Question

find the cubic polynomial whose zero are 5 , 6, -1​

Answer 1

Given:

Zeroes of a cubic polynomial are 5,6,-1

To find:

The corresponding cubic polynomial

Explanation:

Let us think that

$\sf (x-5),(x-6),(x+1)\: are \:the \:factors\\ \sf of\: the\: cubic\: poynomial$

let the cubic polynomial be p(x)

So

$\longrightarrow \sf p(x) = (x - 5)(x - 6)(x + 1) \\ \\ \longrightarrow \sf p(x) = {x}^{2} - 5x - 6x + 30(x + 1) \\ \\ \longrightarrow \sf p(x) = {x}^{2} - 11x + 30(x + 1) \\ \\ \longrightarrow \sf p(x) = {x}^{3} + {x}^{2} - 11 {x}^{2} - 11x + 30x + 30 \\ \\ \longrightarrow \boxed{ \tt p(x) = {x}^{3} - 10 {x}^{2} + 19x + 30}$

Therefore the cubic polynomial is

x³-10x²+19x+30

Extra information:

For a cubic polynomial,

↔Sum of roots α+β+δ=-b/a

↔αβ+βδ+δα=c/a

↔Product of roots αβδ=d/a