SOLUTION: Need help to solve 4608=(36-w)(w)(w+4) I have to solve for w my work shows that the next step is 0=32w^2+144w-w^3-4608 what I do not understand is where the 32 comes from pleas

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Need help to solve 4608=(36-w)(w)(w+4) I have to solve for w my work shows that the next step is 0=32w^2+144w-w^3-4608 what I do not understand is where the 32 comes from pleas      Log On


   

Question 413777: Need help to solve 4608=(36-w)(w)(w+4)
I have to solve for w
my work shows that the next step is 0=32w^2+144w-w^3-4608
what I do not understand is where the 32 comes from please help! thank you!
Answer by jsmallt9(3759) About Me  (Show Source):
You can put this solution on YOUR website!
4608 = (36-w)(w)(w+4)
To solve this equation we need one side to be zero and then factor the other side. We can get one side to be zero easily by subtracting 4608 from each side:
0 = (36-w)(w)(x+4) - 4608
In order to factor the right side, we will need to simplify it first. Since multiplication is Commutative we can multiply (36-w)(w)(w+4) in any order we choose. I choose to multiply the last two factors (using the Distributive Property) first:
0+=+%2836-w%29%28w%5E2%2B4w%29+-+4608
To finish the multiplication we we use FOIL:
0+=+36%2Aw%5E2+%2B+36%2A4w+%2B+%28-w%29%28w%5E2%29+%2B+%28-w%29%2A4w+-+4608
which simplifies as follows:
0+=+36w%5E2+%2B+144w+%2B+%28-w%5E3%29+%2B+%28-4w%5E2%29+-+4608
Combining the like terms, the 36w%5E2 and -4w%5E2 we get:
0+=+33w%5E2+%2B+144w+%2B+%28-w%5E3%29+%2B+%28-4608%29
(As you can see, the 32w%5E2 comes from adding 36w%5E2 and -4w%5E2.)
Rearranging the terms in standard order we have:
0+=+%28-w%5E3%29+%2B+33w%5E2+%2B+144w+%2B+%28-4608%29
Since having a positive leading coefficient makes things easier, let's multiply both sides by -1:
0+=+w%5E3+%2B+%28-32w%5E2%29+%2B+%28-144w%29+%2B+4608
To solve for w we now factor this expression. There is no GCF (other than 1). There are too many terms for factoring by patterns or for trinomial factoring. But it will factor by grouping.

The GCF of the first two terms is w%5E2. The GCF of the last two terms is 144. Factoring the GCF's out of each pair we get:
0+=+w%5E2%28w+%2B+%28-32%29%29+%2B+144%28%28-w%29+%2B+32%29
The "non-GCF" factors are not the same as we hoped. But they are negatives of each other! So if we factor out -144 instead of 144 they will match!
0+=+w%5E2%28w+%2B+%28-32%29%29+%2B+%28-144%29%28w+%2B+%28-32%29%29
Now the "non-GCF" factors match so we can finish factoring:
0+=+%28w+%2B+%28-32%29%29%28w%5E2%2B+%28-144%29%29
The second factor, which can be rewritten as w%5E2-144, is a difference of squares and so it can be factored:
0+=+%28w+%2B+%28-32%29%29%28w%2B12%29%28w-12%29
Now that we're finished factoring we can solve the equation. From the Zero Product Property we know that one of these factors must be zero. So:
w + (-32) = 0 or x+12 = 0 or w-12 = 0
Solving each of these we get:
w = 32 or w = -12 or w = 12