SOLUTION: 4b^3 + 16b^2 + 15b factor

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Question 925884: 4b^3 + 16b^2 + 15b factor
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

4b%5E3%2B16b%5E2%2B15b Start with the given expression.


b%284b%5E2%2B16b%2B15%29 Factor out the GCF b.


Now let's try to factor the inner expression 4b%5E2%2B16b%2B15


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Looking at the expression 4b%5E2%2B16b%2B15, we can see that the first coefficient is 4, the second coefficient is 16, and the last term is 15.


Now multiply the first coefficient 4 by the last term 15 to get %284%29%2815%29=60.


Now the question is: what two whole numbers multiply to 60 (the previous product) and add to the second coefficient 16?


To find these two numbers, we need to list all of the factors of 60 (the previous product).


Factors of 60:
1,2,3,4,5,6,10,12,15,20,30,60
-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60


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


These factors pair up and multiply to 60.
1*60 = 60
2*30 = 60
3*20 = 60
4*15 = 60
5*12 = 60
6*10 = 60
(-1)*(-60) = 60
(-2)*(-30) = 60
(-3)*(-20) = 60
(-4)*(-15) = 60
(-5)*(-12) = 60
(-6)*(-10) = 60

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 16:


First NumberSecond NumberSum
1601+60=61
2302+30=32
3203+20=23
4154+15=19
5125+12=17
6106+10=16
-1-60-1+(-60)=-61
-2-30-2+(-30)=-32
-3-20-3+(-20)=-23
-4-15-4+(-15)=-19
-5-12-5+(-12)=-17
-6-10-6+(-10)=-16



From the table, we can see that the two numbers 6 and 10 add to 16 (the middle coefficient).


So the two numbers 6 and 10 both multiply to 60 and add to 16


Now replace the middle term 16b with 6b%2B10b. Remember, 6 and 10 add to 16. So this shows us that 6b%2B10b=16b.


4b%5E2%2Bhighlight%286b%2B10b%29%2B15 Replace the second term 16b with 6b%2B10b.


%284b%5E2%2B6b%29%2B%2810b%2B15%29 Group the terms into two pairs.


2b%282b%2B3%29%2B%2810b%2B15%29 Factor out the GCF 2b from the first group.


2b%282b%2B3%29%2B5%282b%2B3%29 Factor out 5 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%2B5%29%282b%2B3%29 Combine like terms. Or factor out the common term 2b%2B3


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So b%284b%5E2%2B16b%2B15%29 then factors further to b%282b%2B5%29%282b%2B3%29


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Answer:


So 4b%5E3%2B16b%5E2%2B15b completely factors to b%282b%2B5%29%282b%2B3%29.


In other words, 4b%5E3%2B16b%5E2%2B15b=b%282b%2B5%29%282b%2B3%29.


Note: you can check the answer by expanding b%282b%2B5%29%282b%2B3%29 to get 4b%5E3%2B16b%5E2%2B15b or by graphing the original expression and the answer (the two graphs should be identical).