SOLUTION: Let f(x)=3x^4 + 7x^3 + ax^2 + bx -14 where a and b are constants.If (x-1) is a factor of f(x) and when f(x) is divided by (x+1), the remainder is -12, find the values of a and b. W

Question 1081177: Let f(x)=3x^4 + 7x^3 + ax^2 + bx -14 where a and b are constants.If (x-1) is a factor of f(x) and when f(x) is divided by (x+1), the remainder is -12, find the values of a and b. With these values of a and b,
(A) find a factor of f(x) in the form x+k where k is a postive integer.
(B) write f(x) in the form
f(x)=(x-1)(x+k)Q(x),where Q(x)is a real quadratic.
Hence,show that Q(x) is irreducible.
Answer by josgarithmetic(39623) (Show Source): You can put this solution on YOUR website!
x+1 is a factor.

 1    |    3   7   a   b    -14
      |
      |---------------------------

         3  10   10+a  b+a+10  a+b-4


Remainder must be equal to zero.

Division by x+1 gives remainder of -12.

 -1   |    3   7   a   b    -14
      |
      |---------------------------
          3  4   a-4  b-a+4  a-b-18

Remainder is given as -12.

System to solve for a and b:

Looking at the cubic result for the first synthetic division, you have . If you do synthetic division checking or root of , you will find remainder 0, meaning is also a root.

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