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In CAT and other MBA entrance exams, there are many questions which fall in the domain of equations (quadratic, cubic, quartic..) and the properties of their roots. Deriving these properties will not only help us remember them but also give us the basic concepts to solve all kinds of equations and their problems. In this chapter we shall learn how the properties of roots of a general polynomial equation are derived. So let’s begin!

Let’s take a polynomial P(x) in a single variable x:

Let $P(x) = a_0(x)^n + a_1(x)^{n-1} + a_2(x)^{n-2} + … + a_{n-1}x + a_n$ where $a_1$, $a_2$ etc. are constants (associated with decreasing powers of x) and n, n-1, etc. are the whole number powers of the variable x. The highest power of the variable x is known as the ‘degree’ of the polynomial P(x).

For example, $P(x) = x^5 – 3x + 1$ is a polynomial with degree 5 with $a_0 = 1, a_1 = 0, a_2 = 0, a_3 = 0, a_4 = -3, a_5 = 1$