SOLUTION: 8z^4-32z^3+30z^2

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Question 309944: 8z^4-32z^3+30z^2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming that you want to factor.


8z%5E4-32z%5E3%2B30z%5E2 Start with the given expression


2z%5E2%284z%5E2-16z%2B15%29 Factor out the GCF 2z%5E2


Now let's focus on the inner expression 4z%5E2-16z%2B15




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Looking at 4z%5E2-16z%2B15 we can see that the first term is 4z%5E2 and the last term is 15 where the coefficients are 4 and 15 respectively.

Now multiply the first coefficient 4 and the last coefficient 15 to get 60. Now what two numbers multiply to 60 and add to the middle coefficient -16? Let's list all of the factors of 60:



Factors of 60:
1,2,3,4,5,6,10,12,15,20,30,60

-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 60
1*60
2*30
3*20
4*15
5*12
6*10
(-1)*(-60)
(-2)*(-30)
(-3)*(-20)
(-4)*(-15)
(-5)*(-12)
(-6)*(-10)

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


Now which of these pairs add to -16? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -16

First NumberSecond NumberSum
1601+60=61
2302+30=32
3203+20=23
4154+15=19
5125+12=17
6106+10=16
-1-60-1+(-60)=-61
-2-30-2+(-30)=-32
-3-20-3+(-20)=-23
-4-15-4+(-15)=-19
-5-12-5+(-12)=-17
-6-10-6+(-10)=-16



From this list we can see that -6 and -10 add up to -16 and multiply to 60


Now looking at the expression 4z%5E2-16z%2B15, replace -16z with -6z%2B-10z (notice -6z%2B-10z adds up to -16z. So it is equivalent to -16z)

4z%5E2%2Bhighlight%28-6z%2B-10z%29%2B15


Now let's factor 4z%5E2-6z-10z%2B15 by grouping:


%284z%5E2-6z%29%2B%28-10z%2B15%29 Group like terms


2z%282z-3%29-5%282z-3%29 Factor out the GCF of 2z out of the first group. Factor out the GCF of -5 out of the second group


%282z-5%29%282z-3%29 Since we have a common term of 2z-3, we can combine like terms

So 4z%5E2-6z-10z%2B15 factors to %282z-5%29%282z-3%29


So this also means that 4z%5E2-16z%2B15 factors to %282z-5%29%282z-3%29 (since 4z%5E2-16z%2B15 is equivalent to 4z%5E2-6z-10z%2B15)



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So our expression goes from 2z%5E2%284z%5E2-16z%2B15%29 and factors further to 2z%5E2%282z-5%29%282z-3%29


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Answer:

So 8z%5E4-32z%5E3%2B30z%5E2 competely factors to 2z%5E2%282z-5%29%282z-3%29


In other words, 8z%5E4-32z%5E3%2B30z%5E2=2z%5E2%282z-5%29%282z-3%29