SOLUTION: These need to be factored completely 30z^8 + 44z^5 +16z^2 Could it be 2z^2(3z^ + 2)(5z^3 +4) 24x² + 14xy +2y² (m+n)(x+3) + (m+n)(5+5) Could it be (m+n+3)(x

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: These need to be factored completely 30z^8 + 44z^5 +16z^2 Could it be 2z^2(3z^ + 2)(5z^3 +4) 24x² + 14xy +2y² (m+n)(x+3) + (m+n)(5+5) Could it be (m+n+3)(x      Log On


   

Question 181934: These need to be factored completely
30z^8 + 44z^5 +16z^2 Could it be 2z^2(3z^ + 2)(5z^3 +4)
24x² + 14xy +2y²
(m+n)(x+3) + (m+n)(5+5) Could it be (m+n+3)(x+y+5)
Solve using the principal of zero products
(x+ 1/7)(x-4/5) = 0
Find the x-intercepts for the graph of the equation
Y = x² + 4x -45 Could it be (-9,0,(5,0)
Factor by grouping
-36x² -30x + 36 Could it be -6(3x-2)(2x+3)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'll do the first two, which will hopefully help you with the rest of the problems. If not, then repost.


# 1



30z%5E8%2B44z%5E5%2B16z%5E2 Start with the given expression


2z%5E2%2815z%5E6%2B22z%5E3%2B8%29 Factor out the GCF 2z%5E2


Now let's focus on the inner expression 15z%5E6%2B22z%5E3%2B8




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Looking at 15z%5E6%2B22z%5E3%2B8 we can see that the first term is 15z%5E6 and the last term is 8 where the coefficients are 15 and 8 respectively.

Now multiply the first coefficient 15 and the last coefficient 8 to get 120. Now what two numbers multiply to 120 and add to the middle coefficient 22? Let's list all of the factors of 120:



Factors of 120:
1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120

-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-20,-24,-30,-40,-60,-120 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 120
1*120
2*60
3*40
4*30
5*24
6*20
8*15
10*12
(-1)*(-120)
(-2)*(-60)
(-3)*(-40)
(-4)*(-30)
(-5)*(-24)
(-6)*(-20)
(-8)*(-15)
(-10)*(-12)

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


Now which of these pairs add to 22? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 22


First NumberSecond NumberSum
11201+120=121
2602+60=62
3403+40=43
4304+30=34
5245+24=29
6206+20=26
8158+15=23
101210+12=22
-1-120-1+(-120)=-121
-2-60-2+(-60)=-62
-3-40-3+(-40)=-43
-4-30-4+(-30)=-34
-5-24-5+(-24)=-29
-6-20-6+(-20)=-26
-8-15-8+(-15)=-23
-10-12-10+(-12)=-22




From this list we can see that 10 and 12 add up to 22 and multiply to 120


Now looking at the expression 15z%5E6%2B22z%5E3%2B8, replace 22z%5E3 with 10z%5E3%2B12z%5E3 (notice 10z%5E3%2B12z%5E3 adds up to 22z%5E3. So it is equivalent to 22z%5E3)

15z%5E6%2Bhighlight%2810z%5E3%2B12z%5E3%29%2B8


Now let's factor 15z%5E6%2B10z%5E3%2B12z%5E3%2B8 by grouping:


%2815z%5E6%2B10z%5E3%29%2B%2812z%5E3%2B8%29 Group like terms


5z%5E3%283z%5E3%2B2%29%2B4%283z%5E3%2B2%29 Factor out the GCF of 5z%5E3 out of the first group. Factor out the GCF of 4 out of the second group


%285z%5E3%2B4%29%283z%5E3%2B2%29 Since we have a common term of 3z%5E3%2B2, we can combine like terms

So 15z%5E6%2B10z%5E3%2B12z%5E3%2B8 factors to %285z%5E3%2B4%29%283z%5E3%2B2%29


So this also means that 15z%5E6%2B22z%5E3%2B8 factors to %285z%5E3%2B4%29%283z%5E3%2B2%29 (since 15z%5E6%2B22z%5E3%2B8 is equivalent to 15z%5E6%2B10z%5E3%2B12z%5E3%2B8)



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So our expression goes from 2z%5E2%2815z%5E6%2B22z%5E3%2B8%29 and factors further to 2z%5E2%285z%5E3%2B4%29%283z%5E3%2B2%29


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Answer:

So 30z%5E8%2B44z%5E5%2B16z%5E2 completely factors to 2z%5E2%285z%5E3%2B4%29%283z%5E3%2B2%29




# 2




24x%5E2%2B14xy%2B2y%5E2 Start with the given expression


2%2812x%5E2%2B7xy%2By%5E2%29 Factor out the GCF 2


Now let's focus on the inner expression 12x%5E2%2B7xy%2By%5E2




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Looking at 12x%5E2%2B7xy%2By%5E2 we can see that the first term is 12x%5E2 and the last term is y%5E2 where the coefficients are 12 and 1 respectively.

Now multiply the first coefficient 12 and the last coefficient 1 to get 12. Now what two numbers multiply to 12 and add to the middle coefficient 7? Let's list all of the factors of 12:



Factors of 12:
1,2,3,4,6,12

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

These factors pair up and multiply to 12
1*12
2*6
3*4
(-1)*(-12)
(-2)*(-6)
(-3)*(-4)

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


Now which of these pairs add to 7? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 7

First NumberSecond NumberSum
1121+12=13
262+6=8
343+4=7
-1-12-1+(-12)=-13
-2-6-2+(-6)=-8
-3-4-3+(-4)=-7



From this list we can see that 3 and 4 add up to 7 and multiply to 12


Now looking at the expression 12x%5E2%2B7xy%2By%5E2, replace 7xy with 3xy%2B4xy (notice 3xy%2B4xy adds up to 7xy. So it is equivalent to 7xy)

12x%5E2%2Bhighlight%283xy%2B4xy%29%2By%5E2


Now let's factor 12x%5E2%2B3xy%2B4xy%2By%5E2 by grouping:


%2812x%5E2%2B3xy%29%2B%284xy%2By%5E2%29 Group like terms


3x%284x%2By%29%2By%284x%2By%29 Factor out the GCF of 3x out of the first group. Factor out the GCF of y out of the second group


%283x%2By%29%284x%2By%29 Since we have a common term of 4x%2By, we can combine like terms

So 12x%5E2%2B3xy%2B4xy%2By%5E2 factors to %283x%2By%29%284x%2By%29


So this also means that 12x%5E2%2B7xy%2By%5E2 factors to %283x%2By%29%284x%2By%29 (since 12x%5E2%2B7xy%2By%5E2 is equivalent to 12x%5E2%2B3xy%2B4xy%2By%5E2)



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So our expression goes from 2%2812x%5E2%2B7xy%2By%5E2%29 and factors further to 2%283x%2By%29%284x%2By%29


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Answer:

So 24x%5E2%2B14xy%2B2y%5E2 completely factors to 2%283x%2By%29%284x%2By%29