A cubic equation has a real root 5 and a complex root 4 + 3i. What is the cubic equation.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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A cubic equation can have real as well as complex roots. If the equation has complex roots they always occur in pairs of complex conjugate numbers, a + i*b and a - i*b. This ensures that the coefficients of the equation are real.

If a cubic equation has a complex root 4 + 3i, it also has a root 4 - 3i. There are an infinite number of equations with root 5 and 4 + 3i. The simplest equation is:

(x - 5)(x - (4 +3i))(x - (4 - 3i)) = 0

=> (x - 5)(x - 4 -3i))(x - 4 + 3i)) = 0

=> (x - 5)((x - 4)^2 - (3i)^2)) = 0

=> (x - 5)(x^2 + 16 - 8x + 9) = 0

=> (x - 5)(x^2 - 8x + 25) = 0

=> x^3 - 8x^2 + 25x - 5x^2 + 40x - 125 = 0

=> x^3 - 13x^2 + 65x - 125 = 0

The simplest cubic equation with roots 5 and 4 + 3i is x^3 - 13x^2 + 65x - 125 = 0

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