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

Algebra ->  Polynomials-and-rational-expressions -> 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      Log On


   

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) About Me  (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.
a%2Bb-4=0


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.
a-b-18=-12
cross%28a-b-4=0%29
a-b=-12%2B18
a-b=6


System to solve for a and b: system%28a%2Bb=4%2Ca-b=6%29

system%28a=5%2Cb=-1%29

Looking at the cubic result for the first synthetic division, you have %28x-1%29%283x%5E3%2B10x%5E2%2B15x%2B14%29. If you do synthetic division checking x%2B2 or root of -2, you will find remainder 0, meaning x%2B2 is also a root.

f%28x%29=%28x-1%29%28x%2B2%29%283x%5E2%2B4x%2B7%29


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