Given that (x^2-1) is a factor of x^3+ax^2-x+b.
When, the expression is divided by (x-2), the remainder is 15, find the values of a and b
To find the values of a and b, we can use the factor theorem to help us.
Since (x^2-1) is a factor of x^3+ax^2-x+b, we can write:
x^3+ax^2-x+b = (x^2-1)(x-k) for some constant k
Expanding the right side, we get:
(x^2-1)(x-k) = x^3 - kx^2 - x + k
Now compare the coefficients of the terms on both sides:
1. For x^3: Coefficient on the left side is 1 (from x^3+ax^2-x+b), and on the right side, it is also 1. So, k = 0.
2. For x^2: Coefficient on the left side is a (from x^3+ax^2-x+b), and on the right side, it is -1. So a = -1.
Now we can substitute k = 0 and a = -1 back into the original expression and the value of b.
x^3+ax^2-x+b = x^3 - x = -15
Comparing coefficients of x, we get:
-1 = -1, which is true.
Lastly, since the remainder when it is divided by (x-2) is 15, substitute x = 2 and equate the equation to 15:
2^3 - 1*2^2 - 2 + b = 15
8 - 4 - 2 + b = 15
2 + b = 15
b = 13
Therefore, the values of a and b are:
a = -1
b = 13