For the years between 1980 and 2012, the function c(x)=0.006x^3−0.48x^2+12.8x+85 approximates the number of countries that attended the summer Olympics x years after 1980. Use the polynomial remainder theorem to approximate the number of countries that attended the Olympics in 2004.

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  199

Step-by-step explanation:

2004 is 24 years after 1980. The polynomial remainder theorem tells you the remainder from division by x-24 will be the function value c(24). We can find the remainder/function value by evaluating the function rewritten to Horner form:

  c(x) = ((0.006x -0.48)x +12.8)x +85

  c(24) = ((0.006(24) -0.48)(24) +12.8)(24) +85 = (-.336(24) +12.8)(24) +85

  = 4.736(24) +85 = 198.664

  c(24) ≈ 199

The approximate number of countries attending the 2004 Olympics was 199.

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step-by-step explanation: