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 1081686: 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 Boreal(15235) (Show Source): You can put this solution on YOUR website!
f(x)=3x^4 + 7x^3 + ax^2 + bx -14
f(1)=0=3+7+a+b-14
a+b=4
f(-1)=-12=3-7+a-b-14
a-b=6
2a=10
a=5
b=1

(x+2) is a factor
k=2
(x^2+x-2) divides into 3x^4 + 7x^3 + ax^2 + bx -14 and that quotient is
3x^2+4x+7
(x-1)(x+2)(3x^2+4x+7)
The roots of the quadratic term are complex
the graph of it is

The original polynomial has two real integer roots and two complex roots.
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