# Polynomial question, am I correct?

1. Nov 10, 2007

### JFonseka

[SOLVED] Polynomial question, am I correct?

1. The problem statement, all variables and given/known data
p(z) = z$$^{5}$$-5z$$^{4}$$+18z$$^{3}$$-50z$$^{2}$$+81z-45
Show that z - 1 is a factor of p(z)

2. Relevant equations

None

3. The attempt at a solution

Well since 1 is a root of p(z), then p(1) = 0
Then z - 1 is a factor?

2. Nov 10, 2007

### PhY

p(1) is equal to 0, than yes it is a factor.

3. Nov 10, 2007

### JFonseka

I know it is a factor, but is my method correct for 'showing' that it is so?

4. Nov 10, 2007

### HallsofIvy

Staff Emeritus
That was PhY's answer! (less a stray "e"). Because p(1) is equal to 0, then yes it is a factor.

You are using a very simple fact about polynomials- If P(x) is any polynomial, dividing P(x) by (x-a) results in a quotient, Q(x), and a remainder, r, which, since x-a has degree 1, must be a number: P(x)= Q(x)(x-a)+ r. Now, let x= a: P(a)= Q(x)(a- a)+ r= r. If P(a)= 0 then r must be 0 and so (x- a) is a factor of P(x).

5. Nov 10, 2007

### JFonseka

Thanks for clearing that up.