"Write the polynomial as a product of linear factors" means the same as factor the polynomial. The polynomial:- has no common factor other than 1 (or -1)
- has too many terms for any of the factoring patterns
- has too many terms for trinomial factoring
- will not factor with the "factor by grouping" method.
The only factoring technique left is factoring by trial and error of the possible rational roots. The leading coefficient is 1 and the constant term is -4. This makes the possible rational roots, 1, -1, 2, -2, 4 or -4.
You may or may not know this but when a polynomial has alternating signs (plus then minus then plus then minus, etc.), like P(x), then there will be no negative roots. (If you've never heard of this, then think about what happens to P(x) is x is negative. I think you'll find every term works out negative. And a bunch of negative terms cannot add up to 0.)
So now we're down to 1, 2 or 4 as possible roots. It is easy to see if 1 is a root in any polynomial because 1 to any power is 1. So P(1) = 1 - 8 +17 - 4 which is not zero. So we'll try 2. To check 2 we could either find P(2) directly or we could use synthetic division. (Note: The larger the possible root and/or the more terms there are in the polynomial, the more advantageous it is to use synthetic division.) I'm going to use synthetic division.
2 | 1 -8 17 -4
----
2 -12 10
------------------
1 -6 5 6
The lower right corner is the remainder, which also happens to be P(2). Since it is not zero, 2 is not a root. Let's try 4:
4 | 1 -8 17 -4
----
4 -16 4
------------------
1 -4 1 0
P(4) is zero! So (x-4) is a factor of P(x). Not only that, but the part of the last row in front of the remainder give us the other factor. The "1 -4 1" tells us that is the other factor. So
The problem asks for linear factors. But the is not linear. It is quadratic. And we have run out of possible rational roots. So the remaining two roots are either irrational or complex. To find these roots we will use the Quadratic Formula on :
which simplifies as follows:
which is short for:
or
With these two irrational roots we can now write the missing factors. (Remember, (x - root) is a factor.)
which simplifies: