Consider the polynomial function P(x) = -x^3 - mx^2 + 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.
What are the values of m and n tho?
To begin, we can use the Remainder Theorem to determine the relationship between P(x) and the remainder when divided by (x-2).
The Remainder Theorem states that when a polynomial P(x) is divided by (x-k), the remainder is equal to P(k). In this case, the remainder when P(x) is divided by (x-2) is given as 1, so we have:
P(2) = 1
Substituting x=2 into the polynomial P(x), we get:
-2^3 - m(2)^2 + n(2) - 5m = 1
Simplifying the equation, we have:
-8 - 4m + 2n - 5m = 1
-8 - 9m + 2n = 1
We can also determine that (x+1) is a factor of P(x), which means that P(-1) = 0. Substituting x=-1 into P(x), we get:
-(-1)^3 - m(-1)^2 + n(-1) - 5m = 0
Simplifying the equation, we have:
1 - m - n - 5m = 0
1 - 6m - n = 0
Now, we have a system of two equations:
-8 - 9m + 2n = 1
1 - 6m - n = 0
We can solve this system algebraically by eliminating one variable. Adding the two equations together, we eliminate n:
-8 - 9m + 2n + 1 - 6m - n = 1 + 0
-7 - 15m = 1
-15m = 8
m = -8/15
Substituting the value of m into either of the equations, we can solve for n. Let's choose the second equation:
1 - 6(-8/15) - n = 0
1 + 16/5 - n = 0
5/5 + 16/5 - n = 0
21/5 - n = 0
n = 21/5
Therefore, the values of m and n that satisfy the given conditions are m = -8/15 and n = 21/5.
To determine the values of m and n algebraically, we can use the fact that the remainder when P(x) is divided by (x - 2) is 1. This means that if we substitute x = 2 into P(x), the result should be 1.
Let's substitute x = 2 into P(x) and solve for m and n:
P(2) = -2^3 - m(2^2) + n(2) - 5m
P(2) = -8 - 4m + 2n - 5m
Since the remainder is 1, we have:
-8 - 4m + 2n - 5m = 1
Simplifying the equation, we get:
-9m + 2n = 9
Now, we know that (x + 1) is a factor of P(x). This means that if we substitute x = -1 into P(x), the result should be zero.
Let's substitute x = -1 into P(x) and solve for m and n:
P(-1) = -(-1)^3 - m(-1)^2 + n(-1) - 5m
P(-1) = 1 - m + n - 5m
Since (x + 1) is a factor, we have:
1 - m + n - 5m = 0
Simplifying the equation, we get:
-6m + n = -1
So now we have two equations:
-9m + 2n = 9 (Equation 1)
-6m + n = -1 (Equation 2)
We can solve this system of equations to find the values of m and n.
First, let's multiply Equation 2 by 2 to eliminate the variable n:
-12m + 2n = -2 (Equation 3)
Now, we'll subtract Equation 3 from Equation 1 to eliminate n:
-9m + 2n - (-12m + 2n) = 9 - (-2)
-9m + 2n + 12m - 2n = 9 + 2
3m = 11
Dividing both sides of the equation by 3:
m = 11/3
Now, substitute the value of m in Equation 2 to find n:
-6(11/3) + n = -1
-22 + n = -1
n = 21
Therefore, m = 11/3 and n = 21.
P(2) = -8-4m+2n-5m = -9m+2n-8
Since (x+1) is a factor, P(-1) = 0, so 1-m-n-5m=0
So now just solve
-9m+2n-8 = 1
6m+n = 1
That makes P(x) = -x^3 + 1/3 x^2 + 3x + 5/3
= -1/3 (x+1)(3x^2-4x-5)
so clearly, P(-1) = 0
and yes, P(2) = 1