SOLUTION: Please help! Can't find any examples of these in my book, there's always a different exponent or something.... Factor the polynomial 1.) (32k^3 n^2 m)+(112k^2 nm^2)+(98km^3)

Algebra ->  Expressions-with-variables -> SOLUTION: Please help! Can't find any examples of these in my book, there's always a different exponent or something.... Factor the polynomial 1.) (32k^3 n^2 m)+(112k^2 nm^2)+(98km^3)       Log On


   

Question 191469: Please help! Can't find any examples of these in my book, there's always a different exponent or something....
Factor the polynomial
1.) (32k^3 n^2 m)+(112k^2 nm^2)+(98km^3)
2.) (9k^2)-(16m^2)
3.) (147a^4 b)-(48b^3)
4.) (x^2)-(2/3)x+c
I know I asked more than one but I really need these answers asap I will not ask anymore today. Also I'm on an iPhone and it's already a pain to text on it I couldn't imgine going back and forth on a website. Thank you so much!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'll do the first two to get you started, repost if you need more help.


# 1

32k%5E3n%5E2m%2B112k%5E2nm%5E2%2B98km%5E3 Start with the given expression


2km%2816k%5E2n%5E2%2B56knm%2B49m%5E2%29 Factor out the GCF 2km


Now let's focus on the inner expression 16k%5E2n%5E2%2B56knm%2B49m%5E2




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Looking at 16k%5E2n%5E2%2B56knm%2B49m%5E2 we can see that the first term is 16k%5E2n%5E2 and the last term is 49m%5E2 where the coefficients are 16 and 49 respectively.

Now multiply the first coefficient 16 and the last coefficient 49 to get 784. Now what two numbers multiply to 784 and add to the middle coefficient 56? Let's list all of the factors of 784:



Factors of 784:
1,2,4,7,8,14,16,28,49,56,98,112,196,392

-1,-2,-4,-7,-8,-14,-16,-28,-49,-56,-98,-112,-196,-392 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 784
1*784
2*392
4*196
7*112
8*98
14*56
16*49
28*28
(-1)*(-784)
(-2)*(-392)
(-4)*(-196)
(-7)*(-112)
(-8)*(-98)
(-14)*(-56)
(-16)*(-49)
(-28)*(-28)

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


Now which of these pairs add to 56? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 56

First NumberSecond NumberSum
17841+784=785
23922+392=394
41964+196=200
71127+112=119
8988+98=106
145614+56=70
164916+49=65
282828+28=56
-1-784-1+(-784)=-785
-2-392-2+(-392)=-394
-4-196-4+(-196)=-200
-7-112-7+(-112)=-119
-8-98-8+(-98)=-106
-14-56-14+(-56)=-70
-16-49-16+(-49)=-65
-28-28-28+(-28)=-56



From this list we can see that 28 and 28 add up to 56 and multiply to 784


Now looking at the expression 16k%5E2n%5E2%2B56knm%2B49m%5E2, replace 56knm with 28knm%2B28knm (notice 28knm%2B28knm adds up to 56knm. So it is equivalent to 56knm)

16k%5E2n%5E2%2Bhighlight%2828knm%2B28knm%29%2B49m%5E2


Now let's factor 16k%5E2n%5E2%2B28knm%2B28knm%2B49m%5E2 by grouping:


%2816k%5E2n%5E2%2B28knm%29%2B%2828knm%2B49m%5E2%29 Group like terms


4kn%284kn%2B7m%29%2B7m%284kn%2B7m%29 Factor out the GCF of 4kn out of the first group. Factor out the GCF of 7m out of the second group


%284kn%2B7m%29%284kn%2B7m%29 Combine like terms.


%284kn%2B7m%29%5E2 Condense the terms.


So 16k%5E2n%5E2%2B28knm%2B28knm%2B49m%5E2 factors to %284kn%2B7m%29%5E2

So this also means that 16k%5E2n%5E2%2B56knm%2B49m%5E2 factors to %284kn%2B7m%29%5E2 (since 16k%5E2n%5E2%2B56knm%2B49m%5E2 is equivalent to 16k%5E2n%5E2%2B28knm%2B28knm%2B49m%5E2)



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So our expression goes from 2km%2816k%5E2n%5E2%2B56knm%2B49m%5E2%29 and factors further to 2km%284kn%2B7m%29%5E2


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Answer:

So 32k%5E3n%5E2m%2B112k%5E2nm%5E2%2B98km%5E3 completely factors to 2km%284kn%2B7m%29%5E2





# 2


9k%5E2-16m%5E2 Start with the given expression.


%283k%29%5E2-16m%5E2 Rewrite 9k%5E2 as %283k%29%5E2.


%283k%29%5E2-%284m%29%5E2 Rewrite 16m%5E2 as %284m%29%5E2.


Notice how we have a difference of squares. So let's use the difference of squares formula A%5E2-B%5E2=%28A%2BB%29%28A-B%29 to factor the expression:


%283k%2B4m%29%283k-4m%29 Factor the expression using the difference of squares.


So 9k%5E2-16m%5E2 factors to %283k%2B4m%29%283k-4m%29.


In other words 9k%5E2-16m%5E2=%283k%2B4m%29%283k-4m%29.