Hello,
I've done a few remainder theorem problems in a workbook involving a single factor, i.e., "If polynomial P(x) is divided by x-r, its remainder is P(r)." In such a case, finding the remainder is easy. I need help applying "When polynomial P(x) is divided by D(x), it can be expressed as P(x) = Q(x)*D(x)+R(x)."
The problem at hand is:
A polynomial P(x) has a remainder of 2 when divided by (x-1) and a remainder of -2 when divided by (x+1). If P(x) is divided by (x-1)(x+1), then the remainder is
A. 2
B. x+2
C. 2x+1
D. 2x
E. 2x-1
The solution in my workbook states that the answer is D, where
P(x) = Q(x)(x-1)(x+1) + a(x-1) + 2
P(-1) = -2 = a(-1 -1) +2
a = 2
It then states that
The remainder is 2(x-1) + 2 = 2x when P(x) is divided by (x-1)(x+1)
Could someone help me understand what's going on here? Where is the "a" coming from in the solution? (I understand that the remainder must be first degree since the divisor is a second degree polynomial).
Much appreciated,
ikegrd
I've done a few remainder theorem problems in a workbook involving a single factor, i.e., "If polynomial P(x) is divided by x-r, its remainder is P(r)." In such a case, finding the remainder is easy. I need help applying "When polynomial P(x) is divided by D(x), it can be expressed as P(x) = Q(x)*D(x)+R(x)."
The problem at hand is:
A polynomial P(x) has a remainder of 2 when divided by (x-1) and a remainder of -2 when divided by (x+1). If P(x) is divided by (x-1)(x+1), then the remainder is
A. 2
B. x+2
C. 2x+1
D. 2x
E. 2x-1
The solution in my workbook states that the answer is D, where
P(x) = Q(x)(x-1)(x+1) + a(x-1) + 2
P(-1) = -2 = a(-1 -1) +2
a = 2
It then states that
The remainder is 2(x-1) + 2 = 2x when P(x) is divided by (x-1)(x+1)
Could someone help me understand what's going on here? Where is the "a" coming from in the solution? (I understand that the remainder must be first degree since the divisor is a second degree polynomial).
Much appreciated,
ikegrd