Consider the polynomial function P(x)=-x3-mx2+nx-5m. The remainder when P(x) is divided by (x-2) is 1 and (x+1) is a factor of P(x). Determine m and n algebraically. Show all your work for full marks.
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To determine the values of m and n, we can use the given information that the remainder when P(x) is divided by (x-2) is 1 and (x+1) is a factor of P(x).
1. Remainder when P(x) is divided by (x-2) is 1:
When dividing a polynomial by (x-a), where a is a constant, the remainder is given by substituting the value of a into the polynomial. So, substituting x = 2 into P(x) gives us:
P(2) = -2^3 - m(2^2) + n(2) - 5m
Simplifying this expression:
-8 - 4m + 2n - 5m = 1
-9m + 2n - 8 = 1
-9m + 2n = 9 ... Equation (1)
2. (x+1) is a factor of P(x):
If (x+1) is a factor of P(x), then when we divide P(x) by (x+1), the remainder should be zero. Substituting x = -1 into P(x) gives us:
P(-1) = -(-1)^3 - m(-1)^2 + n(-1) - 5m
Simplifying the expression:
1 - m - n - 5m = 0
-6m - n + 1 = 0
-6m - n = -1 ... Equation (2)
We now have a system of two equations (Equation 1 and Equation 2) with two unknowns (m and n). We can solve this system to find the values of m and n.
Using the method of substitution, we can solve Equation (2) for n in terms of m:
-6m - n = -1
n = -6m + 1
Substituting this expression for n into Equation (1):
-9m + 2(-6m + 1) = 9
-9m - 12m + 2 = 9
-21m = 7
m = -7/21
Simplifying:
m = -1/3
Now, substituting the value of m into Equation (2):
-6(-1/3) - n = -1
2 - n = -1
n = 2 + 1
n = 3
Therefore, the values of m and n are m = -1/3 and n = 3, respectively.