Question

If α and β are zeroes of x²-x-1, then find 1/α + 1/β.

Answer 1

Answer :

1/α + 1/ß = -1

Note:

★ The possible values of the variable for which the polynomial becomes zero are called its zeros .

★ A quadratic polynomial can have atmost two zeros .

★ The general form of a quadratic polynomial is given as ; ax² + bx + c .

★ If α and ß are the zeros of the quadratic polynomial ax² + bx + c , then ;

• Sum of zeros , (α + ß) = -b/a

• Product of zeros , (αß) = c/a

Solution :

Here ,

The given quadratic polynomial is ;

x² - x - 1 .

Now ,

Comparing the given quadratic polynomial with the general quadratic polynomial ax² + bx + c , we have ;

a = 1

b = -1

c = -1

Also ,

It is given that , α and ß are the zeros of the given quadratic polynomial .

Thus ,

=> Sum of zeros = -b/a

=> α + ß = -(-1)/1

=> α + ß = 1

Also ,

=> Product of zeros = c/a

=> αß = -1/1

=> αß = -1

Now ,

=> 1/α + 1/ß = (ß + α)/αß

=> 1/α + 1/ß = (α + ß)/αß

=> 1/α + 1/ß = 1/-1

=> 1/α + 1/ß = -1

Hence , 1/α + 1/ß = -1 .