# Thread: Root Linear Coefficient Theorem

1. ## Root Linear Coefficient Theorem

Hi Forum (I'm new)

Root Linear Coefficient Theorem: "The sum of the reciprocal of the roots of a polynomial equals the negative of the coefficient of the linear term in the Maclaurin series" (Root Linear Coefficient Theorem -- from Wolfram MathWorld)

I get it equals the negative of the ratio of the linear coefficient to the constant term.
$\frac{1}{r_1}+\frac{1}{r_2}+...+\frac{1}{r_n}=\fra c{S_{n-1}}{S_{n}}=\frac{(-1)^{n-1}\frac{a_1}{a_n}}{(-1)^{n}\frac{a_0}{a_n}}=-\frac{a_1}{a_0}$
where $P(x)=a_nx^n+...+a_1x+a_0=a_n(x-r_1)(x-r_2)...(x-r_n)$ and I'm using Vieta's Formulas (Vieta's Formulas)

I'm pretty sure that's right... What went wrong?

2. I believe that you require the polynomial itself is monic, i.e., the leading coefficient is 1; otherwise, "the negative of the coeff. of the linear term" may not even be defined. For instance, f(x)=x^2-3x and g(x)=2x^2-6x, both have precisely the same roots, but the coefficients of x are definitely distinct.

Hope this helps.

3. Thanks for answering. If the constant term is non-zero, can I assume the sum is $-a_1/a_0$ then?

4. Originally Posted by sleepingcat
Thanks for answering. If the constant term is non-zero, can I assume the sum is $-a_1/a_0$ then?
Yes