# f is divisible by g . g = ( x - 1 )^3 f = x^4 + ( m - 1 )*x^3 + ( n + 2 )*x^2 + x + p - 3

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f(x) =x^4+(m-1)x^3+(n+2)x^2+x+p^3.

g(x) = (x-1)^3.

f(x) is is divisible by (x-1)^3.

Solution :

Since f(x) the 4th degree is divisible by (x-1)^3, w

we can write, f(x) as product of (x-1)^3 and ax+b

f(x) = g(x) { ax+b) Or

x^4 + ( m - 1 )*x^3 + ( n + 2 )*x^2 + x + p - 3 = (x-1)^3 (ax+b).

This is an identtity, so we can equate coefficients of equal powers on both sides.

x^4 = ax^4. Or a = 1.

constant term p-3 = -b, or -b =3-p

Therefore ,

x^4 + ( m - 1 )*x^3 + ( n + 2 )*x^2 + x + p - 3 = (x-1)^3*(x+3-p)

Now diffrentiate both sides:

4x^3+3(m-1)x^3 +2(n+2)x+1 = 3(x-1)^2 (x+3-p) +(x-1)^3 ..(1)

If x = 0, 1 = 3 (-1)^2(3-p)+(-1)^3 = 3(3-p) -3 = 9-3p-3 = 6-p.

1= 6-p.

p = 6-1 = 5.

We rewrite (1) with p = 5.

4x^3+3(m-1)x^2+2(n+2)x+1 = 3(x-1)^2(x-2)+(x-1)^3...........(2).

Put x = 1, then 4+3(m-1)+2(n+2)+1 = 0

4+3m-3 +2n+4+1 = 0

3m+2n = -6.................(3)

Differentiate both sides of (2):

{4x^3+3(m-1)x^2+2(n+2)x+1}' = {3(x-1)^2(x-2)+(x-1)^3}'

12x^2 + 6(m-1)x +2(n+2) = {(x-1)^2 (x-1+3x-6)}' = {(x-1)^2*(4x-7)}' = 2(x-1)(4x-7) +(x-1)^2*(4)

12x^2+6(m-1)x +2(n+2) = (x-1){ 2(4x-7)+4(x-1)} = (x-1)(12x-18)

12x^2+6(m-1)x +2(n+2) = 12x^2 -30x +18

Therefore 6(m-1) = -30 and 2(n+2) = 18

m-1 = -30/6 = -5

m= -5+1 = -4.

2(n+2) = 18

n+2= 18/2 = 9

n = 9-2 = 7.

We notice that f is not defined, because we don't know the coefficients of it's expression.

We know that f(x) is divided by g(x)=(x-1)^3.

We'll write the rule of division with reminder:

f = g*c + r

If f is divided by g, that means that the reminder is 0.

f = g*c

We'll calculate the root of g = ( x - 1 )^3

( x - 1 )^3 = 0

x - 1 = 0

x = 1

We notice that if we'll substitute the roots of g(x) in the expression of f(x), we'll clear f(x).

We alsonotice that the value "1" is the multiple root, of 3rd order, of g(x), meaning that:

f(1)=0 <=> f(1)=1+m-1+n+2+1+p-3=0 <=> m+n+p=0

f'(1)=0 <=> f'(1)=4*1+3*(m-1)+2*(n+2)+1=0<=>3m+2n=-6

f"(1)=0<=>f"(1)=12*1+6*(m-1)+2*(n+2)=0<=>6m+2n=-10

f"'(1) different from 0

3m+2n-6m-2n=-6+10

We'll combine like terms from the left side:

-3m=4

We'll divide by -3:

**m=-4/3**

3(-4/3)+2n=-6

-4+2n=-6

2n=-6+4

2n=-2

We'll divide by 2:

** n=-1**

m+n+p=0

(-4/3)+(-1)=-p

**p=7/3**

**f(x) = x^4 + ( -4/3 - 1 )*x^3 + ( -1 + 2 )*x^2 + x + 7/3 - 3**

**f(x) = x^4 + -7x^3/3 + x^2 + x - 2/3**