College algebra Books
I am looking for college algebra books containing as many elementary algebra questions as possible. For example, simplify
$$\frac{x}{1+x^2} - \frac{4}{(1+x^2)^2}, \quad \frac{4x}{1+2x+ x^3} - \frac{10x^2}{x^2+x^3},$$ or factorise $$x^4-2x^2+1, \quad 2x^2 + 4x+2, \quad etc.$$
It would be good if the book contains challenging and non-routine questions.
Problems for my attempts:
Factorize the following polynomial on three factors (polynomials with degree is greater that 0) with integer coefficients. $$x^6+27$$
Let $a$, $b$ and $c$ be positive numbers such that $abc=1$. Prove that: $$\frac{1}{1+a+ab}+\frac{1}{1+b+bc}+\frac{1}{1+c+ca}=1.$$
Solve the following equation. $$\sqrt[3]{2-\sqrt[3]{2-x}}=x.$$
There are hundreds of 1800s algebra texts that have been digitized and are freely available on the internet (in the U.S. at least), and many of those would be useful for what you want. The following are some of the better known “advanced school level” algebra texts:
William Steadman Aldis, A Text Book of Algebra
George Chrystal, Algebra. An Elementary Text-Book, Part I
Henry Sinclair Hall and Samuel Ratcliffe Knight, Higher Algebra
Elias Loomis, A Treatise on Algebra
Charles Smith, A Treatise on Algebra
Isaac Todhunter, Algebra
You'll find that intermediate level and beginning level texts are more numerous, many of which are by the same authors as above as well as by other authors. Also, look for older such books with "Key" in their titles (see this search also), as these will be solution manuals to certain corresponding texts.
Below are two especially nice algebra texts at a lower level (than those above) that I happen to know about.
George Chrystal, Introduction to Algebra
This book, written about a decade after his better known 2-part Algebra. An Elementary Text-Book, is somewhat historically significant because of its advocation of graphical methods in algebra which, in large part due to this book, began being incorporated non-trivially into algebra textbooks beginning around 1900. (I suspect another reason for the introduction of graphical methods in algebra around this time is due to the introduction of graph paper for classroom use around the 1890s.)
Charles Davies, New Elementary Algebra