Graph of 1/sinx.

when x=0, the function goes to infinity and the function is even.

is it correct to say that since it is periodic, the graph of 1/sinx will be the copy of the function, but in the other direction? is it correct for all periodic functions?

Step-by-step explanation:

See the attached for a graph of 1/sin(x) = csc(x). Just as sin(x) is an odd function, so is 1/sin(x).

Any odd function is symmetrical about the origin, so the left side of the graph is a copy of the right side, rotated 180° about the origin ("in the  other direction"). That is, csc(x) = -csc(-x). This is true because the function is odd, not because it is periodic.

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Csc(x) is also a periodic function with a period of 2π. That means ...

  csc(x+2π) ≡ csc(x)

This sort of replication of the function is true for all periodic functions (where the added value, the horizontal translation, is the period of the function).

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You will note that the relation applicable to an odd function (-f(-x) = f(x)) is different from the relation applicable to a periodic function (f(x) = f(x+period)). An odd periodic function, such as csc(x), will be described by both relations.

(x3 + 6x2 + 2x - 6) • (x - 1)

step-by-step explanation:

polynomial long division :

3.2     polynomial long division

dividing :   x4+5x3-4x2-8x+6

                              ("dividend")

by         :     x-1     ("divisor")

dividend     x4   +   5x3   -   4x2   -   8x   +   6

- divisor   * x3     x4   -   x3            

remainder         6x3   -   4x2   -   8x   +   6

- divisor   * 6x2         6x3   -   6x2        

remainder             2x2   -   8x   +   6

- divisor   * 2x1             2x2   -   2x    

remainder               -   6x   +   6

- divisor   * -6x0               -   6x   +   6

remainder                     0

quotient :   x3+6x2+2x-6   remainder:   0