansion of \((3x+2)^4\)

Hint

Full Solution

Question 2

a) Expand (x - 3)4 and simplify your result

b) Hence find the x3 term in (x + 2)(x - 3)4 .

Hint

Full Solution

Question 3

Find the term independent of x in the expansion \((2x- \frac{3}{x^2})^6\)

Hint

Full Solution

 

Question 4

The x term in the expansion \((4+2x)^3(2+ax)^4\) is -4608x

Find a

Hint

Full Solution

Question 5

\(\large(\textbf{a}+2x)^3(4-x)^4 = 6912 + \textbf{b}x +...\)

Find a and b

Hint

Full Solution

MY PROGRESS

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