SOLUTION: How do I factor 16z to the 5th power + 12z to the fourth power - 10z to the third power

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: How do I factor 16z to the 5th power + 12z to the fourth power - 10z to the third power      Log On


   

Question 156977: How do I factor
16z to the 5th power + 12z to the fourth power - 10z to the third power
Found 2 solutions by jim_thompson5910, gonzo:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

16z%5E5%2B12z%5E4-10z%5E3 Start with the given expression


2z%5E3%288z%5E2%2B6z-5%29 Factor out the GCF 2z%5E3


Now let's focus on the inner expression 8z%5E2%2B6z-5




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Looking at 8z%5E2%2B6z-5 we can see that the first term is 8z%5E2 and the last term is -5 where the coefficients are 8 and -5 respectively.

Now multiply the first coefficient 8 and the last coefficient -5 to get -40. Now what two numbers multiply to -40 and add to the middle coefficient 6? Let's list all of the factors of -40:



Factors of -40:
1,2,4,5,8,10,20,40

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

These factors pair up and multiply to -40
(1)*(-40)
(2)*(-20)
(4)*(-10)
(5)*(-8)
(-1)*(40)
(-2)*(20)
(-4)*(10)
(-5)*(8)

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


Now which of these pairs add to 6? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 6


First NumberSecond NumberSum
1-401+(-40)=-39
2-202+(-20)=-18
4-104+(-10)=-6
5-85+(-8)=-3
-140-1+40=39
-220-2+20=18
-410-4+10=6
-58-5+8=3







From this list we can see that -4 and 10 add up to 6 and multiply to -40


Now looking at the expression 8z%5E2%2B6z-5, replace 6z with -4z%2B10z (notice -4z%2B10z adds up to 6z. So it is equivalent to 6z)

8z%5E2%2Bhighlight%28-4z%2B10z%29%2B-5


Now let's factor 8z%5E2-4z%2B10z-5 by grouping:


%288z%5E2-4z%29%2B%2810z-5%29 Group like terms


4z%282z-1%29%2B5%282z-1%29 Factor out the GCF of 4z out of the first group. Factor out the GCF of 5 out of the second group


%284z%2B5%29%282z-1%29 Since we have a common term of 2z-1, we can combine like terms

So 8z%5E2-4z%2B10z-5 factors to %284z%2B5%29%282z-1%29


So this also means that 8z%5E2%2B6z-5 factors to %284z%2B5%29%282z-1%29 (since 8z%5E2%2B6z-5 is equivalent to 8z%5E2-4z%2B10z-5)



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So our expression goes from 2z%5E3%288z%5E2%2B6z-5%29 and factors further to 2z%5E3%284z%2B5%29%282z-1%29


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Answer:


So 16z%5E5%2B12z%5E4-10z%5E3 completely factors to 2z%5E3%284z%2B5%29%282z-1%29

Answer by gonzo(654) About Me  (Show Source):
You can put this solution on YOUR website!
16z%5E5%2B12z%5E4-10z%5E3
you start by factoring out a common term to all which winds up being z%5E3
the formula becomes %28z%5E3%29%2A%2816z%5E2%2B12z-10%29.
16z%5E2%2B12z-10 is a quadratic equation that can be factored by sight (with a little fiddling around cause it wasn't that easy) so the completed factorization becomes %28z%5E3%29%2A%282z-1%29%2A%288z%2B10%29.