Introducing A New Series: Reviewing Algebra II (Part One Algebraic Fractions)

 Algebra II can be said to be the basis for successfully advancing  to more advanced mathematics, whether Calculus, differential equations, linear algebra, numerical analysis or abstract algebra. It certainly is also the foundation for getting anywhere in the STEM field, and especially the quantitative sciences like astronomy, plasma physics, as well as astrophysics and quantum physics. Finally, delving into this background will also equip readers to work more of the Mensa algebra puzzles, problems!

 With that in mind, I decided to start this brief  (not exhaustive) series in Algebra II review with algebraic fractions - often as important in Calculus problems, i.e. introducing partial fractions in a problem,  as in Algebra. Future topics will include: solving exponential equations, radical-surd equations, complex & imaginary equations and solving simultaneous equations.

Basically, an algebraic fraction consists of one algebraic expression divided by another. If numerator and denominator contain similar factor (e.g. x, y, z) the algebraic fraction can be simplified. We say the algebraic fraction has been reduced to lowest terms if both numerator and denominator have no common factors.

Example:   Simplify:

(x + 2) ²  / x²  - 4

Factor numerator and denominator:

(x + 2) (x + 2) /  (x + 2) (x - 2)

Cancel out  (x + 2):

Þ   

(x + 2) / (x - 2)

Example 2:  Simplify:

x²  - 3x - 10/ 2 x² +  8x + 8

Factor numerator and denominator:

(x + 2) (x - 5)/ 2(x + 2) (x + 2) =  x - 5/ 2 (x + 2)  

Note that a negative sign applied to an algebraic fraction can be considered either in the numerator or denominator.  Expressed in a general form:

- x / y  =  x/ - y

Example 3:  Simplify:

 a²  - 4/ 2 - a

Factor the numerator, cancel common factor:

a²  - 4/ 2 - a  =   (a + 2) (a - 2)/ - (a - 2) = (a + 2)/ -1

(Applying the negative sign rule)

Þ   

 - (a + 2)  OR:    - a - 2

Example 4: Simplify:

2x²  - 7 x -+ 6 / 3 + x  - 2x²

Factor numerator and denominator:

Þ   

(2x -  3) / (x - 2) /  (3 - 2x) (1 + x)  =

(2x -  3) / (x - 2) / - (2x - 3) (1 + x)

Note change in denominator from (3 - 2x) to - (2x - 3)

We can then cancel the  (2x - 3) factors in numerator and denominator:

Þ   

- (x -  2) / 1 + x   =  2 - x/ 1 + x

Practice Problems:  

Simplify each of the following:

1) x²  - y²/ y²  - x²

2)  27 a ³ x²y⁵  / 36 a ² x⁴y⁴

3) 2a ^1/2 (x + y)/  5 x a ^3/2

4) (x -1) (2 - x) (3 - x)/ (1 - x) (x - 2) (x - 4)