Simplifying rational expressions and restrictions

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Introduction
Lessons
  1. Why is it important to determine the non-permissible values prior to simplifying a rational expression?
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Examples
Lessons
  1. For each rational expression:
    i) determine the non-permissible values of the variable, then
    ii) simplify the rational expression
    1. 6x34x\frac{{6{x^3}}}{{4x}}
    2. 5xx28x+15\frac{{5 - {x}}}{{{x^2} - 8x + 15}}
    3. x2+13x+40x225\frac{{{x^2} + 13x + 40}}{{{x^2} - 25}}
  2. For each rational expression:
    i) determine the non-permissible values of the variable, then
    ii) simplify the rational expression
    1. 9t316t3t2+4t\frac{{9{t^3} - 16t}}{{3{t^2} + 4t}}
    2. x2+2x3x410x2+9\frac{{{x^2} + 2x - 3}}{{{x^4} - 10{x^2} + 9}}
  3. For each rational expression:
    i) determine the non-permissible values of the variable, then
    ii) simplify the rational expression
    1. x33x\frac{{x - 3}}{{3 - x}}
    2. 5y310y23015y\frac{{5{y^3} - 10{y^2}}}{{30 - 15y}}
    3. 19x26x27x3\frac{{1 - 9{x^2}}}{{6{x^2} - 7x - 3}}
  4. The area of a rectangular window can be expressed as 4x2+13x+34{x^2} + 13x + 3, while its length can be expressed as 4x+14x + 1.
    1. Find the width of the window.
    2. If the perimeter of the window is 68 mm, what is the value of xx?
    3. If a cleaning company charges $3/m2m^2 for cleaning the window, how much does it cost to clean the window?
  5. For each rational expression:
    i) determine the non-permissible values for yy in terms of xx , then
    ii) simplify, where possible.
    1. 2x+y2xy\frac{{2x + y}}{{2x - y}}
    2. x3yx29y2\frac{{x - 3y}}{{{x^2} - 9{y^2}}}
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Topic Notes
A rational expression is a fraction that its numerator and/or denominator are polynomials. In this lesson, we will first learn how to find the non-permissible values of the variable in a rational expression. Then, we will how to simplify rational expressions.
\cdot multiplication rule: xaxb=xa+bx^a \cdot x^b=x^{a+b}

\cdot division rule: xaxb=xab\frac{x^a}{x^b}=x^{a-b}