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polynomial functions

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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.

Jun 4, 2021

#1
+3

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

Jun 4, 2021
#2
+1

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   Jun 4, 2021
edited by CPhill  Jun 4, 2021