Question

hi i need help to work out this problem, thanks in advance.
let p(z)=f(z-1) and f(z)=z^3-8
express p(z) in the form p(z)=z^3+az^2+bz+c
hence solve p(z)=0

Answer 1

p(z) = f(z-1)
therefore:
p(z) = (z-1)^3 - (z-1)
this expands to:
p(z) = (z^2 - 2z + 1)(z-1) - z + 1
= z^3 - z^2 - 2z^2 + 2z + z - 1 - z + 1
= z^3 - 3z^2 + 2z
Now, can you solve the cubic equation:
z^3 - 3z^2 + 2z = 0

Answer 2

ok so how do work out the cubic equation?

Answer 3

Factor it.

Answer 4

could you please show me how?

Answer 5

You know that the factorization has to be of form:
(x + a)(x + b)(x + c)
I recommend this guide: http://www.wikihow.com/Factor-a-Cubic-Polynomial

Answer 6

Yeah. So, to get the solutions for z, just solve the equations:
z - 2 = 0
z - 1 = 0
z = 0

Answer 7

which the equals 2,1,0 is that correct

Answer 8

Yup.

Answer 9

I have a lifetime.

Answer 10

Is there a picture and/or more information?

Answer 11

Is there a definition for OA, OB, OC in the standard basis?

Answer 12

Yeah. So I'm assuming it's defining OA = <2, 0>, OB = <1, 0>, OC = <0, 0>?

Answer 13

yeah that seems right and relevant to the rest of the question

Answer 14

Do you know how to get the polar forms of those vectors?

Answer 15

Do you think I did not explain the previous question properly? If so, I apologize.

Answer 16

no ,not at all you did great but just that question about vectors it would be good if you explained it thanks

Answer 17

Okay, so you have a vector in the rectangular form. Let's take <2, 0>, OA.
Obviously, we know immediately that the angle must be 0. We can check this with:
$$\tan^{-1}(0)$$
Normally we would used atan2, but it's fine. We're taking the inverse tangent of 0, the y-coordinate.

Answer 18

The arctangent of the y-coord gives us the angle, theta. Now to find the magnitude we just use \(\sqrt{x^2 + y^2}\)

Answer 19

sqrt(2^2 + 0^2) = sqrt(4) = 2
So the polar form of the vector is (2, 0 degrees), because the magnitude is 2 and theta = 0

Answer 20

Alright. Can you think of a way to express this in terms of \(\vec i, \vec j\)?