SOLUTION: 1)5x^2-10x Factor 5x(x-5) Is this correct? 2)6x^2y^2+12xy^2+12y^2 Factor 6y^2(x^2+2x+2) Is this correct? 3)3a^3b-3ab^3 Factor 3ab(a^2-b^2) Is this correct?

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: 1)5x^2-10x Factor 5x(x-5) Is this correct? 2)6x^2y^2+12xy^2+12y^2 Factor 6y^2(x^2+2x+2) Is this correct? 3)3a^3b-3ab^3 Factor 3ab(a^2-b^2) Is this correct?       Log On


   

Question 148860: 1)5x^2-10x Factor
5x(x-5) Is this correct?


2)6x^2y^2+12xy^2+12y^2 Factor
6y^2(x^2+2x+2) Is this correct?


3)3a^3b-3ab^3 Factor
3ab(a^2-b^2) Is this correct?


4) a^2+2a-24 Factor
Is this factored already?


5) 4b^2-28b+49
I need help on this one.....


6)3m^3+27m Factor
3m(m^2+9) Is this correct?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!



1)


5x%5E2-10x Start with the given expression


5x%28x-2%29 Factor out the GCF 5x.



2)
Correct. You can check the answer by distributing




3)


Good so far, but can factor a%5E2-b%5E2 into %28a%2Bb%29%28a-b%29

So 3a%5E3b-3ab%5E3 completely factors to 3ab%28a%2Bb%29%28a-b%29




4)



Looking at the expression a%5E2%2B2a-24, we can see that the first coefficient is 1, the second coefficient is 2, and the last term is -24.


Now multiply the first coefficient 1 by the last term -24 to get %281%29%28-24%29=-24.


Now the question is: what two whole numbers multiply to -24 (the previous product) and add to the second coefficient 2?


To find these two numbers, we need to list all of the factors of -24 (the previous product).


Factors of -24:
1,2,3,4,6,8,12,24
-1,-2,-3,-4,-6,-8,-12,-24


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to -24. For instance, 1%2A24=-24, 2%2A12=-24, etc.


Since -24 is negative, this means that one factor is positive and one is negative.


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 2:


First NumberSecond NumberSum
1-241+(-24)=-23
2-122+(-12)=-10
3-83+(-8)=-5
4-64+(-6)=-2
-124-1+24=23
-212-2+12=10
-38-3+8=5
-46-4+6=2



From the table, we can see that the two numbers -4 and 6 add to 2 (the middle coefficient).


So the two numbers -4 and 6 both multiply to -24 and add to 2


Now replace the middle term 2a with -4a%2B6a. Remember, -4 and 6 add to 2. So this shows us that -4a%2B6a=2a.


a%5E2%2Bhighlight%28-4a%2B6a%29-24 Replace the second term 2a with -4a%2B6a.


%28a%5E2-4a%29%2B%286a-24%29 Group the terms into two pairs.


a%28a-4%29%2B%286a-24%29 Factor out the GCF a from the first group.


a%28a-4%29%2B6%28a-4%29 Factor out 6 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


%28a%2B6%29%28a-4%29 Combine like terms. Or factor out the common term a-4

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Answer:


So a%5E2%2B2a-24 factors to %28a%2B6%29%28a-4%29.


Note: you can check the answer by FOILing %28a%2B6%29%28a-4%29 to get a%5E2%2B2a-24 or by graphing the original expression and the answer (the two graphs should be identical).




5)



Looking at the expression 4b%5E2-28b%2B49, we can see that the first coefficient is 4, the second coefficient is -28, and the last term is 49.


Now multiply the first coefficient 4 by the last term 49 to get %284%29%2849%29=196.


Now the question is: what two whole numbers multiply to 196 (the previous product) and add to the second coefficient -28?


To find these two numbers, we need to list all of the factors of 196 (the previous product).


Factors of 196:
1,2,4,7,14,28,49,98,196
-1,-2,-4,-7,-14,-28,-49,-98,-196


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to 196. For instance, 1%2A196=196, 2%2A98=196, etc.


Since 196 is positive, this means that either
a) both factors are positive, or...
b) both factors are negative.


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -28:


First NumberSecond NumberSum
11961+196=197
2982+98=100
4494+49=53
7287+28=35
141414+14=28
-1-196-1+(-196)=-197
-2-98-2+(-98)=-100
-4-49-4+(-49)=-53
-7-28-7+(-28)=-35
-14-14-14+(-14)=-28



From the table, we can see that the two numbers -14 and -14 add to -28 (the middle coefficient).


So the two numbers -14 and -14 both multiply to 196 and add to -28


Now replace the middle term -28b with -14b-14b. Remember, -14 and -14 add to -28. So this shows us that -14b-14b=-28b.


4b%5E2%2Bhighlight%28-14b-14b%29%2B49 Replace the second term -28b with -14b-14b.


%284b%5E2-14b%29%2B%28-14b%2B49%29 Group the terms into two pairs.


2b%282b-7%29%2B%28-14b%2B49%29 Factor out the GCF 2b from the first group.


2b%282b-7%29-7%282b-7%29 Factor out 7 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


%282b-7%29%282b-7%29 Combine like terms. Or factor out the common term 2b-7


%282b-7%29%5E2 Collect and condense the terms.

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Answer:


So 4b%5E2-28b%2B49 factors to %282b-7%29%5E2.


Note: you can check the answer by FOILing %282b-7%29%5E2 to get 4b%5E2-28b%2B49 or by graphing the original expression and the answer (the two graphs should be identical).





6) Correct. You can check the answer by distributing