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

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

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) (Show Source): You can put this solution on YOUR website!

Start with the given expression

Factor out the GCF

Now let's focus on the inner expression

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Looking at we can see that the first term is and the last term is 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 Number Second Number Sum
1 -40 1+(-40)=-39
2 -20 2+(-20)=-18
4 -10 4+(-10)=-6
5 -8 5+(-8)=-3
-1 40 -1+40=39
-2 20 -2+20=18
-4 10 -4+10=6
-5 8 -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 , replace with (notice adds up to . So it is equivalent to )

Now let's factor by grouping:

Group like terms

Factor out the GCF of out of the first group. Factor out the GCF of out of the second group

Since we have a common term of , we can combine like terms

So factors to

So this also means that factors to (since is equivalent to )

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So our expression goes from and factors further to

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Answer:

So completely factors to

Answer by gonzo(654) (Show Source): You can put this solution on YOUR website!

you start by factoring out a common term to all which winds up being
the formula becomes .
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 .

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First Number	Second Number	Sum
1	-40	1+(-40)=-39
2	-20	2+(-20)=-18
4	-10	4+(-10)=-6
5	-8	5+(-8)=-3
-1	40	-1+40=39
-2	20	-2+20=18
-4	10	-4+10=6
-5	8	-5+8=3