Alan

As follows:

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I knew I'd gone wrong somewhere as I did a Monte Carlo simulation which consistently came out at 0.325, rather than 0.324, (i.e. 421/1296, not 420/1296).  However, I couldn't find the extra 1 in the analytical solution!  Thanks for finding it for me Melody!

Probability that the first card is a king of spades, clubs or diamonds, followed by a heart is p1 = 3/52*13/51

Probability that the first card is the king of hearts, followed by a heart is p2 = 1/52*12/51

Total probability = p1 + p2 = (3*13 + 1*12)/(52*51) = 51/(52*51) → 1/52

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Modifying Melody's result a little we have:

1 * * *                     1*4*4*4 = 64 ways, but 4 possible positions for the 1, so 64*4 = 256 ways

1 1 * *                     1*1*4*4 = 16 ways, but 6 possible combinations of the two 1's, so 16*6 = 96 ways

1 1 6 *                     1*1*1*4 = 4  ways, but 6 possible positions for the 1's

and two possible remaining positions for the 6,  so 4*6*2 = 48 ways

1, 1, 1, any              1*1*1*5 = 5 ways, but 4 possible positions for the three 1's, so 5*4 = 20 ways

                                  total   =  256 + 96 + 48 + 20 =  420 ways

Hence probability = 420/1296 → 35/108 ≈ 0.324

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(-1,5):   a(-1)²+b(-1)+c = 0 or a - b + c = 0

(0,5):    a(0)²+b(0)+c = 5 or c = 5

(5,0):   a(5)²+b(5)+c = 0 or 25a + 5b +c = 0

So: 

a - b + 5 = 0.        (1)

25a + 5b + 5 = 0. (2) 

Multiply (1) by 5 and add to (2):

30a + 30 = 0.  So a = -1 

Hence, From (1): 

-1 - b + 5 = 0. So b = 4

a = -1, b = 4, c = 5

I’ll leave you to calculate 100a + 10b + c

I don’t think it says division gives a polynomial Melody! There is a full stop (period) between the function definition and the word Give (which also starts with a capital letter), so I interpreted the sentence beginning with Give as an instruction to the reader, not a continuation of the previous sentence.

Here are some pointers to the right direction. I’ll leave you to fill in the details:

(a)  b = 10*cos(130°)*i + 10*sin(130°)*j

      w = 5*cos(45°)*i + 5*sin(45°)*j

(b) v = b + w.   (add corresponding terms in i and j)

(c) If v = r*i + s*j then |v| = sqrt(r²+s²) and angle = tan^-1(s/r)

The question doesn’t say that f(x) is a polynomial; it just asks for g(x) to be a polynomial, and for the limit of the sum of the two functions to have a zero asymptote.

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"Let f(x) = 3(x^4+x^3+x^2+1)/(x^2+x-2). Give a polynomial g(x) so that f(x) + g(x) has a horizontal asymptote of 0 as x approaches positive infinity."

This graph should help:

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