Find ALL the zeros of the function

Question

4) f(x) = x4 + x3 – 15x2 – 16x – 16

Find ALL the zeros of the function must show work

Answer 1

Rational Root Theorem says....

p = +or- {1,2,4,8,16}

q = +or- {1}

so p/q = p

of these, x=4 and x=-4 are rational solutions

since

f(4) = 256 + 64 - 240 - 64 - 16

= 320 - 240 -64-16

= 0

and

f(-4) = 256 - 64 - 240 + 64 - 16

= 192-240+64-16

= -48 + 64-16

= 0

Synthetic division says:

4 | 1 1 -15 -16 -16

4 20 20 16

_____________________________

1 5 5 4 0

-4 | 1 5 5 4

-4 -4 -4

________________________

1 1 1 0

x^2+x+1 is the quadratic quotient

which has solutions, per quadratic formula

[-1 +or- sqrt( 1^1 - 4(1)(1))]/ 2 =

[ -1 +or- sqrt( 1 - 4)]/2 =

(-1 +or- i sqrt(3))/2

the complete solution set is x= {+or- 4, (-1 +or- i sqrt(3)) / 2 }