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October 10th, 2012, 08:28 AM  #1 
Newbie Joined: Sep 2012 Posts: 8 Thanks: 0  Find f(x) from the following information
Let f(x) be a polynomial of degree less than or equal to 5 which leaves remainders 1 and 1 respectively upon division by and . Find f(x).

October 10th, 2012, 03:23 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 19,282 Thanks: 1680 
f(x) = (3x^5 + 10x^3  15x)/8

October 10th, 2012, 04:00 PM  #3  
Newbie Joined: Oct 2012 Posts: 1 Thanks: 0  Quote:
 
October 10th, 2012, 05:38 PM  #4 
Global Moderator Joined: Dec 2006 Posts: 19,282 Thanks: 1680 
If you meant the divisor to be (x + 2) rather than just x, see the division below. x^2 + 3x  1 _________________ x + 2  x^3 + 5x^2 + 5x + 19 x^3 + 2x^2  3x^2 + 5x 3x^2 + 6x  x + 19 x  2  21 
October 11th, 2012, 04:51 AM  #5  
Newbie Joined: Sep 2012 Posts: 8 Thanks: 0  Re: Quote:
 
October 11th, 2012, 04:58 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 19,282 Thanks: 1680 
I couldn't spot any simple method, so I wrote f(x) = (ax² + bx + c)(x  1)³  1 = (px² + qx + r)(x + 1)³ + 1. On expanding and equating coefficients, one has six equations for the six coefficients of the quadratics. Solving these equations allows f(x) to be found.

October 11th, 2012, 05:14 AM  #7  
Newbie Joined: Sep 2012 Posts: 8 Thanks: 0  Quote:
 
October 11th, 2012, 05:27 AM  #8 
Global Moderator Joined: Dec 2006 Posts: 19,282 Thanks: 1680 
In general, the remainder on dividing a function f(x) by (x  a)^n would be a nonconstant function of x. If that function is R(x), you would know that R(a) = f(a).


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