SOLUTION: “A quartic function in the form f(x) = ax^4 + bx^3 + cx^2 + dx + e is such that the coefficient of the quadratic and linear terms are 10 and -18 respectively. Additionally, f(0)

Algebra ->  Functions -> SOLUTION: “A quartic function in the form f(x) = ax^4 + bx^3 + cx^2 + dx + e is such that the coefficient of the quadratic and linear terms are 10 and -18 respectively. Additionally, f(0)      Log On


   

Question 1147780: “A quartic function in the form f(x) = ax^4 + bx^3 + cx^2 + dx + e is such that the coefficient of the quadratic and linear terms are 10 and -18 respectively. Additionally, f(0) =9 and x=1 is a root with a multiplicity of two. What is the value of (a +b)?
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


The values of c and d are given in the statement of the problem; and f(0)=9 tells us e is 9. So the function is

ax%5E4%2Bbx%5E3%2B10x%5E2-18x%2B9

Synthetic division knowing that x=1 is a root then gives us the answer to the question that is asked. The fact that x=1 is a root of multiplicity 2 is not needed.

   1  |  a     b     10     -18     9
      |
      |        a    a+b   a+b+10  a+b-8
      +---------------------------------
         a    a+b  a+b+10 a+b-8   a+b+1

Since 1 is a root, a%2Bb%2B1+=+0, which means a%2Bb+=+-1

ANSWER: a+b = -1

Answer by ikleyn(52776) About Me  (Show Source):
You can put this solution on YOUR website!
.

It is clear from the condition that c= 10, d= -18  and  e= f(0) = 9.


So, the polynomial is  f(x) = ax^4 + bx^3 + 10x^2 - 18x + 9.


Now, substitute x= 1 into the last expression for the polynomial and take into account that x= 1 is the root.


You will get
   

    f(1) = a + b + 10 - 18 + 9 = 0,


which implies


    a + b = -10 + 18 - 9 = -1.    ANSWER

Solved.