Binomial expansions :Math Reflection By. Annabel Diong 8C

Introduction - A binomial is polynomial expression with 2 Terms. (^ = square) e.g. (×+y), (×^2+1), (×^4-3×) -Binomial expansion refers to a formula by which one can “expand out” expressions like (×+y^2) or (3×+2^5). Where the entire binomial is raised to some power.

General rule for expanding binomials ((a+b)^2=a^2+2ab+b^2) or ((a-b)^2=a^2-2ab +b^2) For example: (0.99)^2 a= 1 b= 0.01 *substitute a,b into formula. (0.99)^2=(1-0.01)(1-0.01)= 1^2-2x1x ^2

Level 1-2 If you were an engineer 100 years ago, explain how our method may have been useful rather than just using long multiplication? -It would have been more useful because the people in the olden times who did not know this formula, and therefore would have to solve the equations presented to them using long multiplication, and would have more likely to make mistakes during the process of expanding. But the math’s world has evolved making it such that there are a number of different ways of finding a solution to a problem.

Level 1-2 continued.. Flaws of long multiplication... A 3 decimal place even more tedious ( as you can see, the 2 decimal place is quite tedious)

Level 1-2 continued… Using the F.O.I.L.( first. Outer. inner. First.) method. (^ = square) (0.99)^2=(1-0.01)(1-0.01)=1^2-2x1x ^2 = (0.99)^3 = (1-0.01)(1-0.01)(1-0.01) = 1^3 – 2×1× ^3 = As we can see, the binomial expansion method is much easier and less tedious to use. It Breaks down the equation into small pieces making it even clearer and easier to solve, compared to the long division method which may allow more room for errors.

Level 3-4 At what point would our method be big and cumbersome?( how many decimal places or What sorts of numbers would make us think twice about using this method?) This depends on how skilled you are, and how far you knowledge can take you, but I would say up to maybe 5 or more decimals places would get quite complicated, but it also depends on The type of number that you use, e.g compared to 0.59 As you can see from the equation, simplifying an equation like this is much easier compared to a number like 0.59

Level 3-4 continued… Also when the exponents are big, e.g. ((×+y)^5=…..). That’s when you would use “Pascal’s Triangle” the numbers shown it the triangle show the coefficients and the numbers beside the triangle ( as seen below) are the exponent’s. e.g. ((×+y)^4=(× +y) (×+y)^3=×^4+4×^3y+6×y^2+4×y^3+y^4). The pattern is that the one’s on the far right and left remains the same, while you add the pair of values from the previous row to get the next, for example, the next row will be , the next after that will be ,and so on..

Level 5-6 Can you give us some detailed explanations and examples of where long multiplication is more efficient than our expansions method? Long multiplication can be is more efficient when the numbers are Accessible, e.g.( 0, 1,2…) therefore easier and faster to multiply than Expand.

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