Question

Write a polynomial i) whose zeroes are -2, 12 ii) whose sum is -34 and product is -8 iii) whose sum is -5 and one of the zero is 4 iv) whose product is -8 and one of the zero is -2.

Answer 1

ZEROS = -2 , 12

(ii) -34, -8

(iii) -5 , 4

(iv) -8 , -2

( I ) the sum of zeros is

$( \alpha + \beta ) = ( - 2 + 12) \\ = 10$

(ii) the product of the zero is

$( \alpha + \beta ) = ( - 2 \times 12) \\ = - 24$

$x {}^{2} - ( \alpha + \beta )x + ( \alpha \beta ) \\ x {}^{2} - ( - 10)x + ( - 24) \\ x {}^{2} - 10x - 24 = 0$

$x {}^{2} - (sum \: of \: zeros \: )x + (product \: of \: zeros) \\ x {}^{2} - ( - 34)x + ( - 8) \\ x {}^{2} + 34x - 8$

$sum \: of \: zeros \: = - 5 \\ product \: of \: zeros \: = ( \alpha \beta ) = ( - 5 \times 4 \\ = - 20$

$x { }^{2} - ( - 5)x + ( - 20) \\ x {}^{2} + 5x - 20$

$product \: of \: the \: zeros \: = - 8 \\ sum \: of \: the \: zeros \: = - 8 + ( - 2) = - 10 \\ = x {}^{2} - ( - 10)x + ( - 8) \\ x {}^{2} + 10 x - 8$