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?

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justaguide | College Teacher | (Level 2) Distinguished Educator

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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

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