Factorisation: last line shows 2/2* (z/(z-(1/2)) * (z/z-(1/4))

[georgedana]

Hi,
I have a problem with the last line of the problem I have shown in the photo.
I am not sure how the second last line becomes the last line.
Any help would be much appreciated.
The last line shows 2/2* (z/(z-(1/2)) * (z/z-(1/4)) if anyone has trouble reading my handwriting
Thanks!

 

[stapel]

I have a problem.... I am not sure how the second last line becomes the last line.

I think your work is as follows:

. . . . .\(\displaystyle H(z)\, =\, \dfrac{1}{1\, -\, 1\dfrac{1}{2}z^{-1}}\, \cdot\, \dfrac{1}{1\, -\, \dfrac{1}{4}z^{-1}}\)

. . . . .\(\displaystyle =\, \dfrac{1\, +\, 2z}{1\, -\, \dfrac{1}{2z}}\, \cdot\, \dfrac{1\, +\, 4z}{1\, -\, \dfrac{1}{4z}}\)

. . . . .\(\displaystyle =\, \dfrac{2z}{2z\, -\, 1}\, \cdot\, \dfrac{4z}{4z\, -\, 1}\)

. . . . .\(\displaystyle =\, \dfrac{2}{2}\, \dfrac{z}{z\, -\, \dfrac{1}{2}}\, \cdot\, \dfrac{z}{z\, -\, 14}\)

I don't understand how the 1.5 in the first denominator in the first line becomes 0.5 in the corresponding denominator in the second line. I don't understand where the "+2z" and "+4z" came from in the second line, nor where they disappear to in the third line. I don't understand the "14" in the last line, nor why the common factors of "2" were pulled out but the common factors of "4" were not. Also, I don't understand where calculus is being applied in any of this.

 

[Otis]

I am not sure how the second [to the] last line becomes the last line

The last line shows 2/2* (z/(z-(1/2)) * (z/z-(1/4)) if anyone has trouble reading my handwriting

Hi.

They factored a 2 out of the top and bottom of the first ratio, and they factored a 4 out of the top and bottom of the second ratio.

Of course, 2/2=1 and 4/4=1, and we don't always show factors of 1.

They chose to show 2/2, and they chose not to show 4/4.

There could be a valid reason for that, but I don't know why.

Cheers