Polynomial Function help

[wind_surfer]

Hello, I just started with polynomial functions and I can't seem to get through a certain practice question. The question asks to state the degree and dominant term of f(x)=2x(x-3)^3(x+1)(4x-2)
I understand I need to take this out of factored form in order to find the degree and dominant term, but every time I try to I end up with f(x)=8x^6-4x^5-4x^4-216x^3-324x^2+108x
I know this is not right when I compare the original graph to my unfactored form. Does anyone know what I am doing wrong? Any help would be appreciated.
This is my work:
f(x)=2x(x-3)^3(x+1)(4x-2)
f(x)=2x(x^3-27)(x+1)(4x-2)
f(x)=2x(x^4+x^3-27x-27)(4x-2)
f(x)=2x(4x^5-2x^4+4x^4-2x^3-108x^2-54x-108x+54)
f(x)=8x^6-4x^5+8x^5-4x^4-216x^3-108x^2-216x^2+108x
f(x)=8x^6-4x^5-4x^4-216x^3-324x^2+108x

Thankyou

 

[HallsofIvy]

:cool:

Hello, I just started with polynomial functions and I can't seem to get through a certain practice question. The question asks to state the degree and dominant term of f(x)=2x(x-3)^3(x+1)(4x-2)
I understand I need to take this out of factored form in order to find the degree and dominant term, but every time I try to I end up with f(x)=8x^6-4x^5-4x^4-216x^3-324x^2+108x
I know this is not right when I compare the original graph to my unfactored form. Does anyone know what I am doing wrong? Any help would be appreciated.
This is my work:
f(x)=2x(x-3)^3(x+1)(4x-2)
f(x)=2x(x^3-27)(x+1)(4x-2)

This is your mistake. Do you see that when x= 4, \(\displaystyle (x- 3)^3= (4- 1)^3= 1^3= 1\) but \(\displaystyle x^3- 27= 4^3- 27= 64- 27= 37\)?
\(\displaystyle (x- 3)^3= x^3- 9x^2+ 27x- 27\), not \(\displaystyle x^3- 27\).

More generally, \(\displaystyle (x- a)^3= x^3- 3ax^2+ 3a^2x- a^3\) not \(\displaystyle x^3- a^3\).

f(x)=2x(x^4+x^3-27x-27)(4x-2)
f(x)=2x(4x^5-2x^4+4x^4-2x^3-108x^2-54x-108x+54)
f(x)=8x^6-4x^5+8x^5-4x^4-216x^3-108x^2-216x^2+108x
f(x)=8x^6-4x^5-4x^4-216x^3-324x^2+108x

Thankyou

 

[wind_surfer]

Thanks for your quick reply! That helped a lot.