SOLUTION: Not understanding, can anyone help? Factor the expression a^2-6ab+96b^2 into a product of binomials.

Question 385276: Not understanding, can anyone help?
Factor the expression a^2-6ab+96b^2 into a product of binomials.

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

Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last coefficient is .

Now multiply the first coefficient by the last coefficient to get .

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

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

Factors of :
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 .
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 :

First Number	Second Number	Sum
1	96	1+96=97
2	48	2+48=50
3	32	3+32=35
4	24	4+24=28
6	16	6+16=22
8	12	8+12=20
-1	-96	-1+(-96)=-97
-2	-48	-2+(-48)=-50
-3	-32	-3+(-32)=-35
-4	-24	-4+(-24)=-28
-6	-16	-6+(-16)=-22
-8	-12	-8+(-12)=-20

From the table, we can see that there are no pairs of numbers which add to . So cannot be factored.

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

Answer:

So doesn't factor at all (over the rational numbers).

So is prime.

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

Also, feel free to check out my tutoring website

Jim
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