The polynomial function P(x) = 4x^4 - 7x^3 + mx^2 + nx + 6 has (x-1) as one of its factors. When it is divided by (x+1), the remainder is 30. Algebraically determine the values of m and n.
+179 If P(x) is divisible by (x-1), then it can be written as (x-1)g(x), where g(x) is a polynomial of degree 3.
Letting x=1 in the equation P(x)=(x-1)g(x), we have P(1)=0, or 4-7+m+n+6=0, simplifying to m+n = -3
If P(x) leaves a remainder of 1 when divided by (x+1), then it can be written as (x+1)q(x)+30, where q(x) is a polynomial of degree 3.
Thus, letting x= -1, we have P(-1) = 30, or 4+7+m-n+6=30, simplifying to m-n = 13
We have the two systems of equations,
m+n = -3
m-n = 13.
m=5
n= -8
+128448 If (x - 1) is a factor......then x = 1 is a root.....therefore
4(1)^4 - 7(1)^3 + m(1)^2 + n(1) + 6 = 0
4 - 7 + 6 + m + n = 0
3 + m + n = 0
m + n = -3 (1)
And by the Remainder Theorem
P(-1) = 30.....so......
4(-1)^4 - 7(-1)^3 + m(-1)^2 + n(-1) + 6 = 30
4 + 7 + 6 + m - n = 30
17 + m - n = 30
m - n = 13 (2)
Add (1) and (2) and we get that
2m = 10
m = 5
And m + n = -3 ⇒ n = -8
