The degree of an equation dictates the number of roots/number of factors an equation has. Since the degree of a cubic equation is 3, we can say that it will have 3 roots, say, α, β and γ. When an equation has 3 roots, you can also say that it has 3 factors viz, (x-α), (x-β), and (x-γ). Since these are the three factors of the equation, the product of these three will give you the cubic equation.
When these three factors are multiplied and simplified, we get,
x^3 – (α+β+γ)x^2 + (αβ + βγ + γα)x – αβγ = 0. Comparing with the standard form,
ax^3 + bx^2 + cx + d = 0, we have
Sum of roots =
[tex](\alpha+\beta+\gamma)=-\frac{b}{a}[/tex]Product of roots =
[tex]\alpha\beta\gamma=-\frac{d}{a}[/tex]Sum of the product of roots taken two at a time
[tex](\alpha\beta+\beta\gamma+\gamma\alpha)=\frac{c}{a}[/tex](i)
a = 1
b = - 8
c = 28
d = -32
[tex]\begin{gathered} \frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma} \\ \frac{\beta\gamma+\alpha\gamma+\alpha\beta}{\alpha\gamma\beta} \\ \frac{c/a}{-d/a} \\ -\frac{c}{d} \\ -\frac{28}{-32} \\ \frac{7}{8} \end{gathered}[/tex]The answer of part (i) is 7/8
(ii)
[tex]\begin{gathered} (\alpha+2)(\beta+2)(\gamma+2) \\ (\alpha\beta+2\beta+2\alpha+4)(\gamma+2) \\ (\alpha\beta\gamma+2\beta\gamma+2\alpha\gamma+4\gamma+2\alpha\beta+4\beta+4\alpha+8) \\ (\alpha\beta\gamma)+2(\beta\gamma+\alpha\beta+\alpha\gamma)+4(\alpha+\beta+\gamma)+8 \\ -\frac{d}{a}+\frac{2c}{a}-\frac{4b}{a}+8 \end{gathered}[/tex]a = 1
b = - 8
c = 28
d = -32
[tex]\begin{gathered} -\frac{-32}{1}+\frac{2\cdot28}{1}-\frac{4\cdot-8}{1}+8 \\ 128 \end{gathered}[/tex]The answer of part (ii) is 128
