If α and β are the zeroes of the polynomial \(2x^2+4x+5\), then find the value of \(α^3+β^3\).

Answer: Option b. 7

Solution:

Step 1: Use the concepts of relationship between roots and coefficients of quadratic polynomial.

Given polynomial is \(2x^2+4x+5\), by comparing it standard quadratic polynomial \(ax^2+bx+c\), we get

\(a=2, \quad b=4, \quad c=5\)

\( \alpha+\beta= \frac{-b}{a}=\frac{-4}{2} = -2 \)            ....(1)

\(\alpha \beta= \frac{c}{a}=\frac{5}{2} \)              ....(2)

Step 2: Use the algebraic identity for the sum of cube of any two numbers

\(α^3+β^3=(α+β)(α^2+β^2-αβ)\)

\(\quad \quad =-2\times(α^2+β^2-αβ)\)          [ Using (1)]

\(\quad \quad =-2\times((α+β)^2-3αβ)\)

[∵ \(α^2+β^2=(α+β)^2-2αβ\)]

\(\quad \quad =-2\times(4-3\times\frac{5}{2})\)       [ Using (1) and (2)]

\(\quad \quad =-2\times(4-\frac{15}{2})\)

\(\quad \quad =-2\times(\frac{8-15}{2})\)

\(\quad \quad =-2\times(\frac{-7}{2})\)

\(\quad \quad =7\)

\(\therefore α^3+β^3 =7\)

Hence, the value of \(α^3+β^3\) is \(7\).