Question

On dividing 2x⁴+x³-3x²+x-10 by a polynomial g(x), the quotient and the remainder are 2x³+7x²+18x+55 and 155 respectively. Find g(x).

Answer 1

Answer:

On dividing x3-3x2+x+2 by a polynomial g(x),the quotient and remainder were x-2 and -2x+4 respectively.find g(x)

f(x) = x³ - 3x² + x + 2

q(x) = x - 2

let say

g(x) = ax² + bx + c

r = - 2x + 4

f(x)  = g(x)q(x) + r

=> x³ - 3x² + x + 2 = (ax² + bx + c)(x - 2) + (-2x + 4)

=> x³ - 3x² + x + 2 =  ax³ + x²(b - 2a) + x(c - 2b) -2c -2x + 4

=> x³ - 3x² + x + 2 =  ax³ + x²(b - 2a) + x(c - 2b - 2)  + (-2c  + 4)

Equating Power terms

a = 1

b - 2a = - 3  => b -2 = -3 => b = -1

c - 2b - 2 = 1 => c + 2 - 2 = 1  => c = 1

or -2c  + 4 = 2  => -2c = -2 => c = 1

g(x) = x² - x + 1

Step-by-step explanation: