POLYNOMIAL FUNCTIONS Solving Polynomial Equations (Page 6 of 6)

Now, perform synthetic division using the two zeros we have identified. The graph behavior suggests that x = -2 should correspond to a single factor, but x = 2/3 appears to be a repeated root. Therefore, use x = 2/3 twice in the synthetic division. The resulting quotient is the quadratic polynomial 9x^2-18x+27 = 9(x^2-2x+3), which can be set equal to zero and solved using the quadratic formula. The two complex solutions to this equation are x = 1+-i*sqrt(2).

Therefore, the solution set to the polynomial equation is {-2, 2/3, 2/3, 1+i*sqrt(2), 1-i*sqrt(2)}. We can also use this information to write the polynomial function in factored form as f(x) = 9(x+2)(x-2/3)^2(x-(1+i*sqrt(2)))(x-(1-i*sqrt(2))).
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