Question

Let f(x)=x^(3)-3x+b and g(x)=x^(2)+bx-3 where b is a real number.What is the sum of all possible values of b for which the equations f(x)=0 and g(x)=0 have a common root?


please answer with explanation.....​

Answer 1

SOLUTION

GIVEN

Let f(x) = x³ - 3x + b and g(x) = x² + bx - 3 where b is a real number.

TO DETERMINE

The sum of all possible values of b for which the equations f(x) = 0 and g(x) = 0 have a common root

EVALUATION

Here the given polynomials are

f(x) = x³ - 3x + b - - - - - (1)

g(x) = x² + bx - 3 - - - - (2)

Let m be the common root of the equations f(x) = 0 and g(x) = 0

Then we have

m³ - 3m + b = 0 - - - - - - (3)

m² + bm - 3 = 0 - - - - - - - (4)

From Equation 3 we get

m( m² - 3) + b = 0 - - - - - - (5)

From Equation 4 we get

m² - 3 = - bm

From Equation 5 we get

- bm² + b = 0

⇒ m² = 1

⇒ m = - 1 , 1

For m = - 1 we get from Equation 3

b = - 2

For m = 1 we get from Equation 3

b = 2

Hence the sum of all possible values of b

= - 2 + 2

= 0

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