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
sum of the real roots =−coefficient of xn−1coefficient of xn
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
3×x2−7×x+12=0⇒{x=−(√95×i−7)6x=(√95×i+7)6}⇒{x=−(−76+1.624465724134i)x=76+1.624465724134i}
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
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 xn-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 !!!
Hi Chris,
The sum of the roots of a polynomial of degree n is
sum of the real roots =−coefficient of xn−1coefficient of xn
but there is a lot more to this Chris.
Have a look at this
http://www.mathsisfun.com/algebra/polynomials-sums-products-roots.html