Simple binomial: (1+0.5x)^4

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Expand (1+0.5x), simplifying the coefficients. 

1. Draw Pascal's triangle to find the coefficients

            1

     1     2     1

   1    3     3     1 

1    4    6     4     1 

As you can see, each row starts and ends with 1. The numbers in between are worked out by adding the two numbers on top. 

For this question, we will be using the 1 4 6 4 1 row because the expression is raised to the power of 4. This expansion will have 5 expressions. 

2.  For each term, both 1 and 0.5 are raised to powers 0 to 4, where the sum of the powers always adds up to 4. In addition, the power of x is increased from 0 to 4 as the term progress. 

(1+0.5x)

1(1)4(0.5)0x04(1)3(0.5)1 x1 + 6(1)2(0.5)x24(1)1(0.5)3 x3+ 1(1)0(0.5)x4

First, we raising 1 to the power of 4, therefore 0.5 is raised to the power of 0. For the next term, the power of 1 decreased by 1 and the power of 0.5 increases by 1, so that the sum of the terms still equates to 4. This is done until we get 5 terms in total. 

3.  The expression can be simplified as followed: 

= 1 + 4(1/2)x + 6(1/4)x+  4(1/8)x3 + (1/16)x

= 1 + 2x + 3/2x+ 1/2x+ 1/16x4

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