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

=>
91

The value of a^2 + b^2 + c^2 = -10 and a^3 + b^3 +
c^3 = 91