Question

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).

Answer 1

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