Find ALL the zeros of the function Find alte zeros of the fi

Find alte zeros of the fiunction for 2r\'-n\'-r ,ar 18) 19) Use the answers from Problem #18 to write the factorization of 2r -r\'-r t4rt2

Solution

Soution:

f(x) = 2x⁴ - x³-x²+4x +2

f(x) =0

2x⁴ - x³-x²+4x +2 = 0

Use the rational root theorem

a₀ = 2 and a_n= 2

let\'s check the following numbers +/1 and +/- 2

as we check x = - 1

we get f(x) = 0

x+1 is factor

(2x⁴ - x³-x²+4x +2) / (x+1) = 2x³-3x²+2x+2

thus f(x) = (2x³-3x²+2x+2 ) (x+1)

let again g(x) = (2x³-3x²+2x+2 )

x = -1/2 we get g(-1/2) = 0

thus g(x) / (2x+1) = (2x³-3x²+2x+2 )/(2x+1) = x² -2x +2

now let

h(x) = x² -2x +2 =0

x = 1+i and x = 1-i

Now

f(x) = (x+1) (2x+1) (2x²-2x+2)

f(x) =(x+1) (2x+1) (x-(1+i)) (x-(1-i)) =0

x = 1 and x = -1/2 and x = 1+i and x = 1-i

all the zeros

(19)

f(x) = 2x⁴ - x³-x²+4x +2 =(x+1) (2x+1) (x-(1+i)) (x-(1-i))

answer