7
Identity transformations of fractional rational expressions
Textbook: pp. 30–33
GoalSimplify fractional rational expressions using operations on algebraic fractions and find their values.
New words
fractional rational expression · kasr-ratsional ifodaidentity transformation · ayniy almashtirishsimplifying an expression · ifodani soddalashtirishorder of operations · amallar tartibi
Explanation
An expression built from several algebraic fractions joined by arithmetic signs is called a fractional rational expression. The denominators must not be zero. We simplify in the usual order: brackets first, then multiplication and division, and addition and subtraction last. At every step we bring fractions to a common denominator, collect the numerator and reduce. Identity transformations do not change the value of the expression for admissible values, so we write the final answer in simple form, keeping the admissibility conditions in mind.
Worked examples
(1/(x - 1) - 1/(x + 1)) · (x² - 1)/2 = ((x + 1) - (x - 1))/(x² - 1) · (x² - 1)/2 = 2/(x² - 1) · (x² - 1)/2 = 1.
(a + b)/(a - b) - (a - b)/(a + b) = ((a + b)² - (a - b)²)/(a² - b²) = 4ab/(a² - b²).
Class activity
“Simplification race”: two teams simplify the same complicated expression; each step must be explained on the board. The team with the correct and clearest solution wins.
Practice
1
Simplify (1/x + 1/y) : ((x + y)/(xy)).
1
2
Simplify (x/y - y/x) · (xy/(x + y)).
x - y
3
In the previous problem, what is the value of the expression for x = 12, y = 5?
7
4
Why must x ≠ 1 be stated when simplifying (x² - x)/(x - 1) to x?
Because the original denominator x - 1 is zero at x = 1, so the expression has no meaning there, although the simplified «x» does.