Algebra basics

[Norloonda]

In my book "Arithmetic and Algebra again......", I covered the previous chapters on "rules" up to the chapter problems leading up to my question. How do you divide a whole number by an algebraic term? The instruction is to "Simplify".


25x-10
5x

The answer from the book is:

5-2
x

While I might understand why the 5-2 goes above the line - why do the variable (x) go below the line or remain in that answer. What should I look at when I see -10/5x.

 

[serds]

The answer is wrong. It is

 

[stapel]

The instruction is to "Simplify".


25x-10
5x

The answer from the book is:

5-2
x

If so, then the book is in error. Polynomial fractions ("rational expressions") simplify in exactly the same way as to "regular" fractions (strictly numerical ones): factor the top and bottom, and cancel if possible.

\(\displaystyle \dfrac{25x\, -\, 10}{5x}\, =\, \dfrac{5(5x\, -\, 2)}{5(x)}\, =\, \dfrac{5x\, -\, 2}{x}\,=\, \dfrac{5x}{x}\, -\, \dfrac{2}{x}\, =\, 5\, -\, \dfrac{2}{x}\)

 

[lookagain]

In my book "Arithmetic and Algebra again......", I covered the previous chapters on "rules" up to the chapter problems leading up to my question. How do you divide a whole number by an algebraic term? The instruction is to "Simplify".


25x-10
5x

The answer from the book is:

5-2
x

While I might understand why the 5-2 goes above the line - why do the variable (x) go below the line or remain in that answer. What should I look at when I see -10/5x.

Norlonda, the answer from the book might actually be wrong (as in a typo). But look at the book's answer again to make sure. Maybe it is \(\displaystyle \ \dfrac{5x - 2}{x}.\)

If so, then the book is in error. Polynomial fractions ("rational expressions") simplify in exactly the same way as to "regular" fractions (strictly numerical ones): factor the top and bottom, and cancel if possible.

\(\displaystyle \dfrac{25x\, -\, 10}{5x}\, =\, \dfrac{5(5x\, -\, 2)}{5(x)}\, =\, \ \ \) > > > \(\displaystyle \ \dfrac{5x\, -\, 2}{x}\,\) < < < \(\displaystyle \ \ =\, \dfrac{5x}{x}\, -\, \dfrac{2}{x}\, =\, 5\, -\, \dfrac{2}{x}\)

I would state that what is highlighted above is what is being sought after with the word "simplify." When you break a fraction apart into the sum/difference of a fraction and something else, it looks to not be simplified. Hypothetically, had it been the other way around, to simplify "5 - 2/x" would have been to combine those two into (5x - 2)/x.