Consider the following Px 3x4 x3 6x2 14x 4 Find all the

Consider the following. P(x) = 3x^4 - x^3 - 6x^2 + 14x - 4 Find all the zeros of the polynomial function. x = Write the polynomial as a product of its leading coefficient and its linear factors. f(x) =

Solution

f(x) = 3x^4 -x^3 -6x^2 +14x -4

f(1) 0

f(2) 0

f(-1)0

f(-2) = 3(-2)^4 -(-2)^3 -6(-2)^2 +14(-2) -4

f(-2) = 3(16) +8 -6(4) -28 -4

f(-2) =48 + 8 -24 -28 -4

f(-2) = 56 -56=0

so -2 is root of f(x)

and f(1/3) =0

f(1/3) = 3(1/3)^4 -(1/3)^3 +6(1/3)^2+14(1/3) -4

f(1/3)=0

we can write f(x) in different terms

f(x) = (3x-1) (x^3 -2x +4)

f(x) = (3x-1) (x+2) (x-(1+i)) (x-(1-i))

so the roots are x = 1/3 ,x=-2 , x= 1+i , x=1-i

P(x) = (3x-1) (x+2) (x-(1+i)) (x-(1-i))