if p(x)=2x^3 +x^2 -8x -4, show that x-2 is a factor of p(x). Hence, factorise p(x) into its linear factors.

Question

anonymous · Answer

i don't understand the method to factorise into linear factors.

hero · Answer

If x - 2 is a factor of p(x) then

2x^3 + x^2 - 8x - 4 = 2x^3 - 4x^2 + 5x^2 - 10x + 2x - 4
                               = 2x^2(x - 2) + 5x(x - 2) + 2(x - 2)
                               = (x - 2)(2x^2 + 5x + 2)

Finish it by factoring 2x^2 + 5x + 2

anonymous · Answer

@Hero how you get 2x^3 - 4x^2 + 5x^2 - 10x + 2x - 4?

hero · Answer

x^2 = -4x^2 + 5x^2
-8x = -10x + 2x

This method is called "term splitting"

hero · Answer

It is a process related to factor by grouping

hero · Answer

Once you split the appropriate terms, you can group them, then factor them two at a time knowing that x - 2 is a factor p(x).

anonymous · Answer

x^2 = -4x^2 + 5x^2
is this has a method? or just think our own? what if i am using -5x^2+6x^2? are the answers will be same?

hero · Answer

Yes it has a method. It comes from knowing that x - 2 is a factor of p(x). You just have to think about it a little more to understand.

anonymous · Answer

ok thank you very much :) , i think i need some time to understand.

raden · Answer

or just use the factor theorem, just show that p(2) = 0

p(x)=2x^3 +x^2 -8x -4
p(2) = 0 or not ?

hero · Answer

It goes without saying @RadEn. They already stated that x - 2 is a factor of p(x) which implies that p(2) = 0

anonymous · Answer

@RadEn yup ,my textbook example got this first step , the thing i don't understand is the factorisation of linear factor

anonymous · Answer

attachment

hero · Answer

The goal is to factor p(x). Figuring that p(2) = 0 only confirms there is no remainder and that x - 2 is definitely a factor. But I demonstrate an approach to actual factoring.

anonymous · Answer

okay let me try it :)

hero · Answer

Good luck

imstuck · Answer

You could also use synthetic division to show that x-2 is a factor...you would also come up with the resulting polynomial that can also be factored using the quadratic formula...it's easy that way!

hero · Answer

You could, but if you learn the method I've shown, then you'll have a method to factor rather than divide. It's not difficult at all once you understand it.

anonymous · Answer

@Hero  my textbook answer is (2x+1)(x+2)(x-2) , how to get that?

hero · Answer

Did you finish factoring 2x^2 + 5x + 2 ?

anonymous · Answer

yup it is (2x+1)(x+2)

anonymous · Answer

oh i know it , just move forward it -.-