A-Math Remainder Theorem

• Jt061952

  15 May 09, 16:14

  I am trying to work on this problem:

  Find the value of a and b for which x^2 + 4x -1 is a factor of x^3 + ax^2 - 5x  + b

  The roots of x^2 + 4x -1 are 0.236.... and 4.236... Because of these decimal numbers, I cannot get a and b as whole numbers.  a = 3, b = 1.  Is there a better way?  Thanks. jt

• Forbiddensinner

  15 May 09, 16:53

  Originally posted by Jt061952:

  I am trying to work on this problem:

  Find the value of a and b for which x^2 + 4x -1 is a factor of x^3 + ax^2 - 5x  + b

  The roots of x^2 + 4x -1 are 0.236.... and 4.236... Because of these decimal numbers, I cannot get a and b as whole numbers.  a = 3, b = 1.  Is there a better way?  Thanks. jt

  Divide ( x^3 + ax^2 - 5x + b ) by ( x^2 +4x - 1 ) using long division method.

  You should get quotient ( x + a - 4 ) and a negligible remainder, as it won't help you in solving the problem.

  Next, you multiply the quotient ( x + a - 4 ) with the divisor ( x^2 + 4x -1 ), and you should get ( x^3 + ax^2 + (4a-17)x + 4 -a ).

  Finally, use the new cubic equation you have and equal it to the original cubic equation, and after simplifying the numbers on both side ( NOTE : DO NOT BRING ANYTHING OVER FROM EITHER SIDE ), you should get:

  -5x + b = (4a - 17)x + (4-a)

  Now, you can directly compare coefficients to get

  -5 = 4a - 17 and b = 4 - a

  Hence, a=3 and b=1.

   

• nightzip

  15 May 09, 17:00

  If x^2 + 4x -1 is a factor of x^3 + ax^2 - 5x + b, then finding roots for x^2 + 4x -1, using discriminant formula, x=-2 +/- sqrt (5) Let f(-2+sqrt(5))= (-2+sqrt(5))^3 + a(-2+sqrt(5))^2 - 5(-2+sqrt(5)) + b = 0 Similarly, for f(-2-sqrt(5)) = (-2-sqrt(5))^3 + a(-2-sqrt(5))^2 - 5(-2-sqrt(5)) + b = 0 Since two equations, two unknowns, you can solve for a and b.
• Jt061952

  15 May 09, 17:26

  Thanks, that was quick.  Now I know how to do both methods.  Appreciate it.

• wee_ws

  15 May 09, 18:23

  Hi,

    There is an even faster approach:

    (x^2 + 4x - 1)(x - b) = x^3 + ax^2 - 5x + b.

    Comparing coefficients of x^2, we obtain -b + 4 = a -- (1).

    Comparing coefficients of x, we obtain -4b - 1 = -5 -- (2).

    Now we can solve for a and b.

    Thanks!

  Cheers,
  Wen Shih

• Forbiddensinner

  15 May 09, 18:38

  Originally posted by wee_ws:

  Hi,

    There is an even faster approach:

    (x^2 + 4x - 1)(x - b) = x^3 + ax^2 - 5x + b.

    Comparing coefficients of x^2, we obtain -b + 4 = a -- (1).

    Comparing coefficients of x, we obtain -4b - 1 = -5 -- (2).

    Now we can solve for a and b.

    Thanks!

  Cheers,
  Wen Shih

  Nice, as expected from a homework forum mod.

  To the TS, the reason why this method can be used here is because the original cubic equation did not contain any other numbers besides 'b', hence wee made use of this to determine (x-b) as one of the roots.

• Jt061952

  15 May 09, 18:48

  Thanks Wen Shih, your method is more elgant.  I find substituting sqrt to be very tedious.  For a 5 marks problem, it should not take that long. 

• nightzip

  15 May 09, 22:07

  Wow Wen Shih, that a very good solution. However, didnt quite understand how (x-b) can be so quickly seen as a root?
• Lee012lee

  16 May 09, 00:35

  Originally posted by Jt061952:

  I am trying to work on this problem:

  Find the value of a and b for which x^2 + 4x -1 is a factor of x^3 + ax^2 - 5x  + b

  The roots of x^2 + 4x -1 are 0.236.... and 4.236... Because of these decimal numbers, I cannot get a and b as whole numbers.  a = 3, b = 1.  Is there a better way?  Thanks. jt

  Purepls aka Jt061952 huh ?

  Same question asked twice.

  Once, the question is asked in the thread "AMaths" by Purepls

  Again, the same question in this thread "AMaths Remainder Theorem" by
  Jt061952 .

  -------------------------------------------------------------------------------------

  (1)   Find the value of a and b for which x^2 - 2x - 3 is a factor of

         ax^3 - 8x^2 +bx + 6

  (2)   Find the value of a and b for which x^2 + 4x -1 is a factor of

         x^3 + ax^2 - 5x  + b

  (3)   Find the value of a and b for which x^2 + x + 4 is a factor of

         x^3 + ax^2 - 5x  + b

  --------------------------------------------------------------------------------

  All three questions look alike.

  Using Nightzip's method, it will be lengthy to solve question 2 and the answers of a and b will not be exact too. While nightzip's method will not be able to solve  question 3.

  Mr Wee and Mikethm method can solve question 3, 2 and 1.

   

• wee_ws

  16 May 09, 12:45

  Originally posted by nightzip:

  Wow Wen Shih, that a very good solution.
  However, didnt quite understand how (x-b) can be so quickly seen as a root?

  Hi,

    Thanks for your encouragement :)

    (x - b) has to be a factor because the expression has b as a constant.

    Thanks again.

  Cheers,
  Wen Shih