Question

if one of the zeros of the cubic polynomial x cube + ax square + bx + c is equal to minus 1 then prove that the product of the other two zeros is b minus a + 1 ​

Answer 1

$\underline{\textsf{Given:}}$

$\textsf{One of the zeros of}\;\mathsf{x^3+ax^2+bx+1=0}$

$\mathsf{is -1}$

$\underline{\textsf{To prove:}}$

$\textsf{Product of the two zeros is b-a+1 }$

$\underline{\textsf{Solution:}}$

$\textsf{Since one of the zeros is -1,}$

$\textsf{Sum of the coefficients of odd powers of x}$

$\textsf{=Sum of the coefficient of even powers of x}$

$\implies\mathsf{1+b=a+c}$........(1)

$\textsf{Let the zeros of the given polynomial be l,m and n}$

$\textsf{Product of the zeros}\mathsf{=\dfrac{c}{-1}}$

$\mathsf{lmn=-c}$

$\mathsf{(-1)mn=-c}$

$\mathsf{mn=c}$

$\textsf{Using (1), we get}$

$\mathsf{mn=b-a+1}$

$\implies\boxed{\textsf{Product of other two zeros = b-a+1}}$