How many real roots does x^4-2*x^3-13*x^2+38*x-24 have

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The expression given is: x^4 -2x^3 - 13x^2 + 38x - 24

x^4 -2x^3 - 13x^2 + 38x - 24

The equation has 4 roots and they give a product of -24. To determine the roots factorize x^4 -2x^3 - 13x^2 + 38x - 24.

x^4 - 2x^3 - 13x^2 + 26x + 12x - 24

=> x^3(x - 2) - 13x(x - 2) + 12(x - 2)

=> (x - 2)(x^3 - 13x + 12)

=> (x - 2)(x^3 - x^2 + x^2 - x - 12x + 12)

=> (x - 2)(x^2(x - 1) + x(x - 1) - 12(x - 1))

=> (x - 2)(x - 1)(x^2 + x - 12)

=> (x - 2)(x - 1)(x^2 + 4x - 3x - 12)

=> (x - 2)(x - 1)(x(x + 4) - 3(x + 4))

=> (x - 2)(x - 1)(x - 3)(x + 4)

This gives the roots of the expression as x = 2, x = 1, x = 3 and x = -4

All the 4 roots of x^4 -2x^3 - 13x^2 + 38x - 24 are real.

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