The equation x^3 + 2x^2 +7x -19 = 0 has roots a, b, c. Compute the values of a^2 + b^2 + c^2 and a^3 + b^3 + c^3. Can this be done without solving?
Yes, we can find the values of the required polynomials without solving the equation. The values of the roots of an equation are related to the coefficient of the powers of x by Viete's formulas.
In the case of a cubic equation ax^3 + bx^2 + cx + d = 0, if the roots are x1, x2 and x3, we have
x1 + x2 + x3 = -b/a
x1*x2 + x2*x3 + x3*x1 = c/a
x1*x2*x3 = -d/a
Here, the equation we have is x^3 + 2x^2 +7x -19 = 0 and it has roots a, b, c.
a + b + c = -2
ab + bc + ac = 7
abc = 19
To find a^2 + b^2 + c^2 we use the relation : a^2 + b^2 + c^2 = (a + b + c) ^2 - 2(ab + bc + ac)
=> (-2)^2 -2*7
=>4 - 14
=> - 10
For a^3 + b^3 + c^3, use the relation a^3 + b^3 + c^3 = (a + b + c)^3 - 3(a + b + c)(ab + bc + ac) + 3abc
=> (-2)^3 -3*(-2)*7 + 3*19
=> -8 + 42 + 57
The value of a^2 + b^2 + c^2 = -10 and a^3 + b^3 + c^3 = 91