Polynomial question, am I correct?

[JFonseka]

[SOLVED] Polynomial question, am I correct?

Homework Statement

p(z) = z$$^{5}$$-5z$$^{4}$$+18z$$^{3}$$-50z$$^{2}$$+81z-45
Show that z - 1 is a factor of p(z)

Homework Equations

None

The Attempt at a Solution

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

 

[PhY]

Homework Statement

p(z) = z$$^{5}$$-5z$$^{4}$$+18z$$^{3}$$-50z$$^{2}$$+81z-45
Show that z - 1 is a factor of p(z)

Homework Equations

None

The Attempt at a Solution

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

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

 

[JFonseka]

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

 

[HallsofIvy]

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).

 

[JFonseka]

Thanks for clearing that up.