Question

find the cubic polynomial its zeros are 6, - 5 and 2 respectively

Answer 1

To find a cubic polynomial by its zeroes, we need 3 things.

1) Sum of all zeroes.

2) Sum of all zeroes multiplied to 1 another zero.

3) Product of zeroes.

Let's do it.

1) Sum of zeroes = 6 + ( -5 ) + 2

                            = 6 - 5 + 2

                            = 8 - 5
 
                            = 3.

2) Sum of all zeroes multiplied to 1 another zero .

= { 6 x ( - 5 ) } + { ( - 5 ) x 2 } + ( 6 x 2 )

= -30 + ( - 10 ) + 12

= -30 - 10 + 12

= -40 + 12

= -28

3.) Product of zeroes.

= 6 x ( -5 ) x 2

= -60.

Formula for a cubic polynomial is :

= x³ - ( Sum of zeroes ) x² + ( Sum of all zeroes multiplied to 1 another zero )x - Product of zeroes 

= x³ - ( 3 ) x² + ( -28 ) x - ( - 60 )

= x³ - 3x² - 28x + 60.

So, required polynomial is ( x³ - 3x² - 28x + 60 ).