Question

If alpha and beta are the zeroes of the polynmial x^2+3x-4 then find a quadratic polynomial whose zeroes are 1/alpha and 1/ beta?

Answer 1

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Answer 2

Answer:

x² - 3x/4 - 1/4

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 .

★ 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

★ If α and ß are the zeros of any quadratic polynomial , then it is given by ;

x² - (α + ß)x + αß

Solution:

Here,

The given quadratic polynomial is ;

x² + 3x - 4 .

Now,

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

a = 1

b = 3

c = -4

Also,

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

Thus,

Sum of zeros of the given quadratic polynomial will be ;

α + ß = -b/a = -3/1 = -3

Also,

Product of zeros of the given quadratic polynomial will be ;

αß = c/a = -4/1 = -4

Now,

According to the question , 1/α and 1/ß are the zeros of the required quadratic polynomial .

Thus ,

Sum of zeros of the required quadratic polynomial will be ;

1/α + 1/ß = (ß + α)/αß = -3/-4 = 3/4

Also ,

Product of zeros of the required quadratic polynomial will be ;

(1/α)×(1/ß) = 1/αß = 1/-4 = -1/4

Now ,

The required quadratic polynomial will be given as ; x² - (1/α + 1/ß)x + 1/αß

ie ; x² - (3/4)x + (-1/4)

ie ; x² - 3x/4 - 1/4

Hence,

The required quadratic polynomial is ;

x² - 3x/4 - 1/4