Question

Which is a factor of z3 - z2 - 9z + 9?

Answer 1

A)
(z - 1)



B)
(z + 1)



C)
(z - 9)



D)
(z + 9)

Answer 2

Well, there's nothing that you can factor out of all of the terms, so it won't be a single variable or a single number. I would guess that you're going to have a \(z+3\) or \(z-3\) or maybe both as factors. \(z+1\) or \(z-1\) might also show up. And \(z+9\) or \(z-9\) might also be possible, although I think not as likely. The reason I can say this is that the presence of the \(z^3\) as the highest power means we'll have 3 things being multiplied together: \[(z+a)(z+b)(z+c)\]or 2 things:\[(z^2 + ...+a)(z+b)\]
and those constants at the end of each term all get multiplied together to produce that \(+9\). The only way to do that is with 1,3,9, either all 3 positive, or 2 negative and 1 positive.

Answer 3

z-1 ?

Answer 4

Now, you have a list of answer possibilities to choose from. Here's a trick:

each true factor is an expression which can be set equal to 0 and solved for the value of the variable. If you substitute that value into the polynomial, the polynomial will evaluate to 0!

So, \(z+1\) is one possibility. Let's try it out:
\[z+1 = 0\]\[z=-1\]Plug \(z=-1\) into the polynomial and see if you get 0:
\[z^3 - z^2 - 9z + 9 = (-1)^3-(-1)^2 - 9(-1) + 9 = -1-1+9+9 = 16\]So \(z+1\) is NOT a factor.

Answer 5

You suggest \(z-1\) as a possibility, let's check it out:
\[z-1 = 0\]\[z = 1\]
\[(1)^3 - (1)^2 -9(1) + 9 = 1-1-9+9 = 0\]Bingo!

Answer 6

At this point, to find the other factors, you could synthetic or long division to divide the polynomial by \((z-1)\) and get a simpler polynomial and repeat the process until you have found all the factors.

Answer 7

At some point, you will often find that you have a polynomial that you can factor by just looking at it. This one, for example, after you divide out the \((z-1)\) factor is \[z^2-9\]which is a difference of squares and thus can be factored as \[z^2-9 = (z-\sqrt{9})(z+\sqrt{9}) = (z-3)(z+3)\]

So \(z=\pm 3\) gives the other two factors. Quickly checking:

\[(3)^3 - (3)^2 -9(3) + 9 = 27 - 9 - 27 = 0\checkmark\]
\[(-3)^3 - (-3)^2 - 9(-3) + 9 = -27-9 + 27 + 9 = 0\checkmark\]