SOLUTION: Please help me factor this trinomail: {{{y^2(y+1)-4y(y+1)-21(y+1)}}} Thank You.

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Question 206639This question is from textbook Introductory Algebra
: Please help me factor this trinomail:
y%5E2%28y%2B1%29-4y%28y%2B1%29-21%28y%2B1%29
Thank You. This question is from textbook Introductory Algebra

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

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y%5E2%28y%2B1%29-4y%28y%2B1%29-21%28y%2B1%29 Start with the given expression.


%28y%2B1%29%28y%5E2-4y-21%29 Factor out the GCF y%2B1



Now let's factor y%5E2-4y-21:


Looking at the expression y%5E2-4y-21, we can see that the first coefficient is 1, the second coefficient is -4, and the last term is -21.


Now multiply the first coefficient 1 by the last term -21 to get %281%29%28-21%29=-21.


Now the question is: what two whole numbers multiply to -21 (the previous product) and add to the second coefficient -4?


To find these two numbers, we need to list all of the factors of -21 (the previous product).


Factors of -21:
1,3,7,21
-1,-3,-7,-21


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


These factors pair up and multiply to -21.
1*(-21) = -21
3*(-7) = -21
(-1)*(21) = -21
(-3)*(7) = -21

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -4:


First NumberSecond NumberSum
1-211+(-21)=-20
3-73+(-7)=-4
-121-1+21=20
-37-3+7=4



From the table, we can see that the two numbers 3 and -7 add to -4 (the middle coefficient).


So the two numbers 3 and -7 both multiply to -21 and add to -4


Now replace the middle term -4y with 3y-7y. Remember, 3 and -7 add to -4. So this shows us that 3y-7y=-4y.


y%5E2%2Bhighlight%283y-7y%29-21 Replace the second term -4y with 3y-7y.


%28y%5E2%2B3y%29%2B%28-7y-21%29 Group the terms into two pairs.


y%28y%2B3%29%2B%28-7y-21%29 Factor out the GCF y from the first group.


y%28y%2B3%29-7%28y%2B3%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.


%28y-7%29%28y%2B3%29 Combine like terms. Or factor out the common term y%2B3



So this means that %28y%2B1%29%28y%5E2-4y-21%29 factors to %28y%2B1%29%28y-7%29%28y%2B3%29


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Answer:


So y%5E2%28y%2B1%29-4y%28y%2B1%29-21%28y%2B1%29 completely factors to %28y%2B1%29%28y-7%29%28y%2B3%29.


In other words, y%5E2%28y%2B1%29-4y%28y%2B1%29-21%28y%2B1%29=%28y%2B1%29%28y-7%29%28y%2B3%29.



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