-16=(x+3)^3(x^2)

Question

I need help on solving this equation to find x.

Answer 1

-16=(x+3)^3(x^2)

Start by breaking down the problem:
Step One:  -16 = (x+3)(x+3)(x+3)(x^2)  (First, break it down and use the foil method)  

Broken down:

 (x+3) * (x+3)

= x^2 + 3x + 3x+ 9

= x^2 + 6x + 9

Now add that extra (x+3):

=(x^2+ 6x + 9)(x+3)

= (x^2 + 6x + 9)(x) + (x^2 + 6x+ 9)(3)

=x^3 + 6x^2 + 9x + 3x^2 + 18x + 27 (So put it in order by the largest polynomial)

=x^3 + 9x^2 + 27x + 27

Now add that extra (x^2)

=(x^3 + 9x^2 + 27x +27)(x^2)

=x^5 + 9x^4 + 27x^3 + 27x^2)

Now add the: -16 =

-16 = x^5 + 9x^4 + 27x^3 + 27x^2

Consider that -16 = (-4)(4). Because you need the negative answer, plug in -4 for x into the broken down polynomial:

-16 = (-4)(4)  = x^5 + 9x^4 + 27x^3 + 27x^2

If you plug in -4 for x you get: (-4^5) + 9(-4^4) + 27(-4^3) + 27 (-4^2) Simplified:

-16 = -1024 + 2304 + -1728 + 432 Simplified:

-16 = -16

 

 

 

 

Answer 2

A polynomial of degree 5.  There will be 5 roots.  Just wondering for a minute, the roots of + 16 are ±4.  +4 will not work, but -4 turns out to be a root.

x^5  +  9x^4  +27x^3  +  27x^2  +16  =  0

The remaining roots must be solved with a program.

They are -2.7 ± 1.187i

                 .2 ± .6487i