SOLUTION: Expand (1/2-2x)^5 up to the term in x^3. If the coefficient of x^2 in the expansion of (1+ax+3x^2)(1/2-2x)^5 is 13/2, find the coefficient of x^3.

Question 1185961: Expand (1/2-2x)^5 up to the term in x^3. If the coefficient of x^2 in the expansion of (1+ax+3x^2)(1/2-2x)^5 is 13/2, find the coefficient of x^3.
Answer by greenestamps(13198) (Show Source): You can put this solution on YOUR website!

(1) Expand (1/2-2x)^5 up to the term in x^3

+ ...

ANSWER: + ...

(2) Find a, given that the coefficient of x^2 in is 13/2

The contributions to the x^2 coefficient come from the constant term in the first polynomial times the coefficient of x^2 in the second, the coefficient of the x term in the first polynomial times the coefficient of the x term in the second, and the coefficient of the x^2 term in the first polynomial times the constant term in the second:

ANSWER: a=-9/4

(3) Find the coefficient of x^3 in (1+ax+3x^2)(1/2-2x)^5

The contributions to the x^3 coefficient come from the constant term in the first polynomial times the coefficient of the x^3 term in the second, the coefficient of the x term in the first polynomial times the coefficient of the x^2 term in the second, and the coefficient of the x^2 term in the first polynomial times the coefficient of the x term in the second:

ANSWER: -265/8

The answers are confirmed with this completely expanded polynomial as calculated on wolframalpha.com:

1/32 - (89 x)/128 + (13 x^2)/2 - (265 x^3)/8 + 100 x^4 - 182 x^5 + 192 x^6 - 96 x^7

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