International Baccalaureate Mathematics

Syllabus Content

The binomial theorem including the expansion of (a+b)ⁿ,n ∈ N. Use of Pascal's triangle and ⁿC_r

Here are some specific activities, investigations or visual aids we have picked out. Click anywhere in the grey area to access the resource.

Pascal's Triangle Get to know this famous number pattern with some revealing learning activities

Binomial Theorem Video You may have learnt about the binomial expansion in class some time ago so here's a reminder to bring you up to speed.

Binomial Theorem Exercises in the process of expanding powers of binomial expressions and finding specific coefficients.

Here are some exam-style questions on this statement:

"If $(x+5)^{10}$ is expanded" ... more

"If $(2x+7)^{6}$ is expanded" ... more

"If you expanded $(2x-3)^{15}$, the term containing $x^6$ can be written as $\binom{15}{a}\times(2x)^b\times(-3)^c$" ... more

"In the expansion of $ (x+j)^{9}$ where $j \in \mathbb{R}$, the coefficient of the term in $x^7$ is 144." ... more

"The expansion of $(x + g)^7$, where $g \in \mathbb{Q}^+$, can be written as:" ... more

"The constant term in the expansion of $x^4(2x^2+\frac{m}{x})^7$ is 896" ... more

"Consider the expansion of $ (3x+ \frac{c}{x})^8$ where $ c \gt 0 $. " ... more

"Consider the expansion of $ (7-x^2)^{n-1}$ where $n \in \mathbb{Z}^+$. " ... more

See all these questions

Click on a topic below for suggested lesson Starters, resources and activities from Transum.

Algebra Pupils begin their study of algebra by investigating number patterns. Later they construct and express in symbolic form and use simple formulae involving one or many operations. They use brackets, indices and other constructs to apply algebra to real word problems. This leads to using algebra as an invaluable tool for solving problems, modelling situations and investigating ideas. If this topic were split into four sub topics they might be: Creating and simplifying expressions; Expanding and factorising expressions; Substituting and using formulae; Solving equations and real life problems; This is a powerful topic and has strong links to other branches of mathematics such as number, geometry and statistics. See also "Number Patterns", "Negative Numbers" and "Simultaneous Equations".

Probability Probability is a measure of the weight of evidence, and is arrived at through reasoning and inference. In simple terms it is a measure or estimation of likelihood of the occurrence of an event. The word probability comes from the Latin word probabilitas which is a measure of the authority of a witness in a legal case. Some of the earlier mathematical studies of probability were motivated by the desire to be more profitable when gambling. Today however the practical uses of probability theory go far beyond gambling and are used in many aspects of modern life. We believe that even adults can, in many cases, have a poor intuition regarding the effects of probability. These activities are designed to help pupils calculate but also get a 'feel' for the principles of probability.

Furthermore

Official Guidance, clarification and syllabus links:

Counting principles may be used in the development of the theorem.

$^nC_r$ should be found using both the formula and technology.

Example: Find $r$ when $^6C_r=20$, using a table of values generated with technology.

Formula Booklet:

Binomial theorem $n \in \mathbb{N} $

$$ (a + b)^n = a^n +\;^nC_1 a^{n-1}b + \ldots + \;^nC_r a^{n-r}b^r + \ldots + b^n $$ $$ ^nC_r = \frac{n!}{r!(n-r)!} $$

Example

To expand $ (2x + 3y)^5 $ using the binomial theorem, we can use the formula:

$$ (a + b)^n = a^n + nC_1 a^{n-1}b + \ldots + nC_r a^{n-r}b^r + \ldots + b^n $$

Substituting $ a = 2x $, $ b = 3y $, and $ n = 5 $ into the formula, we get:

$$ (2x + 3y)^5 = (2x)^5 + 5C_1 (2x)^4(3y) + 5C_2 (2x)^3(3y)^2 + 5C_3 (2x)^2(3y)^3 + 5C_4 (2x)(3y)^4 + (3y)^5 $$

Simplifying further:

$$ (2x + 3y)^5 = 32x^5 + 160x^4 \cdot 3y + 80x^3 \cdot 9y^2 + 40x^2 \cdot 27y^3 + 10x \cdot 81y^4 + 243y^5 $$

Which results in:

$$ (2x + 3y)^5 = 32x^5 + 480x^4y + 720x^3y^2 + 1080x^2y^3 + 810xy^4 + 243y^5 $$

This video on the Binomial Theorem is from Revision Village and is aimed at students taking the IB Maths AA Standard level course

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