Question

if alpha and beta are the zeros of the quadratic polynomial 3 X square + 2 x minus 2 then find the value of alpha cube + beta cube​

Answer 1

Step-by-step explanation:

Given:-

α and β are the zeros of the quadratic polynomial 3 x^2 + 2 x - 2

To find :-

Find the value of α^3 + β^3 ?

Solution:-

Given quadratic polynomial is 3 x^2 + 2 x - 2

In Comparing this with the standard quadratic Polynomial ax^2+bx+c

a = 3

b = 2

c= -2

We know that

Sum of the zeroes = α + β = -b/a

=> α + β = -2/3 --------(1)

Product of the zeroes = αβ = c/a

=> αβ = -2/3 ---------(2)

We Know that

(a+b)^2 = a^2+2ab+b^2

a^2+b^2 = (a+b)^2 -2ab

α^2 + β^2 = (α + β )^2 - 2αβ

=>α^2 + β^2 = (-2/3)^2-2(-2/3)

=> α^2 + β^2 = (4/9)+(4/3)

=>α^2 + β^2 = (4+12)/9

=>α^2 + β^2 = 16/9-----------(3)

Now α^3 + β^3

It is in the form of a^3+b^3

We know that

a^3+b^3 = (a+b)(a^2-ab+b^2)

α^3 + β^3 = (α+ β )(α^2 -αβ +β^2)

=> α^3 + β^3= (α+ β )(α^2 +β^2-αβ )

=>α^3 + β^3= (-2/3) [(16/9)-(-2/3)]

=>α^3 + β^3= (-2/3)[(16/9)+(2/3)]

=>α^3 + β^3= (-2/3)[(16+6)/9]

=> α^3 + β^3= (-2/3)(22/9)

=>α^3 + β^3= (-2×22)/(3×9)

=>α^3 + β^3= -44/27

Answer:-

The value of α^3 + β^3 for the given problem is

-44/27

Used formulae:-

• the standard quadratic Polynomial ax^2+bx+c

• Sum of the zeroes = α + β = -b/a

• Product of the zeroes = αβ = c/a

• (a+b)^2 = a^2+2ab+b^2

• a^3+b^3 = (a+b)(a^2-ab+b^2)