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Sum And Product Of The Roots

Theorem: If f(x)=a_{n}x^n + a_{n-1}x^{n-1} + \cdots + a_{0}x^0 is a polynomial,\\[12pt] then the normalized coefficients with alternate '+'and '-' signs, i.e., -\dfrac{a_{n-1}}{a_{n}}, \dfrac{a_{n-2}}{a_{n}}, \cdots , (-1)^n\dfrac{a_{0}}{a_{n}} gives the different sums and products of the roots of the equation f(x)=0.

Prerequisites:
Factor Representation of a Polynomial (proof)

Proof:
Let the given polynomial be:

\qquad\quad f(x)=a_{n}x^n + a_{n-1}x^{n-1} + \cdots + a_{0}x^0  \qquad\qquad\qquad\qquad\qquad\qquad\qquad\cdots (1)

This can be written in the form of factor representation as:

\qquad\quad f(x) = (x-\alpha_1)(x-\alpha_2) \cdots (x-\alpha_n) g(x)\qquad\qquad\qquad\qquad\qquad\qquad\;\;\cdots (2)

Here \;\alpha_1, \alpha_2, \cdots ,\alpha_n are the roots of the equation f(x)=0.
Also, \;deg\big(g(x)\big) = n-n = 0. This implies that g(x) is a constant = g (say).

Expanding equation (2):

\qquad\quad f(x) = gx^n - g\left(\alpha_1+\alpha_2+\cdots +\alpha_n\right)x^{n-1} + g\left(\alpha_1\alpha_2+\alpha_2\alpha_3+\cdots +\alpha_{n-1}\alpha_n\right)x^{n-2}\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad + \cdots + (-1)^n g\left(\alpha_1\alpha_2\cdots\alpha_n\right)

Comparing coefficients with equation (1),

\qquad\quad g = \alpha_n\\ \qquad\quad\displaystyle\sum\limits_{i=1}^n \alpha_i = \alpha_1 +\alpha_2+\cdots +\alpha_n = - \dfrac{a_{n-1}}{a_{n}}\\ \\ \qquad\quad\displaystyle\sum\limits_{i\neq j} \alpha_i\alpha_j =\alpha_1\alpha_2+\alpha_2\alpha_3+\cdots +\alpha_{n-1}\alpha_n = \dfrac{a_{n-2}}{a_{n}}\\ \\ \qquad\quad\;\;\vdots\\[6pt] \\ \qquad\quad\displaystyle\prod\limits_{i=1}^n \alpha_i = \alpha_1\alpha_2\cdots\alpha_n = (-1)^n\dfrac{a_{0}}{a_{n}}

Hence the result.


Recommended:
Condition for roots of quadratic equation
Quadratic Formula
Factor Theorem

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