Help me with step by step to finding the factors of f(x) = x^3 + 3x^2 – 10x – 24 - medal reward

Question

anonymous · Answer

The factors are (x-3), (x+2) and (x+4) but I need step by step help

debbieg · Answer

It kind of depends on what methods you are expected to use. are you allowed to incorporate the use of a graph?

debbieg · Answer

Because all REAL roots correspond to an x-intercept. So inspection of the graph would reveal these roots, or at least give you an idea of where the roots are, which you can always confirm by substituting in that value of x.

anonymous · Answer

divide x^3 + 3x^2 – 10x – 24 by (x-3) and i'm left with x^2+6x+8 
f(x) = (x-3)(x^2 + 6x + 8)
= (x-3)(x^2 + 2x + 4x+ 8)
= (x-3)[ x(x+2) + 4(x+2) ]
= (x-3)[ (x+2)(x+4) ]
= (x-3)(x+2)(x+4) 
?

debbieg · Answer

Or, once you have a "suspected" root, you can use synthetic division to verify that it is a root, and also to give you the quotient, which would then be a quadratic that can be factored.

debbieg · Answer

OK, that works fine. Once you have one factor, you can always factor it out by dividing. Either polynomial long division, or synthetic division (easier, if you know the method).

anonymous · Answer

So is my step by step correct?

debbieg · Answer

The key is figuring out what that first factor is... which is why I asked about using a graph. If you understand that each factor (x-c) gives a root x=c which gives an x-intercept, then the graph will help you find any real, rational roots.

Let me look at it closer, hang on...

debbieg · Answer

Yes, it looks fine.

debbieg · Answer

Once you find that first factor (I'm not sure if that was given to you? or how you found it)  then do exactly as you did: divide by that factor, leaving you with a quadratic, which you can then either factor, or use the quadratic formula if necessary to find the remaining roots (and hence, the remaining factors).

agent0smith · Answer

If you've used the rational roots theorem, you can use that here, there aren't too many roots to check... factors of the constant term, 24, over factors of the leading coefficient, 1,

so plus or minus 24, 12, 8, 6, 4, 3, 2, 1

Oh i guess that is a few... this is where a graph can speed things up (but you've already solved it anyway)

anonymous · Answer

Could I get help with another?

debbieg · Answer

Sure, but you should really post as a new question. :)