to divide a function f(x+3) by (x2+2x-3), we see that the quotient is Q(x) and the remainder is
(x+2). If f(0)=a and f(4)=b, what is the value of (a+b)?
To divide a function f(x+3) by (x^2+2x-3), we can use polynomial long division. The quotient Q(x) would be the result of the long division, and the remainder is (x+2).
Given that f(0) = a, we can substitute x = 0 into the original function:
f(0+3) = f(3) = a
Given that f(4) = b, we can substitute x = 4 into the original function:
f(4+3) = f(7) = b
Therefore, we have f(3) = a and f(7) = b.
By using the quotient and remainder from the division, we can write: f(x+3) = Q(x) * (x^2+2x-3) + (x+2)
To find the value of (a+b), we need to evaluate f(3) and f(7) using the given information:
f(3) = Q(3) * (3^2+2*3-3) + (3+2)
f(3) = Q(3) * (9+6-3) + 5
Since the remainder is (x+2), we can substitute x = -2 into the dividend:
f(3) = Q(3) * (9+6-3) + (3+2)
f(3) = Q(3) * (12) + 5
Similarly,
f(7) = Q(7) * (7^2+2*7-3) + (7+2)
f(7) = Q(7) * (49+14-3) + 9
Therefore, we now have the equations:
f(3) = Q(3) * (12) + 5 = a
f(7) = Q(7) * (60) + 9 = b
To find (a+b), we need to substitute Q(3) and Q(7) with their respective values in terms of a and b:
a = Q(3) * (12) + 5
Q(3) = (a-5)/12
b = Q(7) * (60) + 9
Q(7) = (b-9)/60
Substituting Q(3) and Q(7) into the equation for (a+b):
(a+b) = ((a-5)/12) * (12) + 5 + ((b-9)/60) * (60) + 9
(a+b) = a - 5 + 5 + b - 9 + 9
(a+b) = a + b
Therefore, the value of (a+b) is (a + b).