The standard form of a cubic equation is a\(x^3\) + b\(x^2\) + cx + d = 0 where a≠0 and a, b, c and d are all real numbers.
Remember that the GMAT doesn’t really expect you to know how to solve complex cubic equations. It probably expects you to know some standard results about a cubic equation, which can be easily understood if you are strong with Quadratic equations.
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,
a\(x^3\) + b\(x^2\) + cx + d = 0, we have
Sum of roots = (α+β+γ) = -\(\frac{b}{a}\)
Product of roots = αβγ = -\(\frac{d}{a}\)Sum of product of roots taken two at a time = (αβ + βγ + γα) = \(\frac{c}{a}\) On the other hand, let us simplify the expression given to us i.e. (a+1)(b+1)(c+1) which yields a+b+c+ab+bc+ca+abc+1.
Since a,b and c are the roots of the equation \(x^3\)-4\(x^2\)+3x+5, we can say,
Sum of the roots = (a+b+c) = -(-\(\frac{4}{1}\)) = 4
Product of the roots = abc = -(\(\frac{5}{1}\)) = -5
Sum of product of roots taken two at a time = (ab+bc+ca) = (\(\frac{3}{1}\)) = 3.
Therefore, the value of the desired expression = 4+3-5+1 = 3.
The correct answer option is E.
Hope that helps!
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