Find all complex zeros of the given polynomial function and

Find all complex zeros of the given polynomial function, and write the polynomial in completely factored form. f(x) = -2x^3 - 7x^2 - 166x + 85 Find the complex zeros of f. Repeat any zeros if their multiplicity is greater than 1. x = 1/2, -2 + 9i, -2 -9i Use the complex zeros to factor f. f(x) =

Solution

We know that the complex zeros of a polynomial occur in conjugate pairs. Thus, if f(x) = -2x³-7x²-166x +85 , then f(x) must have at least 1 real zero. As per the Rational Zeros Theorem , if f(x) is a polynomial with integer coefficients and if p/q is a zero of f(x) ,then p is a factor of the constant term of f(x) and q is a factor of the leading coefficient of f(x) . The factors of 85 are ±1, ±5 and ±17, and the factors of -2 are ±1 and ±2. Further, f(1/2) = -2/8 -7/4 -166*1/2 +85 = -1/4 -7/4 – 83+85 = -2-83+85 = 0. Hence ½ is a zero of f(x) so that 2x-1 is a factor of f(x). Then f(x) = -x²(2x-1)-4x(2x-1)-85(2x-1) = (2x-1)(-x²-4x-85). It means that the remaining zeros of f(x) are zeros of -x²-4x-85. On using the quadratic formula, these zeros are [ -(-4)± {(-4)²-4*(-1)*(-85)}]/2*(-1) = [4± ( 16-340)]/(-2) = [4± (-324)]/(-2) = (-4 ± 18i)/(-2) = -2± 9i. Thus, the zeros of f(x) are x =1/2, -2+9i and -2-9i. Also, f(x) = (2x-1)(x+2-9i)(x+2+9i).