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Supreme Value Validation (Posted on 2013-09-08) Difficulty: 3 of 5
P1(x) = x2 + (k-29)x - k and, P2(x) = 2x2+(2k-43)x + k, where each of P1(x) and P2(x) is a factor of a cubic polynomial P(x).

Determine the maximum value of k.

No Solution Yet Submitted by K Sengupta    
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Different approach, different answer | Comment 2 of 3 |
P(x) is a cubic and when divided by a quadratic will give a result with x = degree 1.

So I set P(x) = P1(x) * (ax-b) = P2(x) * (cx-d), multiplied them out and equated coefficients of like powers.

The highest degree term and the constant term are easy. ax^3 = 2cx^3 give a = 2c and bk = -dk (and here I assumed k<>0) give b = -d.  

Equating coefficients of x^2 gives d = 5c.  

Plugging in these values and equating coefficients for x gives k = 30.

P1(x) = x^2 + x - 30 = (x+6)(x-5), 
P2(x) = 2x^2 + 17x + 30 = (x+6)(2x+5), 
(ax-b) = c(2x+5), 
(cx-d) = c(x - 5),
and P(x) = c * (2x^3 + 7x^2 - 55x - 150). 

 

  Posted by xdog on 2013-09-09 00:16:00
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