Binomial Expansion: Introduction and Examples
Explore binomial expansion basics, Pascal's Triangle, and examples like expanding expressions with ascending or descending powers. Understand the coefficients, powers of terms, and how to find specific terms in the expansion. Get a glimpse of binomial expansion before delving deeper into Year 12 mathematics.
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Starter Expand ? + ?0 1 ? ? ? ? ? a) b) Expand ? + ?1 c) Expand ? + ?2 d) Expand ? + ?3 e) Expand ? + ?4 1? + 1? 1?2 + 2?? + ?2 1?3 + 3?2? + 3??2 + 1?3 1?4 + 4?3? + 6?2?2 + 4??3 + ?4 What do you notice about: ? The coefficients: The powers of ? and ?: They follow Pascal s triangle (we ll explore on next slide). Power of ? decreases each time (starting at the power) Power of ? increases each time (starting at 0) ?
Induction lesson: Binomial Expansion Note: Please don t worry if you cannot complete everything. It will be covered in Year 12 properly with your teacher. This lesson is just to give you a flavour of a Year 12 topic. Email Miss Filgate if you have any questions: Laura.Filgate@thebicesterschool.org.uk
Pascals Triangle The second number of each row tells us what row we should use for an expansion. In Pascal s Triangle, each term (except for the 1s) is the sum of the two terms above. 1 So if we were expanding 2 + ?4, the power is 4, so we use this row. 1 1 Tip: I highly recommend memorising each row up to what you see here. 1 2 1 1 3 3 1 1 4 6 4 1 1 1 10 1 5 10 We ll see later WHY each row gives us the coefficients in an expansion of ? + ??
Example Next have descending or ascending powers of one of the terms, going between 0 and 4 (note that if the power is 0, the term is 1, so we need not write it). Find the expansion of 2 + 3?4 2 + 3?4= 1 (24) 23 22 21 3?1 3?2 3?3 3?4 +4 +6 +4 +1 First fill in the correct row of Pascal s triangle. And do the same with the second term but with powers going the opposite way, noting again that the power of 0 term does not appear. Simplify each term (ensuring any number in a bracket is raised to the appropriate power) = 16 + 96? + 216?2+ 216?3+ 81?4 Tip: Initially write one line per term for your expansion (before you simplify at the end), as we have done above. There will be less faffing trying to ensure you have enough space for each term.
Another Example (1 2?) is the same as (1 + 2? ), so we expand as before, but use 2? for the second term. 1 2?3= (13) 12 1 1 2?1 2?2 2?3 +3 +3 +1 = 1 6? + 12?2 8?3 Tip: If one of the terms in the original bracket is negative, the terms in your expansion will oscillate between positive and negative. If they don t (e.g. two consecutive negatives), you ve done something wrong!
Getting a single term in the expansion The coefficient of ?2 in the expansion of 2 ??5 is 720. Find the possible value(s) of the constant ?. The 5 row in Pascal s triangle is 1 5 10 10 5 1. If we count the 1 as the 0thterm , we want the 2nd term, which is 10. Since we want the ?2 term: The power of ?? must be 2. The power of 2 must be 3 (the two powers must add up to 5). ? ? Therefore term is: 10 ??223= 80?2?2 80?2= 720 ?2= 9 ? = 3 ?
Test Your Understanding Edexcel C2 ? ?
Answers https://activeteach-prod.resource.pearson- intl.com/r00/r0066/r006620/r00662094/current/alevelsb_p1_ex8a.pdf Ignore Q2 e), f), g), h) and Q3 e), f), g), h) solutions. Email Miss Filgate if you have any questions or you would like to be sent any extension work: Laura.Filgate@thebicesterschool.org.uk
Compulsory induction work You need to complete the induction booklet before you start in September. There are also additional worksheets that you can complete. Email Miss Filgate if you cannot find them on the school website: Laura.Filgate@thebicesterschool.org.uk
