Binomial theorem (AASL 1.5)

Author:Cliff Packman

Inquiry questions

Factual Inquiry Questions

What is the Binomial Theorem, and for what types of expressions is it used?
How can you calculate a specific term in a binomial expansion using the Binomial Theorem?

Conceptual Inquiry Questions

Why does the Binomial Theorem work for any power of a binomial expression, and how does Pascal's Triangle relate to it?
How does the concept of combinations play a role in the coefficients of the terms in a binomial expansion?

Debatable Inquiry Questions

Is the Binomial Theorem more important for its theoretical implications in mathematics or for its practical applications?
Can the principles of the Binomial Theorem be extended to non-binomial expressions, and what would be the implications of such an extension?
How has the understanding and application of the Binomial Theorem evolved with the development of modern algebra and computational tools?

Mini-Investigation: Pascal's Triangle and the Binomial Theorem Objective:Understand the relationship between Pascal's Triangle, binomial coefficients, and the binomial expansion.

Introduction: Examine the image of Pascal's Triangle provided in the applet. Notice the sum of the two numbers directly above each number. This triangle is more than a mathematical curiosity; it reveals patterns and relationships! Activity 1: Exploring Pascal's Triangle 1. Finding the Next Row: Add the next row to Pascal's Triangle. Each new number is the sum of the two numbers above it. 2. Edge Numbers: Observe the pattern of the numbers on the edge of the triangle. Predict the edge numbers for the next row. 3. Row Sums: Add up the numbers in each row. Look for a pattern related to powers of 2. 4. Triangle Symmetry: Determine if Pascal's Triangle is symmetrical and prove your findings. What other patterns can be observed in the numbers in Pascal's triangle?

Activity 2: Understanding nCr (Combinations)1. Formula Familiarity: Write the nCr formula and discuss each part of the formula.2. Calculating Combinations: Use the nCr formula to calculate the numbers in a row of Pascal's Triangle. Compare with the applet. Explain how nCr can be used to find a particular value in Pascal's triangle

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Activity 3: Connecting to the Binomial Expansion 1. Expansion Basics: Change the value of n in "Expand (a + b)^n" on the applet and observe the expansion's relation to Pascal's Triangle.2. Coefficient Connection: Identify the coefficients in the binomial expansion and their positions in Pascal's Triangle.3. Pattern Prediction: Predict the binomial expansion of (a + b)^5 and verify using the formula or the applet.

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Conclusion:Reflect on how Pascal's Triangle, nCr, and the binomial expansion are related. Discuss the patterns discovered and their implications. Extension:- Explore diagonal patterns in Pascal's Triangle. - Investigate the Fibonacci sequence within Pascal's Triangle. - Determine the number of different pathways from the top to the bottom of the triangle. Sharing Your Findings: After completing the investigation, share and discuss your findings. Explore other areas where these patterns might appear. Post any interesting facts or discoveries here.

Part 2 - Applying the binomial theorem

Watch following video before attempting the mini-quiz questions.

Alternatively if you feel confident with the theory, look at these worked examples of exam-style questions, Binomial expansion - Basic https://youtu.be/6hay4gEK-y8 Binomial expansion with negative term https://youtu.be/qkkA8xkINVo Binomial expansion past paper question https://youtu.be/Pl5BGmxQYvs Binomial expansion past paper question - challenging version https://youtu.be/HugZvjtv4Io

What is the coefficient of in the expansion of ?

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