let P(x)=0 be a polynomial equation of least possible degree with rational coefficients having $\displaystyle \sqrt[3]7+\sqrt[3]49$ as one of its roots

.then the product of all the roots of P(x)=0 is

Printable View

- Oct 4th 2012, 09:52 AMayushdadhwalproduct of roots
let P(x)=0 be a polynomial equation of least possible degree with rational coefficients having $\displaystyle \sqrt[3]7+\sqrt[3]49$ as one of its roots

.then the product of all the roots of P(x)=0 is - Oct 4th 2012, 10:28 AMHallsofIvyRe: product of roots
You left out an awful lot of information. What kind of course did you see this problem in? What do you know about such problems? What ideas do you have about it? Do you have, for example, a "guess" as to what degree this polynomial should be?

- Oct 4th 2012, 12:06 PMjohnsomeoneRe: product of roots
Let $\displaystyle a = 7^{\frac{1}{3}}$. Let $\displaystyle b = a + a^2$.

Note $\displaystyle a^3 = 7$, which will be used to simplify powers of $\displaystyle a$ greater than 2.

$\displaystyle b^3 = (a + a^2)^3 = (a)^3(1+a)^3 = 7(a^3 + 3a^2 + 3a + 1) = 7(7 + 3a^2 + 3a + 1)$

$\displaystyle = 21a^2 + 21a + 56 = 21(a^2 + a) + 56 = 21b + 56$.

Can you complete it from here? - Oct 5th 2012, 09:00 AMayushdadhwalRe: product of roots
sir this is the exact wording given in book.one thing i am sure that they are working on real field.