Maths Reflection: Binomial Expansion

1. Maths Reflection: Binomial Expansion By: Maisha Loveday 8C

2. Introduction The binomial expansion method is sometimes better than using long multiplication for various reasons like the amount of time needed to work each question out and how many steps or rules there are in each method. But in some cases, using long multiplication is the safest way to go. Nowadays, we have calculators that work out the answer easily and quickly but even if we use calculators we should be able to know the working out to understand how to do it using both methods. Binomial Expansion General Rule: (a+b)2 = a2 + 2ab + b2 and (a-b) 2 = a2 – 2ab + b2

3. Level 1-2: How is Binomial Expansion Useful • This method is more useful than long multiplication because it’s quicker and simple. It’s also easier to check. The method doesn’t really change with different numbers. Also, you could do all the calculations in your head. In long multiplication, there are a lot of steps and one error could mess up the answer. Then, you have to go through the whole process and look for what went wrong. This could affect several other numbers. • Long Multiplication Disadvantage Example: 0.99 x 0.99 = o.99 X 0.99 --------- 8.91 89.10 + 000.00 ------------- 098.01 This shows that there are a lot of steps to long multiplication and it’s pretty easy to make a simple mistake. Binomial expansion doesn’t have as many steps and there are ways to check if the method is going the way it should be.

4. Level 1-2: Binomial Expansion and Engineering Engineers have to deal with a lot of mechanics and if you didn’t have calculators 100 years ago, without the binomial expansion method it would be very pressuring on engineers to their job because they have to plan and create things we use everyday using long multiplication. They would probably had to have a lot of people check each answer and method to ensure that the answer is correct. With binomial expansion there would be a less chance of getting a wrong answer and if a lot of people still needed to check the results it would be quicker and easier to because the method is so effective. Checking Binomial Expansion Example: (101)2 = (100+1)(100+1) = 10000 + 100+100+1 = 10201 RULE (a+b)2 = (a+b)(a+b) = a2 + ab + ab + b2 = a2 + 2ab + b2 F irst O uter I nner L ast Middle numbers on a perfect square are the same

5. Level 3-4: What Makes the Method Harder Just like long multiplication, working with long numbers take a long time to figure out and there’s more risk of making a mistake. Numbers that have more than 4digits can be difficult to work with especially if each digit is larger than 5 because that way when they are multiplied there is more a chance of getting a 10+. Other sorts of numbers that are hard to multiply are ones with a decimal point because it would take longer to work out because of the different places. Also, when a number has a very high exponent it would make the method big and cumbersome because of the extra multiplying. This would make you think twice about using binomial expansion. When you have to use these types of numbers the method gets more difficult because you might have to use long multiplication as well.

6. Level 3-4: Comparing Long Multiplication and Binomial Expansion with Long Numbers Binomial Expansion Example: Long Multiplication Example: (1231)2 = (1230+1)(1230+1) = 1512900+1230+1230+1 = 1515361 This method might be better for these kinds of numbers because you don’t have to do it all in your head and you can actually write down if you have to carry numbers and the layout is effective because when you add them up each digit is directly under each other (1231)2 1231 x 1231 --------------- 1231 36930 246200 + 1231000 ---------------- 1515361 This method isn’t as effective as it would be with shorter numbers that you can do in your head. In fact, you might actually need to do long multiplication as well as this method to get the answers for the second step. Also, adding the numbers up when the layout goes across the page instead of down it might be more confusing because it would be harder to figure out which places are lined up to each other.

7. Level 5-6: Long Multiplication more Efficient than Binomial Expansion Long multiplication can be more efficient than binomial expansion in some cases. For example, working out equations with numbers that have a lot of digits in your head can be confusing and there’s more of a likeliness of you making errors. When using long multiplication, the downwards layout where each digit is placed directly on top of each other can help and prevent errors or confusion. Also, when you get negative and positive numbers in long multiplication you know to add the negative sign (-) to the answer whereas in the binomial expansion, it might only be one number that has to be negative and this could be easy to forget or miss thus creating errors. Even though the long multiplication method is a longer process it’s easier to mark where you have to carry numbers you can also see each step and how it works. Binomial expansion is more like a version of long multiplication without the physical working out, there for it can be double the work.

8. Level 5-6: Long Multiplication Examples (11)2 1 1 x 1 1 ----------- 1 1 + 1 1 0 ----------- 1 2 1 (-11)2 -11 x -11 ---------- -11 + -110 ---------- -121 This shows the effective layout which can help the working out because the digits are lined up perfectly under each other. This shows how when the negative or positive sign changes, they all change making it easier to pick out mistakes. This also helps mark it.

9. Conclusion In some cases, binomial expansion is the most effective method. If the numbers used are small and you can’t waste time, that would be the best way to solve an equation. On the other hand, if you have a lot of time then the safest method to use is long multiplication. Really, the key is to know which method would be best or more effective to use in situations.