SOLUTION: If F(x) = ax²+bx+c, and F(x+5) = x²+9x-7 What is the sum of a+b+c?

Algebra ->  Functions -> SOLUTION: If F(x) = ax²+bx+c, and F(x+5) = x²+9x-7 What is the sum of a+b+c?      Log On


   

Question 1149396: If F(x) = ax²+bx+c, and F(x+5) = x²+9x-7
What is the sum of a+b+c?
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


F%28x%2B5%29+=+a%28x%2B5%29%5E2%2Bb%28x%2B5%29%2Bc

= a%28x%5E2%2B10x%2B25%29%2Bb%28x%2B5%29%2Bc

= %28a%29x%5E2%2B%2810a%2Bb%29x%2B%2825a%2B5b%2Bc%29

Given F(x+5) = x^2+9x-7, equate the coefficients of each term:

quadratic coefficients: a+=+1
linear coefficients: 10a%2Bb+=+9
constants: 25a%2B5b%2Bc+=+-7

Solve that system of 3 equations to find the values of a, b, and c to answer the question.

Answer by ikleyn(52797) About Me  (Show Source):
You can put this solution on YOUR website!
.

The sum a + b + c  of the coefficients of the polynomial  F(x) = ax^2 + bx + c  is equal

to the value of the polynomial at x= 1.



Therefore, to get the value of the sum,  calculate F(1).

To do it, put x= -4 in the expression F(x+5) = x^2 +9x - 7.



You will get then


    a + b + c = F(1) = F(-4+5) = (-4)^2 + 9*(-4) - 7 = 16 - 36 - 7 = -27.    ANSWER

Solved.

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Every time, when the problem asks you about the sum of the coefficients of a polynomial,

            REMEMBER  (!)

that this sum is the value of the polynomial at x= 1.



Every time, when the problem asks you about the alternate sum of the coefficients of a polynomial,

            REMEMBER  (!)

that this sum is the value of the polynomial at x= -1.