Question

If GCD of f(x) = x ^ 3 + c * x ^ 2 - x + 2c and g(x) = x ^ 2 + cx - 2 is a linear polynomial then the value of c is​

Answer 1

Given : GCD of f(x) = x ^ 3 + c * x ^ 2 - x + 2c and g(x) = x ^ 2 + cx - 2 is a linear polynomial

To Find : the value of c is

Solution:

Let say  x  -  a  is the GCD of f(x) and g(x)

f(x) = x³ + cx² - x + 2c

g(x) = x² + cx  - 2

=> f(a) = 0  and g(a) = 0

f(a) = 0

=> a³ +  ca² - a + 2c = 0   Eq1

g(a) = 0

=>  a² + ca - 2 = 0

Multiplying with a

=> a³ + ca²  - 2a = 0    Eq2

Eq1 - Eq2

=>  a  + 2c  = 0

=> a = - 2c

 a² + ca - 2 = 0

=> (-2c)² + c(-2c) - 2 = 0

=> 4c² - 2c² - 2 = 0

=> 2c² = 2

=> c² = 1

=>  c = ±1    and a  = -/+ 2  hence factor (x + 2) and ( x - 2)

case  1 : c = 1

f(x) = x³ +  x² - x + 2   = (x + 2) (x²  -x + 1)

g(x) = x² + cx  - 2  = x² + x - 2  =  ( x + 2)(x - 1)

x + 2 is common factor

case  1 : c = -1

f(x) = x³ -  x² - x - 2   = (x - 2) (x² + x + 1)

g(x) = x² - cx  - 2  = x² + x - 2  =  ( x - 2)(x + 1)

x-  2 is common factor

Value of c are ±1

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