SOLUTION: Factor completely. 25w^2 + 16 - 40w I haven't done these type of problems for a while now and forgot how to do them. Please help.

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Factor completely. 25w^2 + 16 - 40w I haven't done these type of problems for a while now and forgot how to do them. Please help.      Log On


   

Question 132417: Factor completely.
25w^2 + 16 - 40w
I haven't done these type of problems for a while now and forgot how to do them. Please help.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
25w%5E2+%2B+16+-+40w Start with the given expression


25w%5E2-+40w+%2B+16+ Rearrange the terms




Looking at 25w%5E2-40w%2B16 we can see that the first term is 25w%5E2 and the last term is 16 where the coefficients are 25 and 16 respectively.

Now multiply the first coefficient 25 and the last coefficient 16 to get 400. Now what two numbers multiply to 400 and add to the middle coefficient -40? Let's list all of the factors of 400:



Factors of 400:
1,2,4,5,8,10,16,20,25,40,50,80,100,200

-1,-2,-4,-5,-8,-10,-16,-20,-25,-40,-50,-80,-100,-200 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 400
1*400
2*200
4*100
5*80
8*50
10*40
16*25
20*20
(-1)*(-400)
(-2)*(-200)
(-4)*(-100)
(-5)*(-80)
(-8)*(-50)
(-10)*(-40)
(-16)*(-25)
(-20)*(-20)

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to -40? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -40

First NumberSecond NumberSum
14001+400=401
22002+200=202
41004+100=104
5805+80=85
8508+50=58
104010+40=50
162516+25=41
202020+20=40
-1-400-1+(-400)=-401
-2-200-2+(-200)=-202
-4-100-4+(-100)=-104
-5-80-5+(-80)=-85
-8-50-8+(-50)=-58
-10-40-10+(-40)=-50
-16-25-16+(-25)=-41
-20-20-20+(-20)=-40



From this list we can see that -20 and -20 add up to -40 and multiply to 400


Now looking at the expression 25w%5E2-40w%2B16, replace -40w with -20w%2B-20w (notice -20w%2B-20w adds up to -40w. So it is equivalent to -40w)

25w%5E2%2Bhighlight%28-20w%2B-20w%29%2B16


Now let's factor 25w%5E2-20w-20w%2B16 by grouping:


%2825w%5E2-20w%29%2B%28-20w%2B16%29 Group like terms


5w%285w-4%29-4%285w-4%29 Factor out the GCF of 5w out of the first group. Factor out the GCF of -4 out of the second group


%285w-4%29%285w-4%29 Since we have a common term of 5w-4, we can combine like terms

So 25w%5E2-20w-20w%2B16 factors to %285w-4%29%285w-4%29


So this also means that 25w%5E2-40w%2B16 factors to %285w-4%29%285w-4%29 (since 25w%5E2-40w%2B16 is equivalent to 25w%5E2-20w-20w%2B16)


note: %285w-4%29%285w-4%29 is equivalent to %285w-4%29%5E2 since the term 5w-4 occurs twice. So 25w%5E2-40w%2B16 also factors to %285w-4%29%5E2



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



Answer:
So 25w%5E2-40w%2B16 factors to %285w-4%29%5E2