Given the identity x^4-2x^3+ax^2-2=x(x-1)(x+1)Q(x)+4x^2+bx+c , where Q(x)is a polynomial,
(i)state the degree of Q(x)
(ii)find the values of a,b and c ,and hence state the remainder when x^4-2x^3+ax^2-2 is divided by x^3-x .
To find the degree of Q(x), we need to determine the highest power of x that is present in Q(x). By comparing the given identity with the form of polynomial division, we can see that Q(x) is the quotient resulting from dividing x^4 - 2x^3 + ax^2 - 2 by x(x - 1)(x + 1). The remainder is 4x^2 + bx + c.
(i) Since the remainder, 4x^2 + bx + c, has a maximum degree of 2 (power of x), we can conclude that the degree of Q(x) is 4 - 2 = 2.
(ii) To find the values of a, b, and c, we can equate the coefficients of like terms on both sides of the given identity.
Comparing the coefficients of x^4, we have:
1 = 1 * Q(x), which implies Q(x) = 1.
Comparing the coefficients of x^3 on both sides:
0 = -2 * Q(x) - Q(x - 1) - Q(x + 1).
Since we know Q(x) = 1, we can substitute it in:
0 = -2 - Q(x - 1) - Q(x + 1).
Comparing the coefficients of x^2 on both sides:
a = 4 + Q(x) = 4 + 1 = 5.
Finally, comparing the coefficients of x on both sides:
0 = -2(1 - Q(x - 1))(1 + Q(x + 1)).
Therefore, we now have the values of a, b, and c:
a = 5,
b = 0,
c = -2(1 - Q(x - 1))(1 + Q(x + 1)).
To find the remainder when x^4 - 2x^3 + ax^2 - 2 is divided by x^3 - x, we substitute x^3 - x into the expression for the remainder:
R(x) = 4x^2 + bx + c
= 4x^2 + 0x + (-2(1 - Q(x - 1))(1 + Q(x + 1)))
= 4x^2 - 2(1 - Q(x - 1))(1 + Q(x + 1)).
Thus, the remainder when x^4 - 2x^3 + ax^2 - 2 is divided by x^3 - x is 4x^2 - 2(1 - Q(x - 1))(1 + Q(x + 1)).