SOLUTION: 46. Factor completely. -3t^3+ 3t^2-6t 60. Factor polynomial completely. 10a^2+ab-2b^2 80. Factor completely. 4m^2+20m+25 90. Factor each polynomial completely, given tha

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: 46. Factor completely. -3t^3+ 3t^2-6t 60. Factor polynomial completely. 10a^2+ab-2b^2 80. Factor completely. 4m^2+20m+25 90. Factor each polynomial completely, given tha      Log On


   

Question 181957: 46. Factor completely. -3t^3+ 3t^2-6t

60. Factor polynomial completely. 10a^2+ab-2b^2
80. Factor completely. 4m^2+20m+25
90. Factor each polynomial completely, given that the binomial Following it is a factor of the polynomial. x^3-4x^2-3x-10, x-5
102. Solve each equation. t2+1=13/6t
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'll do the first three to get you started:


# 46


-3t%5E3%2B3t%5E2-6t Start with the given expression


-3t%28t%5E2-t%2B2%29 Factor out the GCF -3t


So -3t%5E3%2B3t%5E2-6t factors to -3t%28t%5E2-t%2B2%29

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# 60


Looking at 10a%5E2%2Bab-2b%5E2 we can see that the first term is 10a%5E2 and the last term is -2b%5E2 where the coefficients are 10 and -2 respectively.

Now multiply the first coefficient 10 and the last coefficient -2 to get -20. Now what two numbers multiply to -20 and add to the middle coefficient 1? Let's list all of the factors of -20:



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

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

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

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


Now which of these pairs add to 1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 1

First NumberSecond NumberSum
1-201+(-20)=-19
2-102+(-10)=-8
4-54+(-5)=-1
-120-1+20=19
-210-2+10=8
-45-4+5=1



From this list we can see that -4 and 5 add up to 1 and multiply to -20


Now looking at the expression 10a%5E2%2Bab-2b%5E2, replace ab with -4ab%2B5ab (notice -4ab%2B5ab adds up to ab. So it is equivalent to ab)


10a%5E2%2Bhighlight%28-4ab%2B5ab%29%2B-2b%5E2


Now let's factor 10a%5E2-4ab%2B5ab-2b%5E2 by grouping:


%2810a%5E2-4ab%29%2B%285ab-2b%5E2%29 Group like terms


2a%285a-2b%29%2Bb%285a-2b%29 Factor out the GCF of 2a out of the first group. Factor out the GCF of b out of the second group


%282a%2Bb%29%285a-2b%29 Since we have a common term of 5a-2b, we can combine like terms


So 10a%5E2-4ab%2B5ab-2b%5E2 factors to %282a%2Bb%29%285a-2b%29


So this also means that 10a%5E2%2Bab-2b%5E2 factors to %282a%2Bb%29%285a-2b%29 (since 10a%5E2%2Bab-2b%5E2 is equivalent to 10a%5E2-4ab%2B5ab-2b%5E2)



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Answer:
So 10a%5E2%2Bab-2b%5E2 factors to %282a%2Bb%29%285a-2b%29


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# 80



Looking at 4m%5E2%2B20m%2B25 we can see that the first term is 4m%5E2 and the last term is 25 where the coefficients are 4 and 25 respectively.

Now multiply the first coefficient 4 and the last coefficient 25 to get 100. Now what two numbers multiply to 100 and add to the middle coefficient 20? Let's list all of the factors of 100:



Factors of 100:
1,2,4,5,10,20,25,50

-1,-2,-4,-5,-10,-20,-25,-50 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 100
1*100
2*50
4*25
5*20
10*10
(-1)*(-100)
(-2)*(-50)
(-4)*(-25)
(-5)*(-20)
(-10)*(-10)

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


Now which of these pairs add to 20? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 20

First NumberSecond NumberSum
11001+100=101
2502+50=52
4254+25=29
5205+20=25
101010+10=20
-1-100-1+(-100)=-101
-2-50-2+(-50)=-52
-4-25-4+(-25)=-29
-5-20-5+(-20)=-25
-10-10-10+(-10)=-20



From this list we can see that 10 and 10 add up to 20 and multiply to 100


Now looking at the expression 4m%5E2%2B20m%2B25, replace 20m with 10m%2B10m (notice 10m%2B10m adds up to 20m. So it is equivalent to 20m)

4m%5E2%2Bhighlight%2810m%2B10m%29%2B25


Now let's factor 4m%5E2%2B10m%2B10m%2B25 by grouping:


%284m%5E2%2B10m%29%2B%2810m%2B25%29 Group like terms


2m%282m%2B5%29%2B5%282m%2B5%29 Factor out the GCF of 2m out of the first group. Factor out the GCF of 5 out of the second group


%282m%2B5%29%282m%2B5%29 Since we have a common term of 2m%2B5, we can combine like terms

So 4m%5E2%2B10m%2B10m%2B25 factors to %282m%2B5%29%282m%2B5%29


So this also means that 4m%5E2%2B20m%2B25 factors to %282m%2B5%29%282m%2B5%29 (since 4m%5E2%2B20m%2B25 is equivalent to 4m%5E2%2B10m%2B10m%2B25)


note: %282m%2B5%29%282m%2B5%29 is equivalent to %282m%2B5%29%5E2 since the term 2m%2B5 occurs twice. So 4m%5E2%2B20m%2B25 also factors to %282m%2B5%29%5E2



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Answer:
So 4m%5E2%2B20m%2B25 factors to %282m%2B5%29%5E2