Question

A School has decided to award prizes to their students for three values honesty
(alpha), punctuality (beta ) and obedience (gama). School decided to give 2 prizes for
punctuality, 3 prizes for honesty and 1 prize for obedience. Find a cubic
polynomial whose zeroes are depicting the number of prizes for honesty,
punctuality and obedience.

Answer 1

Answer:

Hint: The question is related to polynomial. We have to make the cubic polynomial using the zeros given in the question. Use the sum of zeroes, product of the zeroes and sum of the product of the zero’s formula. Zeroes of the cubic polynomials are α,β,γα,β,γ. Here αα is equal to 3 ,ββ is equal to 5 and γγ is equal to -2. In the cubic polynomial the coefficient of x3x3 is a, coefficient of x2x2 is b, coefficient of xx is c and the consent term is d. use the formula to get your cubic polynomial equation.

Complete step by step solution:

Given that the zeroes of the cubic polynomial are 3, 5 and-2 that means (x+3),(x+5),(x−2)(x+3),(x+5),(x−2)

We know that the zeroes of cubic polynomial is denoted by α,β,γα,β,γ

Here we know that the sum of the zeroes is

sum of zeroes = coefficient of x2coefficient of x3sum of zeroes = coefficient of x2coefficient of x3

α+β+γ=−baα+β+γ=−ba

Putting the vales of the zero in above formula

3+5−2=−bc3+5−2=−bc

Solve the left-hand side

⇒6=−bc⇒6=−bc

Product of the zeroes is

product of the zeroes = consent termcoefficient of x3product of the zeroes = consent termcoefficient of x3

α×β×γ=−daα×β×γ=−da

Putting the values of the zeroes

3×5×(−2)=−da3×5×(−2)=−da

Solve the left-hand side

⇒−30=−da⇒−30=−da

Sum of the product of zeroes is

sum of product of zeroes = coefficient of xcoefficient of x3sum of product of zeroes = coefficient of xcoefficient of x3

α×β+β×γ+α×γ=caα×β+β×γ+α×γ=ca

Putting the values of zeroes

3×5+5×(−2)+3×(−2)=ca3×5+5×(−2)+3×(−2)=ca

Solve the left-hand side

⇒15−10−6=ca⇒15−10−6=ca

⇒−1=ca⇒−1=ca

On comparing the above solutions, we get the cubic polynomial

a=1,b=−6,c−1,d=30a=1,b=−6,c−1,d=30

Equation

∴x3−6x2−x+30=0∴x3−6x2−x+30=0

Hence the cubic polynomial equation is x3−6x2−x+30=0x3−6x2−x+30=0.