o divide a function f(x+3) by (x
2+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)? (A) 2 (B) 4 (C) 8 (D) 10
To divide the function f(x+3) by (x^2+2x-3), we can use long division or synthetic division:
(x^2+2x-3) goes into f(x+3) a certain number of times, giving us the quotient Q(x), and there is a remainder of (x+2).
If f(0)=a, then plugging in x=0 in f(x+3), we get:
f(3) = a
If f(4)=b, then plugging in x=4 in f(x+3), we get:
f(7) = b
Since we know that the remainder is (x+2), we can also plug in x=-2 in f(x+3) (because (x+2) divided by (x^2+2x-3) would give us a remainder of (x+2) at x=-2):
f(1) = -2
Now, we can use these three equations to solve for a and b.
First, we can find a by using the equation f(3) = a:
f(3) = a
f(x+3) = Q(x)(x^2+2x-3) + (x+2)
f(3+3) = Q(3)(3^2+2(3)-3) + (3+2)
f(6) = Q(3)(12) + 5
f(6) = 12Q(3) + 5
But we also know that f(x+3) = Q(x)(x^2+2x-3) + (x+2), so we can also plug in x=0 into this equation:
f(0+3) = Q(0)(0^2+2(0)-3) + (0+2)
f(3) = -3Q(0) + 2
Since f(3) = a, we can set these two equations equal to each other:
12Q(3) + 5 = -3Q(0) + 2
Similarly, we can find b by using the equation f(7) = b:
f(7) = b
f(x+3) = Q(x)(x^2+2x-3) + (x+2)
f(7+3) = Q(7)(7^2+2(7)-3) + (7+2)
f(10) = Q(7)(72) + 9
f(10) = 72Q(7) + 9
But we also know that f(x+3) = Q(x)(x^2+2x-3) + (x+2), so we can also plug in x=4 into this equation:
f(4+3) = Q(4)(4^2+2(4)-3) + (4+2)
f(7) = 14Q(4) + 6
Since f(7) = b, we can set these two equations equal to each other:
72Q(7) + 9 = 14Q(4) + 6
Now we have two equations with two variables (Q(3) and Q(7)). Solving these equations will give us the values of Q(3) and Q(7).
After finding the values of Q(3) and Q(7), we can plug these values back into either equation (12Q(3) + 5 = -3Q(0) + 2 or 72Q(7) + 9 = 14Q(4) + 6) to solve for either Q(0) or Q(4).
Finally, we plug the values of Q(0) or Q(4) into the equation f(1) = -2 to solve for a (by plugging x=-2 into f(x+3), we can find f(1)).
Once we have a, we can find b by plugging x=4 in f(x+3), giving us f(7) = b.
Then, we find the sum of a and b: a + b, which gives us the final answer. However, the current information is not sufficient to solve for (a+b), so the answer is (E) insufficient information.