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Find the coefficient of x^3 in (1 + x)^4 (1 + x^2)^5.

 May 14, 2020
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Using combinations,

    the first factor becomes:   (1 + x)4  =  1 + 4x + 6x2 + 4x3 + x4

    the second factor becomes:  (1 + x2)5  =  1 + 5x2 + 10x4 + 10x6 + 5x8 + 1

 

The only way that you can get an x3 term is to

    multiply the 4x in the first factor times the 5x2 in the second factor to get 20x3 

    and to multiply the 4x3 in the first factor times the 1 in the second factor to get 4x3

 

Adding these two terms together:  20x3 + 4x3  =  24x3

 May 14, 2020

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