What are the roots of the polynomial equation x3 - 10x=-3x2+ 24?

x = -2, 3, -4.

Step-by-step explanation:

x^3 - 10x = -3x2 + 24 = 0

x^3 + 3x^2 - 10x - 24 = 0

The last term is 24 and the coefficient of x^3 is 1 so +/- 1,  +-2, +/- 3 and +/- 4 could be among the roots ( by The Rational Root Theorem).

Let x = 1

f(x) = 1^3 +3(1)^2 - 10 - 24 = -30 so it's no t .

f(-1) = -1 + 3 + 10 - 24 = -12  so it's not -1.

Let  x = 2:

f(2) =  2^3 +3*4 - 20 - 24 = -32  so its not 2

f(-2) = -8 + 12 + 20 - 24 = 0  so x = -2 is a root

and therefore x+ 2 is a factor and we divide:

x + 2 ) x^3 + 3x^2 - 10x - 24   (  x^2 + x - 12   < The quotient

          x^3 + 2x^2

                      x^2 - 10x

                       x^2 + 2x

                              -12x - 24

                              -12x - 24

Now x^2 + x - 12 = (x - 3)(x + 4)

So (x + 2)(x - 3)(x + 4) = 0

This gives

x = -2, 3, -4.

–4, –2, and 3

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