Simplify rational expressions with domain restrictions

• Identify values that make an original denominator zero before simplifying.
• Factor numerators and denominators so common multiplicative factors become visible.
• Preserve excluded values after cancellation and verify equivalence at allowed inputs.

Build the idea

Worked example

1. Step 1

   Factor the denominator as the quantity x minus three times the quantity x plus two. The original expression is undefined at three and negative two.

   $$x^2-x-6=(x-3)(x+2)$$

2. Step 2

   Recognize a difference of squares in the numerator. This exposes x minus three as a common factor of the whole numerator and denominator.

   $$x^2-9=(x-3)(x+3)$$

3. Step 3

   Cancel the nonzero factor x minus three on the original domain. The shortened quotient still excludes both original denominator zeros.

   $$\frac{x+3}{x+2},\quad x\ne3,-2$$

• Cancelling the x terms directly from x squared minus nine and x squared minus x minus six.

  Terms in sums cannot cancel across a fraction. Factor each entire polynomial first and cancel only a common multiplicative factor.

• Dropping x equals three from the exclusion list after its factor cancels.

  The original formula remains undefined at three, so the cancelled factor creates a removable hole rather than restoring that input.

Simplify the quantity x squared plus five x plus six, divided by the quantity x squared plus four x plus three, and state all original exclusions.

This reviewed reading order covers the lesson explanation, equations, example, and checks without claiming that a video exists.

1. Part 1

   Lesson overview

   A rational expression is a quotient of polynomials. Factoring reveals common multiplicative factors that may cancel, but values excluded by the original denominator remain excluded. Cancellation is division by a common nonzero factor. It works across a product, not across separate terms joined by addition or subtraction, so factoring must usually happen before anything disappears. Identify values that make an original denominator zero before simplifying.

2. Part 2

   Visual model and equations

   The original denominator controls the domain. A cancelled factor still records an input where the original expression attempted division by zero, even if the shortened formula has a finite value there. Imagine the graph of a rational expression matching a simpler line everywhere except at one open circle. Cancellation explains the shared line, while the original denominator explains the missing point. The quantity x squared minus nine divided by the quantity x minus three equals the product of x minus three and x plus three, divided by x minus three. x plus three, with x not equal to three.

   • The quantity x squared minus nine divided by the quantity x minus three equals the product of x minus three and x plus three, divided by x minus three.
   • x plus three, with x not equal to three.
3. Part 3

   Worked example

   Simplify the quantity x squared minus nine, divided by the quantity x squared minus x minus six, and state every excluded value. Find denominator zeros first, factor both polynomials, cancel only a complete common factor, and carry the original exclusions to the result. Factor the denominator as the quantity x minus three times the quantity x plus two. The original expression is undefined at three and negative two. Recognize a difference of squares in the numerator. This exposes x minus three as a common factor of the whole numerator and denominator. Cancel the nonzero factor x minus three on the original domain. The shortened quotient still excludes both original denominator zeros. The expression simplifies to the quantity x plus three divided by the quantity x plus two, with x not equal to three or negative two. At x equals one, the original value is negative eight divided by negative six, or four thirds. The simplified value is four divided by three, matching on an allowed input.

   • x squared minus x minus six equals the quantity x minus three times the quantity x plus two.
   • x squared minus nine equals the quantity x minus three times the quantity x plus three.
   • The quantity x plus three divided by the quantity x plus two, with x not equal to three or negative two.
4. Part 4

   Checks, practice, and scope

   A simplified expression is therefore an equivalent rule only on the original domain. Writing the excluded values beside the result keeps the algebraic formula and its allowed inputs together. Cancelling the x terms directly from x squared minus nine and x squared minus x minus six. Terms in sums cannot cancel across a fraction. Factor each entire polynomial first and cancel only a common multiplicative factor. Dropping x equals three from the exclusion list after its factor cancels. The original formula remains undefined at three, so the cancelled factor creates a removable hole rather than restoring that input. Simplify the quantity x squared plus five x plus six, divided by the quantity x squared plus four x plus three, and state all original exclusions. Factor the numerator as the quantity x plus two times the quantity x plus three, and the denominator as the quantity x plus one times the quantity x plus three. The result is the quantity x plus two divided by the quantity x plus one, with x not equal to negative three or negative one. Determine excluded values from the original denominator before cancellation. Only common factors, not pieces of sums, may cancel across a quotient. A simplified rule and its original domain together describe the equivalent expression. This lesson treats algebraic simplification, not equations with rational expressions or asymptotic graph analysis. Factoring over real or complex numbers may require methods beyond the integer examples used here.

Scope and limitations

• This lesson treats algebraic simplification, not equations with rational expressions or asymptotic graph analysis.
• Factoring over real or complex numbers may require methods beyond the integer examples used here.