I am self-learning precalc (Precalculus Demystified) and found the following problem on page 170 :
Completely factor the the polynomial. $P(x) = x^3 - 5x^2 + 5x + 3; c = 3$ is a zero.
Since $c = 3$ is a zero, I know $x - 3$ is a factor (and remainder is zero). So I go ahead and do the division and end up with the following :
$(x - 3)(x^2 - 2x - 1)$
However, this needs to be factored even further, so I do the square and end up with :
$(x - 3)((x - 1)^2 - 2)$
Now I'm scratching my head because it doesn't look like the nice factors I'm used to seeing, so I check my progress in the book and they say the following :
"In order to factor $x^2 - 2x - 1$, we must first find its zeros"
Now I understand that in order to find the intercepts on a graph I must find its zeros, but what does finding its zeros have to do with factoring this equation? I'm completely lost at this point because though I can take it at face value and proceed, I need to understand the why before I feel comfortable plowing ahead or I'm going to dig myself into an even deeper pool of confusion. :(
Any thoughts? Thanks in advance!
Update : Due to the answers I'm getting, I need to clarify my question to pinpoint my actual confusion.
Thank you for your quick replies! I'm so sorry but I'm a little slow in understanding the implication. My problem is that I don't understand how factoring and finding zeros is related. For me, factoring is finding what are the root divisors of a number or equation. For example, 3 is a factor of 9 because 3 * 3 = 9. When I look for factors of 9, I don't think about finding zeros... I don't even know what that means! I just look to see what multiplied by what gives me 9. In an equation such as x^2 + 2xy + y^2, I understand how to factor this to (x + y)^2... this I understand the why of and how to do it. But in all the times I have done it, never once have I thought about "finding the zero" of anything nor have I approached it thinking about zeros at all. This may betray an immense ignorance of some basic understanding on my part, but this is why I'm asking this question... I would love to know what I'm missing and what zeros have to do with anything. For finding the x/y intercepts, yes, I understand how zero relates... for factoring I don't. :(