SOLUTION: I need help on factoring a trinomial. {{{r^3-2r^2-48r}}}

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: I need help on factoring a trinomial. {{{r^3-2r^2-48r}}}      Log On


   

Question 197362: I need help on factoring a trinomial.
r%5E3-2r%5E2-48r
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

r%5E3-2r%5E2-48r Start with the given expression


r%28r%5E2-2r-48%29 Factor out the GCF r


Now let's focus on the inner expression r%5E2-2r-48




------------------------------------------------------------



Looking at r%5E2-2r-48 we can see that the first term is r%5E2 and the last term is -48 where the coefficients are 1 and -48 respectively.

Now multiply the first coefficient 1 and the last coefficient -48 to get -48. Now what two numbers multiply to -48 and add to the middle coefficient -2? Let's list all of the factors of -48:



Factors of -48:
1,2,3,4,6,8,12,16,24,48

-1,-2,-3,-4,-6,-8,-12,-16,-24,-48 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -48
(1)*(-48)
(2)*(-24)
(3)*(-16)
(4)*(-12)
(6)*(-8)
(-1)*(48)
(-2)*(24)
(-3)*(16)
(-4)*(12)
(-6)*(8)

note: remember, the product of a negative and a positive number is a negative number


Now which of these pairs add to -2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -2

First NumberSecond NumberSum
1-481+(-48)=-47
2-242+(-24)=-22
3-163+(-16)=-13
4-124+(-12)=-8
6-86+(-8)=-2
-148-1+48=47
-224-2+24=22
-316-3+16=13
-412-4+12=8
-68-6+8=2



From this list we can see that 6 and -8 add up to -2 and multiply to -48


Now looking at the expression r%5E2-2r-48, replace -2r with 6r-8r (notice 6r-8r combines back to -2r. So it is equivalent to -2r)

r%5E2%2Bhighlight%286r-8r%29-48


Now let's factor r%5E2%2B6r-8r-48 by grouping:


%28r%5E2%2B6r%29%2B%28-8r-48%29 Group like terms


r%28r%2B6%29-8%28r%2B6%29 Factor out the GCF of r out of the first group. Factor out the GCF of -8 out of the second group


%28r-8%29%28r%2B6%29 Since we have a common term of r%2B6, we can combine like terms

So r%5E2%2B6r-8r-48 factors to %28r-8%29%28r%2B6%29


So this also means that r%5E2-2r-48 factors to %28r-8%29%28r%2B6%29 (since r%5E2-2r-48 is equivalent to r%5E2%2B6r-8r-48)



------------------------------------------------------------




So our expression goes from r%28r%5E2-2r-48%29 and factors further to r%28r-8%29%28r%2B6%29


------------------
Answer:

So r%5E3-2r%5E2-48r completely factors to r%28r-8%29%28r%2B6%29