Answer:
(x-4)(5x³+18x²+54x+215)+860
Step-by-step explanation:
This follows that when the fourth degree polynomial is divided by the monomial(x-4),it gives a third degree polynomial and some remainder r
That is:
5x⁴-2x³-18x²-x/x-4=(ax³+bx²+cx+d)+r
if so,then
5x⁴-2x³-18x²-x=(x-4)(ax³+bx²+cx+d)+r
5x⁴-2x³-18x²-x=ax⁴+bx³+cx²+dx-4ax³-4bx²-4cx-4d+r
5x⁴-2x³-18x²-x=ax⁴+x³(b-4a)+x²(c-4b)+x(d-4c)-4d+r
We can now equate the similar terms thus,the terms with the same power of x on both sides
5x⁴=ax⁴,a=5
Again,
b-4a=-2,but a=5
b-4(5)=-2
b-20=-2
b=18
Again,
c-4b=-18(but b=18)
c-4(18)=-18
c-72=-18
c=54
Again,
d-4c=-1(but c=54)
d-4(54)=-1
d-216=-1
d=215
Finally,
-4d+r=0(but d=215)
-4(215)+r=0
r-860=0
r=860
This means that when 5x⁴-2x³-18x²-x is divided by x-4,it gives5x³+18x²+54x+215 with a remainder of 860
Now rewriting it in the form as required by the question=(x-4)(5x³+18x²+54x+215)+860
