To divide the function f(x-1) by (x^2+2x-3), we find 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 b/a? (A)5 (B)4 (C3 (D)2
I don't understand the logic behind this
To find the value of b/a, we need to substitute the values of f(0) and f(-4) into the expression.
We are given that f(x-1) divided by (x^2+2x-3) results in a quotient Q(x) and a remainder of (x-2). This can be represented as:
f(x-1)/(x^2+2x-3) = Q(x) + (x-2)/(x^2+2x-3)
Let's substitute x=0 and x=-4 into this expression and simplify to find the values of f(0) and f(-4).
For x = 0:
f(0-1)/(0^2+2(0)-3) = Q(0) + (0-2)/(0^2+2(0)-3)
f(-1)/(0+0-3) = Q(0) + (-2)/(-3)
f(-1)/(-3) = Q(0) + 2/3
f(-1) = -3Q(0) + 2/3
For x = -4:
f(-4-1)/((-4)^2+2(-4)-3) = Q(-4) + (-4-2)/((-4)^2+2(-4)-3)
f(-5)/((-4)^2-8-3) = Q(-4) + (-6)/((-4)^2-8-3)
f(-5)/((-4)^2-8-3) = Q(-4) - 6/((-4)^2 - 8 - 3)
f(-5)/5 = Q(-4) - 6/5
Now, let's compute the value of b/a, which is f(-4)/f(0).
b/a = (f(-5)/5) / (f(-1))
To simplify further, we substitute the expression for f(-5) and f(-1) that we found earlier:
b/a = ((-3Q(0) + 2/3)/5) / (-3Q(0) + 2/3)
Now, we can cancel out the common factors:
b/a = (-3Q(0) + 2/3) / (-3Q(0) + 2/3)
b/a = 1
Therefore, the value of b/a is 1.
In this case, none of the options (A)5, (B)4, (C)3, or (D)2 is the correct answer.
Geez - just like the other one.
f(x-1) = (x-1)^2 + 4(x-1)
so f(x) = Q(x) * x(x+4) + x-2
Now do this just like the other
f(0) = 0-2 = -2
f(-4) = -4-2 = -6