Question

Find the value of A and B if (X + 1) and (x -1) are the factors of the polynomial x^4+ax^3-3x^2+2x+b by using factor theorem

Answer 1

Given Equation is f(x) = x^4 + ax^3 - 3x^2 + 2x + b.

By remainder theorem, we get

x + 1 = 0

x = -1.

Now substitute x = -1 in f(x), we get

f(-1) = (-1)^4 + a(-1)^3 - 3(-1)^2 + 2(-1) + b = 0

       = 1 - a - 3 - 2 + b = 0

      = b - a - 4 = 0

       b - a = 4.  ------- (1)




By remainder theorem, we get

x - 1 = 0

x = 1.

Substitute x = 1 in f(x), we get

f(1) = x^4 + ax^3 - 3x^2 + 2x + b

     = (1)^4 + a(1)^3 - 3(1)^2 + 2(1) + b = 0

     = 1 + a - 3 + 2 + b = 0

     = a + b = 0   -------------- (2)


On solving (1) & (2), we get

b - a = 4

b + a = 0

------------------

2b = 4

b = 2


Substiitute b = 2 in (1), we get

b - a = 4

2 - a = 4

-a = 2

a = -2.


Hope this helps!