SOLUTION: Factor the expression correctly 32b^2 - 20b - 3. i dont know how to factor it if all the numbers have no GCF. Can u please help me how to factor it.

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Factor the expression correctly 32b^2 - 20b - 3. i dont know how to factor it if all the numbers have no GCF. Can u please help me how to factor it.      Log On


   

Question 416132: Factor the expression correctly 32b^2 - 20b - 3. i dont know how to factor it if all the numbers have no GCF. Can u please help me how to factor it.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 32b%5E2-20b-3, we can see that the first coefficient is 32, the second coefficient is -20, and the last term is -3.


Now multiply the first coefficient 32 by the last term -3 to get %2832%29%28-3%29=-96.


Now the question is: what two whole numbers multiply to -96 (the previous product) and add to the second coefficient -20?


To find these two numbers, we need to list all of the factors of -96 (the previous product).


Factors of -96:
1,2,3,4,6,8,12,16,24,32,48,96
-1,-2,-3,-4,-6,-8,-12,-16,-24,-32,-48,-96


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


These factors pair up and multiply to -96.
1*(-96) = -96
2*(-48) = -96
3*(-32) = -96
4*(-24) = -96
6*(-16) = -96
8*(-12) = -96
(-1)*(96) = -96
(-2)*(48) = -96
(-3)*(32) = -96
(-4)*(24) = -96
(-6)*(16) = -96
(-8)*(12) = -96

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -20:


First NumberSecond NumberSum
1-961+(-96)=-95
2-482+(-48)=-46
3-323+(-32)=-29
4-244+(-24)=-20
6-166+(-16)=-10
8-128+(-12)=-4
-196-1+96=95
-248-2+48=46
-332-3+32=29
-424-4+24=20
-616-6+16=10
-812-8+12=4



From the table, we can see that the two numbers 4 and -24 add to -20 (the middle coefficient).


So the two numbers 4 and -24 both multiply to -96 and add to -20


Now replace the middle term -20b with 4b-24b. Remember, 4 and -24 add to -20. So this shows us that 4b-24b=-20b.


32b%5E2%2Bhighlight%284b-24b%29-3 Replace the second term -20b with 4b-24b.


%2832b%5E2%2B4b%29%2B%28-24b-3%29 Group the terms into two pairs.


4b%288b%2B1%29%2B%28-24b-3%29 Factor out the GCF 4b from the first group.


4b%288b%2B1%29-3%288b%2B1%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.


%284b-3%29%288b%2B1%29 Combine like terms. Or factor out the common term 8b%2B1


===============================================================


Answer:


So 32b%5E2-20b-3 factors to %284b-3%29%288b%2B1%29.


In other words, 32b%5E2-20b-3=%284b-3%29%288b%2B1%29.


Note: you can check the answer by expanding %284b-3%29%288b%2B1%29 to get 32b%5E2-20b-3 or by graphing the original expression and the answer (the two graphs should be identical).


If you need more help, email me at jim_thompson5910@hotmail.com

Also, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Thank you

Jim