Binomial Expansions Reflection. What is a binomial? A binomial is a mathematical expression of two unlike terms with coefficients and which is raised.

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Presentation transcript:

Binomial Expansions Reflection

What is a binomial? A binomial is a mathematical expression of two unlike terms with coefficients and which is raised to at least the power of 1. Examples of binomials are (a+b), (x+2)^2, (a+c)^3, (y-4)^6. To expand a binomial there are many different methods, long multiplication, Pascal’s triangle and the method that I will be talking about, binomial expansions (the FOIL method).

Pascal's triangle Pascal's triangle looks like this Starting from the first line, that is from the line 1 1, (this stands for an expression raised to the power of 1), the coefficient of the terms after the expansion of the binomial will be 1 1.

Long multiplication Here is an example to explain how to go through this method. for (a+c)^3, we start out by writing (a+c)^3= (a+c)(a+c)(a+c) to give (a+c)^3= (a^2+ 2ac+ c^2)(a+c) to give (a+c)^3= a^3+3a^2c+3ac^2+c^3. And then we have expanded (a+c)^3.

Binomial expansion (FOIL) The foil method is a process used in algebra to multiply two binomials. The word FOIL stands for F=first, O=outer, I=inner, L=last. If you were to expand (2+3)^2 ((2+3)(2+3)) you would use the FOIL method to know which numbers to multiply. So first you would: 2x2+2x3+3x2+3x3. and in the end you would end up with the answer of 25.

Level 1-2 The process becomes harder and time consuming as the exponent of the binomial becomes higher (6th, 10th, 20th power, etc...), for instance like the case of (y- 4)^6. Expanding an expression like this going through the long multiplication process is not only hard and complicated, but also time wasting. Furthermore, there is a greater risk of making great errors and getting the wrong answers. The binomial expansion method would be useful here.

Level 3-4 It would take too much time to work out 13^18 using the binomial expansion method. There should be an easier way to expand numbers with big exponents. By using a calculator, the method becomes easier and faster. When the number you want to expand has reached the 5 th or 6 th decimal place, for example , the method will become very cumbersome, and very long. There is also a big chance that your final answer will be wrong.

Level 5-6 Long multiplication comes useful when expanding numbers with small exponents.