SOLUTION: The factoring Strategy Factor each polynomial completely. If a polynomial is prime, say so. 66. 8b2 + 24b + 18

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: The factoring Strategy Factor each polynomial completely. If a polynomial is prime, say so. 66. 8b2 + 24b + 18       Log On


   

Question 167956: The factoring Strategy
Factor each polynomial completely. If a polynomial is prime, say so.
66. 8b2 + 24b + 18
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

8b%5E2%2B24b%2B18 Start with the given expression


2%284b%5E2%2B12b%2B9%29 Factor out the GCF 2


Now let's focus on the inner expression 4b%5E2%2B12b%2B9




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Looking at the expression 4b%5E2%2B12b%2B9, we can see that the first coefficient is 4, the second coefficient is 12, and the last term is 9.


Now multiply the first coefficient 4 by the last term 9 to get %284%29%289%29=36.


Now the question is: what two whole numbers multiply to 36 (the previous product) and add to the second coefficient 12?


To find these two numbers, we need to list all of the factors of 36 (the previous product).


Factors of 36:
1,2,3,4,6,9,12,18,36
-1,-2,-3,-4,-6,-9,-12,-18,-36


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


These factors pair up and multiply to 36.
1*36
2*18
3*12
4*9
6*6
(-1)*(-36)
(-2)*(-18)
(-3)*(-12)
(-4)*(-9)
(-6)*(-6)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 12:


First NumberSecond NumberSum
1361+36=37
2182+18=20
3123+12=15
494+9=13
666+6=12
-1-36-1+(-36)=-37
-2-18-2+(-18)=-20
-3-12-3+(-12)=-15
-4-9-4+(-9)=-13
-6-6-6+(-6)=-12



From the table, we can see that the two numbers 6 and 6 add to 12 (the middle coefficient).


So the two numbers 6 and 6 both multiply to 36 and add to 12


Now replace the middle term 12b with 6b%2B6b. Remember, 6 and 6 add to 12. So this shows us that 6b%2B6b=12b.


4b%5E2%2Bhighlight%286b%2B6b%29%2B9 Replace the second term 12b with 6b%2B6b.


%284b%5E2%2B6b%29%2B%286b%2B9%29 Group the terms into two pairs.


2b%282b%2B3%29%2B%286b%2B9%29 Factor out the GCF 2b from the first group.


2b%282b%2B3%29%2B3%282b%2B3%29 Factor out 3 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%2B3%29%282b%2B3%29 Combine like terms. Or factor out the common term 2b%2B3


%282b%2B3%29%5E2 Simplify

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So our expression goes from 2%284b%5E2%2B12b%2B9%29 and factors further to 2%282b%2B3%29%5E2


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Answer:

So 8b%5E2%2B24b%2B18 completely factors to 2%282b%2B3%29%5E2