Question

EXAMPLE 6
Find the quadratic polynomial, the sum of whose zeros is V2 and
their product is –12. Hence, find the zeros of the polynomial.
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Answer 1

Answer:

Required quadratic polynomial :

x² - √2x - 12

Zeros : x = 3√2 , - 2√2

Points to be noted:

• If A and B are the zeros of any quadratic polynomial then it is given as ;

x² - (A + B)x + A•B .

• The possible values of variables ( unknown ) for which the polynomial becomes zero are called its zeros.

• To find the zeros of a polynomial, equate it to zero .

Solution:

Let A and B be the zeros of the required polynomial.

Now,

It is given that , sum of zeros is √2 .

Thus , A + B = √2

Also,

It is given that , product of is –12 .

Thus, A•B = –12

Thus,

The required quadratic polynomial will be given as ; x² - (A+B)x + A•B

ie ; x² - √2x + (-12)

ie ; x² - √2x - 12

Hence,

The required quadratic polynomial is :

x² - √2x - 12 .

Now,

In order to find the zeros of the polynomial, let's equate it to zero .

=> x² - √2x - 12 = 0

=> x² - 3√2x + 2√2x - 12 = 0

=> x(x - 3√2) + 2√2(x - 3√2) = 0

=> (x - 3√2)(x + 2√2) = 0

=> x = 3√2 , - 2√2

Hence,

The zeros of the obtained quadratic polynomial are : x = 3√2 , - 2√2 .