SOLUTION: given that x=3+i is a root of p(x)=3x^4+ax^3+34x^2+bx-20 find the value of a and b if when p(-1)=34. then factor the polynomial completely

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: given that x=3+i is a root of p(x)=3x^4+ax^3+34x^2+bx-20 find the value of a and b if when p(-1)=34. then factor the polynomial completely      Log On


   

Question 1167586: given that x=3+i is a root of p(x)=3x^4+ax^3+34x^2+bx-20 find the value of a and b if when p(-1)=34. then factor the polynomial completely
Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


The information p(-1)=34 is not needed to solve the problem....

Since x=3+i is a root, x=3-i is another root. The quadratic factor corresponding to those two roots is x%5E2-6x%2B10

Given the leading coefficient and constant term of the polynomial, we know the factorization of the polynomial is of the form

%28x%5E2-6x%2B10%29%283x%5E2%2Bnx-2%29

Performing that multiplication yields

3x%5E4%2B%28n-18%29x%5E3%2B%2828-6n%29x%5E2%2B%2810n%2B12%29x-20

We can solve for n knowing that the coefficient of the quadratic term is 34

28-6n+=+34
6n+=+-6
n+=+-1

We can then find the values of a and b, which are the coefficients of the cubic and linear terms of the polynomial:

a = n-18 = -19
b = 10n+12 = 2

ANSWERS: a = -19; b = 2