On dividing 2x⁴+x³-3x²+x-10 by a polynomial g(x), the quotient and remainder are 2x³+7x²+18x+55 and 155 respectively. Find g(x).
Answers
g(x) = (x-3)
Step-by-step explanation:
Given :-
On dividing 2x⁴+x³-3x²+x-10 by a polynomial g(x), the quotient and remainder are 2x³+7x²+18x+55 and 155 respectively.
To find :-
Find g(x) ?
Solution :-
Given that :
On dividing 2x⁴+x³-3x²+x-10 by a polynomial g(x), the quotient and remainder are 2x³+7x²+18x+55 and 155 respectively.
The Dividend = 2x⁴+x³-3x²+x-10
Let p(x) = 2x⁴+x³-3x²+x-10
The divisor = g(x)
The quotient = 2x³+7x²+18x+55
Let q(x) = 2x³+7x²+18x+55
The remainder = 155
Let r(x) = 155
We know that
The Division Rule in Polynomials
p(x) = g(x)×q(x) + r(x)
According to the given problem
2x⁴+x³-3x²+x-10=g(x)(2x³+7x²+18x+55)+ +155
2x⁴+x³-3x²+x-10-155=g(x)(2x³+7x²+18x+55)
=>2x⁴+x³-3x²+x-165=g(x)(2x³+7x²+18x+55)
=> g(x)(2x³+7x²+18x+55) = 2x⁴+x³-3x²+x-165
=>g(x)=(2x⁴+x³-3x²+x-165 )/(2x³+7x²+18x+55)
Now,
2x³+7x²+18x+55)2x⁴+x³-3x² +x -165(x-3
2x⁴+7x³+18x²+55x
(-) (-) (-) (-)
__________________
0 -6x³ -21x² -54x -165
-6x³ -21x² -54x - 165
(+) (+) (+) (+)
____________________
0
_____________________
=> g(x) = (2x⁴+x³-3x²+x-165 )/(2x³+7x²+18x+55)
=> g(x) = (x-3)
Therefore, g(x) = (x-3)
Answer:-
The divisor or the value of g(x) for the given problem is (x-3)
Used formulae:-
The Division Rule in Polynomials is
p(x) = g(x)×q(x) + r(x)
p(x) = Dividend
g(x) = Divisor
q(x) = Quotient
r(x) = Remainder