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 \(7\).