Today in highschool we were doing a chapter called "Roots of polynomials" where we learnt something new and interesting which is :
$ax^2+bx+c=0$ Has roots $\alpha$ , $\beta$ Then:
$$\alpha + \beta= -b/a$$
$$\alpha \beta=c/a$$
$ax^3+bx^2+cx+ d=0$ Has roots $\alpha$ , $\beta$ , $\gamma$. Then:
$$\alpha+ \beta + \gamma = -b/a$$
$$\alpha \beta + \alpha \gamma + \beta \gamma= c/a$$
$$\alpha \beta \gamma= -d/a$$
My curiosity turned to, what happens in 4th degree power polynomial. We haven't learnt in class (not in our syllabus). But is there something like a general formula for this? Coz I'm sure people who learn higher powers cannot memorise all the powers and remember when it becomes – or +. (For me I can memorize these because It's only a few set of equations and two different polynomials), what happens in higher powers and how does one memorise ? What is the general formula, if there is any?
And also : what happen in fourth power that is for :
$$ax^4+bx^3+cx^2+dx+e=0$$
After looking at the first comment I understood that it's Vietas formulas. And I checked in out in Wikipedia. The formulas are complicated looking, but I understood after looking for a while. But there are dots in the middle which means more equations. I tried this with my 3rd power and it works fine, but the question remains "How to do for higher degree polynomials. I don't know what are the formulas in the middle (the dots going downwards in the middle). I believe there are n number of formulas for n powers, here there is only three. Which I already knew, please help me
Best Answer
I'll give examples, then you'll understand the general pattern better.
Take a degree four polynomial. I'll denote a general fourth-degree polynomial by $P_4$. A general $5^{\text{th}}$ degree polynomial as $P_5$ and so on...
Your $P_4$ is: $ a_0x^4 + a_1x^3 + a_2x^2 + a_3x^1 + a_4$
Then, I'll get $4$ formulas:
$r_0 + r_1 + r_2 + r_3 = -\dfrac{a_1}{a_0}$
$r_0r_1 + r_0r_2 + r_0r_3 + r_1r_2 + r_1r_3 + r_2r_3 = +\dfrac{a_2}{a_0}$
$r_0r_1r_2 + r_0r_1r_3 + r_0r_2r_3 + r_1r_2r_3= -\dfrac{a_3}{a_0}$
$ r_0r_1r_2r_3 = + \dfrac{a_4}{a_0}$
The idea is, first you add products of single roots. That is, the first formula. Then, you add products containing 2 roots - second formula. Then you add products containing 3 roots - third formula. Then add products of 4 roots - fourth formula. And on the RHS, the denominator is always the coeffecient of the heighest power, and the signs alternate. The first formula always has the negative sign. The numerator is the coefficient matching with the numbers of roots in the multiplication, i. e. for product containing 4 roots, numerator should be A4.