Factoring: How do you factor: 4x^3 + 8x^2​ - x - 2

[heather_brett]

How do you factor:

4x³ + 8x²​ - x - 2

 

[Deleted member 4993]

How do you factor:

4x³ + 8x²​ - x - 2

Have you been taught "rational factor" theorem of polynomials?

However, by inspection you can see that:

4x³ + 8x²​ - x - 2 = 4x² *x +4x² * 2 - (x + 2)

and continue....

 

[Harry_the_cat]

One of the things you can try when factorising an expression with 4 terms is to group them in two pairs.

For example:

4x³ + 8x²​ - x - 2 = (4x³ + 8x²​) + (-x - 2)

and then factorise each pair

= 4x² (x + 2) + -1( x + 2) ..... and hope you have a common factor!! (doesn't always work obviously)

= ( 4x² - 1 )( x + 2 ) .... and then factorise further if possible ... here you have the difference of two squares

= (2x + 1)(2x - 1)(x + 2)

 

[stapel]

How do you factor:

4x³ + 8x²​ - x - 2

One of the things you can try when factorising an expression with 4 terms is to group them in two pairs.

Note to "heather_brett": If your class hasn't covered this technique, or in case you missed the significance, let me tell you: This method is important! Make sure you understand this method! It will come up on your next test!

 

[Ishuda]

How do you factor:

4x³ + 8x²​ - x - 2

Another way: First, to make discussion easier, call the expression f [f(x)], i.e.
f(x) = 4x³ + 8x²​ - x - 2
Look at f; if x is very large f is always positive, if x is very small [large negative] f is always negative, so you know there is at least one zero for f. Start at x=0 and work your way out on both sides
x=0 then f=-2
x=1 then f=+9 [so a zero between 0 and 1]
x=-1 then f=+3 [so a zero between -1 and 0]

Now we know there is a zero between 0 and 1, -1 and 0, and -1 and minus infinity [f is positive for one of the pair and negative for the other]. Because this is a 'test question', I would suspect that the easiest zero to find is the one less that -1, so try -2, -3, -4, ... until you find it. Luckily when x=-2, f=0 so you don't have to do much. Now you know [x-(-2)]=x+2 if a factor of f [of course you may have seen this immediately and not had to go through all of this]. So you know
f(x) = (x+2)*something
and that should get you started.