SOLUTION: Factor completely. Say if prime: A. 40x^3-64x^2y B. 125r^3-8s^3 C. 13-14y+y^2 D. 5c^2+2c-3 E. 9x^2-30x+25 F. 16u^4-v^4

Algebra ->  Distributive-associative-commutative-properties -> SOLUTION: Factor completely. Say if prime: A. 40x^3-64x^2y B. 125r^3-8s^3 C. 13-14y+y^2 D. 5c^2+2c-3 E. 9x^2-30x+25 F. 16u^4-v^4      Log On


   

Question 208123: Factor completely. Say if prime:
A. 40x^3-64x^2y
B. 125r^3-8s^3
C. 13-14y+y^2
D. 5c^2+2c-3
E. 9x^2-30x+25
F. 16u^4-v^4
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.

A)


40x%5E3-64x%5E2y Start with the given expression.


8x%5E2%285x-8y%29 Factor out the GCF 8x%5E2.



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Answer:


So 40x%5E3-64x%5E2y factors to 8x%5E2%285x-8y%29.


In other words, 40x%5E3-64x%5E2y=8x%5E2%285x-8y%29.






B)



125r%5E3-8s%5E3 Start with the given expression.


%285r%29%5E3-%282s%29%5E3 Rewrite 125r%5E3 as %285r%29%5E3. Rewrite 8s%5E3 as %282s%29%5E3.


%285r-2s%29%28%285r%29%5E2%2B%285r%29%282s%29%2B%282s%29%5E2%29 Now factor by using the difference of cubes formula. Remember the difference of cubes formula is A%5E3-B%5E3=%28A-B%29%28A%5E2%2BAB%2BB%5E2%29


%285r-2s%29%2825r%5E2%2B10rs%2B4s%5E2%29 Multiply

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Answer:

So 125r%5E3-8s%5E3 factors to %285r-2s%29%2825r%5E2%2B10rs%2B4s%5E2%29.


In other words, 125r%5E3-8s%5E3=%285r-2s%29%2825r%5E2%2B10rs%2B4s%5E2%29




C)


13-14y%2By%5E2 Start with the given expression.


y%5E2-14y%2B13 Rearrange the terms.




Looking at the expression y%5E2-14y%2B13, we can see that the first coefficient is 1, the second coefficient is -14, and the last term is 13.


Now multiply the first coefficient 1 by the last term 13 to get %281%29%2813%29=13.


Now the question is: what two whole numbers multiply to 13 (the previous product) and add to the second coefficient -14?


To find these two numbers, we need to list all of the factors of 13 (the previous product).


Factors of 13:
1,13
-1,-13


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


These factors pair up and multiply to 13.
1*13 = 13
(-1)*(-13) = 13

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -14:


First NumberSecond NumberSum
1131+13=14
-1-13-1+(-13)=-14



From the table, we can see that the two numbers -1 and -13 add to -14 (the middle coefficient).


So the two numbers -1 and -13 both multiply to 13 and add to -14


Now replace the middle term -14y with -y-13y. Remember, -1 and -13 add to -14. So this shows us that -y-13y=-14y.


y%5E2%2Bhighlight%28-y-13y%29%2B13 Replace the second term -14y with -y-13y.


%28y%5E2-y%29%2B%28-13y%2B13%29 Group the terms into two pairs.


y%28y-1%29%2B%28-13y%2B13%29 Factor out the GCF y from the first group.


y%28y-1%29-13%28y-1%29 Factor out 13 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.


%28y-13%29%28y-1%29 Combine like terms. Or factor out the common term y-1


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Answer:


So 13-14y%2By%5E2 factors to %28y-13%29%28y-1%29.


In other words, 13-14y%2By%5E2=%28y-13%29%28y-1%29.