Find the complex zeros of the polynomial function fx Write f

Find the complex zeros of the polynomial function f(x). Write f is factor form. f(x) = 2x^4 + 2x^3 - 11x^2 + x - 6 f(x) = 2x^4 - 2x^3 - 11x^2 - x - 6 Maximum = 4, Positive = 3, 1 Negative = 1

Solution

consider the eqn : 2x⁴+2x³-11x²+x -6 =0 to find the zeros

put x=2 : 32+16 -44+2-6 =0

hence x=2 is one root

by the method of synthetic division we can find the other roots

2 | 2 2 -11 1 -6

4 12 2 6

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2 6 1 3 { 0} = remainder is 0

(x-2) is one factor the other factors can be obtained from

2x³+6x²+x+3 = 2x²( x+3) +1(x+3) = (x+3) (2x²+1)

= (x+3) { (2x)^1/2+ i )} {  (2x)^1/2- i )} where i is the complex no i= (-1)^1/2

the factors of the polynomial f(x) = (x-2) (x+3) { (2x)^1/2+ i )} {  (2x)^1/2- i )}