What are the sum (S) and product (P) of the roots of the equation 3x^2 - 7x + 12 = 0?

 Hi Chris,

The sum of the roots of a polynomial of degree n is   

$$\mbox{sum of the real roots }= -\frac{\mbox{coefficient of }x^{n-1}}{\mbox{coefficient of }x^n}$$

but there is a lot more to this Chris.

Have a look  at this

http://www.mathsisfun.com/algebra/polynomials-sums-products-roots.html

3x^2 - 7x + 12 = 0    this doesn't factor......using the onsite solver, we have

$${\mathtt{3}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{7}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{12}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{95}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,-\,}}{\mathtt{7}}\right)}{{\mathtt{6}}}}\\
{\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{95}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,\small\textbf+\,}}{\mathtt{7}}\right)}{{\mathtt{6}}}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}\left({\mathtt{\,-\,}}{\frac{{\mathtt{7}}}{{\mathtt{6}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1.624\: \!465\: \!724\: \!134}}{i}\right)\\
{\mathtt{x}} = {\frac{{\mathtt{7}}}{{\mathtt{6}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1.624\: \!465\: \!724\: \!134}}{i}\\
\end{array} \right\}$$

The sum of the roots =  (14 / 6) =( 7 / 3 )

The product of the roots is

(1/36)[(7 - (√95)i] [ (7 + (√95)i ] =

(1/36) [49 - 95i²] =

(1/36) [ 49 + 95] =

(1/36)[144] =  4

What are the sum (S) and product (P) of the roots of the equation 3x^2 - 7x + 12 = 0 ?

$$\small{\text{
$
\begin{array}{rcl}
3x^2 - 7x + 12 & = & 0 \quad | \quad :3 \\ \\
x^2 - \frac{7}{3}x + \frac{12}{3} & = & 0 \\ \\
x^2 \underbrace{- \frac{7}{3}}_{ = -S } x + \underbrace{ 4 }_{=P }& = & 0 \\\\
$ sum (S) of the roots: $x_1+x_2& = &+ \frac{7}{3} \\\\
$ product (P) of the roots: $ x_1*x_2& = & 4
\end{array}
$
}}$$

Wow!!.....thanks, heureka.....that's a new one on me !!!....I never recognized it before.....  DOH !!!

Hi Chris,

What Heureka has done has many implications.  it can be used on polynomials of degree higher than 2.

although I'd need to 'think' with it if I was asked a question with a higher degree.

I think it can be used in probablility too although again I am very rusty   

Think of it like this:

(x - x₁)(x - x₂) = 0

x² - (x₁ + x₂)x +x₁x₂ = 0

.

Ah.....that makes it much clearer.....!!!

Melody...just playing around with this in WolframAlpha....it appears that in a polynomial of degree n, that the sum of the roots will always be the coefficient on the x^n-1 term, unless the sum of the roots = 0. In that case, that power (naturally) isn't represented in the polynomial. The ending constant will (again, naturally) be the product of the roots !!!

Ah...I see....my guess was wrong....thanks for that link....I'm going to look at that very closely...!!!