SOLUTION: Please help Im stuck on these problems. Factoring 15a^3-30a^4-5a^5 x^2-8+7x 4a-5b+6c 49b^2-18+21b

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Please help Im stuck on these problems. Factoring 15a^3-30a^4-5a^5 x^2-8+7x 4a-5b+6c 49b^2-18+21b      Log On


   

Question 173375: Please help Im stuck on these problems. Factoring
15a^3-30a^4-5a^5
x^2-8+7x
4a-5b+6c
49b^2-18+21b
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
15a%5E3-30a%5E4-5a%5E5 Start with the given expression.


-5a%5E5-30a%5E4%2B15a%5E3 Rearrange the terms in descending order


-5a%5E3%28a%5E2%2B6a-3%29 Factor out the GCF -5a%5E3


Now let's try to factor the inner expression a%5E2%2B6a-3




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Looking at a%5E2%2B6a-3 we can see that the first term is a%5E2 and the last term is -3 where the coefficients are 1 and -3 respectively.

Now multiply the first coefficient 1 and the last coefficient -3 to get -3. Now what two numbers multiply to -3 and add to the middle coefficient 6? Let's list all of the factors of -3:



Factors of -3:
1,3

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

These factors pair up and multiply to -3
(1)*(-3)
(-1)*(3)

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
1-31+(-3)=-2
-13-1+3=2



None of these pairs of factors add to 6. So the expression 1a%5E2%2B6a-3 cannot be factored

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Answer:

So -5a%5E5-30a%5E4%2B15a%5E3 factors to -5a%5E3%28a%5E2%2B6a-3%29





# 2


x%5E2-8%2B7x Start with the given expression.


x%5E2%2B7x-8 Rearrange the terms.




Looking at the expression x%5E2%2B7x-8, we can see that the first coefficient is 1, the second coefficient is 7, and the last term is -8.


Now multiply the first coefficient 1 by the last term -8 to get %281%29%28-8%29=-8.


Now the question is: what two whole numbers multiply to -8 (the previous product) and add to the second coefficient 7?


To find these two numbers, we need to list all of the factors of -8 (the previous product).


Factors of -8:
1,2,4,8
-1,-2,-4,-8


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


These factors pair up and multiply to -8.
1*(-8)
2*(-4)
(-1)*(8)
(-2)*(4)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 7:


First NumberSecond NumberSum
1-81+(-8)=-7
2-42+(-4)=-2
-18-1+8=7
-24-2+4=2



From the table, we can see that the two numbers -1 and 8 add to 7 (the middle coefficient).


So the two numbers -1 and 8 both multiply to -8 and add to 7


Now replace the middle term 7x with -x%2B8x. Remember, -1 and 8 add to 7. So this shows us that -x%2B8x=7x.


x%5E2%2Bhighlight%28-x%2B8x%29-8 Replace the second term 7x with -x%2B8x.


%28x%5E2-x%29%2B%288x-8%29 Group the terms into two pairs.


x%28x-1%29%2B%288x-8%29 Factor out the GCF x from the first group.


x%28x-1%29%2B8%28x-1%29 Factor out 8 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.


%28x%2B8%29%28x-1%29 Combine like terms. Or factor out the common term x-1

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Answer:


So x%5E2%2B7x-8 factors to %28x%2B8%29%28x-1%29.


Note: you can check the answer by FOILing %28x%2B8%29%28x-1%29 to get x%5E2%2B7x-8 or by graphing the original expression and the answer (the two graphs should be identical).




# 3


Since there is nothing in common between the terms of 4a-5b%2B6c and there are no exponents, this means that we cannot factor 4a-5b%2B6c






# 4


49b%5E2-18%2B21b Start with the given expression.


49b%5E2%2B21b-18 Rearrange the terms.




Looking at the expression 49b%5E2%2B21b-18, we can see that the first coefficient is 49, the second coefficient is 21, and the last term is -18.


Now multiply the first coefficient 49 by the last term -18 to get %2849%29%28-18%29=-882.


Now the question is: what two whole numbers multiply to -882 (the previous product) and add to the second coefficient 21?


To find these two numbers, we need to list all of the factors of -882 (the previous product).


Factors of -882:
1,2,3,6,7,9,14,18,21,42,49,63,98,126,147,294,441,882
-1,-2,-3,-6,-7,-9,-14,-18,-21,-42,-49,-63,-98,-126,-147,-294,-441,-882


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


These factors pair up and multiply to -882.
1*(-882)
2*(-441)
3*(-294)
6*(-147)
7*(-126)
9*(-98)
14*(-63)
18*(-49)
21*(-42)
(-1)*(882)
(-2)*(441)
(-3)*(294)
(-6)*(147)
(-7)*(126)
(-9)*(98)
(-14)*(63)
(-18)*(49)
(-21)*(42)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 21:


First NumberSecond NumberSum
1-8821+(-882)=-881
2-4412+(-441)=-439
3-2943+(-294)=-291
6-1476+(-147)=-141
7-1267+(-126)=-119
9-989+(-98)=-89
14-6314+(-63)=-49
18-4918+(-49)=-31
21-4221+(-42)=-21
-1882-1+882=881
-2441-2+441=439
-3294-3+294=291
-6147-6+147=141
-7126-7+126=119
-998-9+98=89
-1463-14+63=49
-1849-18+49=31
-2142-21+42=21



From the table, we can see that the two numbers -21 and 42 add to 21 (the middle coefficient).


So the two numbers -21 and 42 both multiply to -882 and add to 21


Now replace the middle term 21b with -21b%2B42b. Remember, -21 and 42 add to 21. So this shows us that -21b%2B42b=21b.


49b%5E2%2Bhighlight%28-21b%2B42b%29-18 Replace the second term 21b with -21b%2B42b.


%2849b%5E2-21b%29%2B%2842b-18%29 Group the terms into two pairs.


7b%287b-3%29%2B%2842b-18%29 Factor out the GCF 7b from the first group.


7b%287b-3%29%2B6%287b-3%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.


%287b%2B6%29%287b-3%29 Combine like terms. Or factor out the common term 7b-3

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Answer:


So 49b%5E2%2B21b-18 factors to %287b%2B6%29%287b-3%29.


Note: you can check the answer by FOILing %287b%2B6%29%287b-3%29 to get 49b%5E2%2B21b-18 or by graphing the original expression and the answer (the two graphs should be identical).