SOLUTION: Factor completely. 15x^2 � 21x + 6

Question 117273: Factor completely. 15x^2 � 21x + 6
Found 2 solutions by bucky, jim_thompson5910:
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
Given to factor:
.

.
Notice first that each of the numbers in the three terms is divisible by 3, so the factor 3
can be removed. This results in:
.

.
So now let's concentrate on factoring the terms inside the parentheses. Note that can
only be factored into . Therefore, if the trinomial in the parentheses can be
factored, the factors will be of the form:
.
(5x ______)*(x _____)
.
Next notice that the last term in the trinomial is +2. It's factors can only be +2 and +1 or
-2 and -1 because they must multiply together to give +2. Since the middle term of the
trinomial is -7 we have to have minuses in the factors of the trinomial. So we assume that
the factors of +2 that we will use are -2 and -1. This leaves two possibilities:
.
(5x - 1)*(x - 2)
.
or
.
(5x - 2)*(x - 1)
.
When you multiply out the (5x - 1)*(x - 2) you get which reduces to
so this combination does not work.
.
When you multiply out the (5x - 2)*(x - 1) you get and this does reduce
to , just as we need.
.
So now we can say that the given:
.

.
factors to:
.

.
None of the expressions inside the parentheses can be factored, so this is the final answer.
.
Hope this helps to see how the given trinomial can be factored using a "logical" analysis and
a little bit of trial.
.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

Looking at we can see that the first term is and the last term is where the coefficients are 15 and 6 respectively.

Now multiply the first coefficient 15 and the last coefficient 6 to get 90. Now what two numbers multiply to 90 and add to the middle coefficient -21? Let's list all of the factors of 90:

Factors of 90:
1,2,3,5,6,9,10,15,18,30,45,90

-1,-2,-3,-5,-6,-9,-10,-15,-18,-30,-45,-90 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 90
1*90
2*45
3*30
5*18
6*15
9*10
(-1)*(-90)
(-2)*(-45)
(-3)*(-30)
(-5)*(-18)
(-6)*(-15)
(-9)*(-10)

note: remember two negative numbers multiplied together make a positive number

Now which of these pairs add to -21? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -21

First Number Second Number Sum
1 90 1+90=91
2 45 2+45=47
3 30 3+30=33
5 18 5+18=23
6 15 6+15=21
9 10 9+10=19
-1 -90 -1+(-90)=-91
-2 -45 -2+(-45)=-47
-3 -30 -3+(-30)=-33
-5 -18 -5+(-18)=-23
-6 -15 -6+(-15)=-21
-9 -10 -9+(-10)=-19

From this list we can see that -6 and -15 add up to -21 and multiply to 90

Now looking at the expression , replace with (notice adds up to . So it is equivalent to )

Now let's factor by grouping:

Group like terms

Factor out the GCF of out of the first group. Factor out the GCF of out of the second group

Since we have a common term of , we can combine like terms

So factors to

So this also means that factors to (since is equivalent to )

-------------------------------
Answer:

So factors to
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First Number	Second Number	Sum
1	90	1+90=91
2	45	2+45=47
3	30	3+30=33
5	18	5+18=23
6	15	6+15=21
9	10	9+10=19
-1	-90	-1+(-90)=-91
-2	-45	-2+(-45)=-47
-3	-30	-3+(-30)=-33
-5	-18	-5+(-18)=-23
-6	-15	-6+(-15)=-21
-9	-10	-9+(-10)=-19