Why Factor Before Simplifying?
A rational expression is a quotient of polynomials. Sometimes its numerator and denominator look complicated, even though they contain a shared factor. Factoring can reveal that shared factor and let us write the quotient more simply.
The order matters: first identify the values excluded by the original denominator, then factor, and only then cancel shared factors. The restrictions belong to the original expression and must be kept with the simplified result.
Domain Restrictions Come First
For a rational expression \(\frac{P(x)}{Q(x)}\), where \(P\) and \(Q\) are polynomials, the expression is defined only at real values of \(x\) for which \(Q(x)\ne 0\). To find its restrictions, solve \(Q(x)=0\) before doing any cancellation.
Factoring the denominator can help identify its zeros. A zero of any denominator factor is excluded. If that factor is later cancelled, the excluded input is still listed as a restriction for the original expression.
For this topic, simplifying means rewriting the quotient in a more compact factored form. It does not mean changing which inputs were allowed in the original expression.
Why Cancelling a Shared Factor Works
Suppose the numerator and denominator factor as \(g(x)A(x)\) and \(g(x)B(x)\). At any input where the denominator is nonzero, both \(g(x)\) and \(B(x)\) are nonzero. The common factor can then be divided out:
This calculation explains the cancellation step. It applies to a factor that multiplies the whole numerator and the whole denominator. It does not apply to terms joined by addition or subtraction. The restrictions are still determined from the original denominator.
A Reliable Procedure
Set the original denominator equal to zero and exclude its real solutions.
Use an appropriate factoring method to express each as a product.
Look for identical factors multiplying the entire numerator and denominator.
Cancel each shared factor, then write the simplified expression together with the original restrictions.
Substitute an input that is not restricted into both forms and compare their values.
Worked Examples
Worked Example: A Difference of Squares
Simplify \(\frac{x^2-9}{x^2+3x}\), retaining the original restrictions.
Step 1: Find the restriction. The original denominator is \(x^2+3x\). Set it equal to zero and factor:
Thus \(x=0\) or \(x=-3\) makes the denominator zero. The original expression requires \(x\ne 0\) and \(x\ne -3\).
Step 2: Factor the numerator and denominator. The numerator is a difference of squares:
Step 3: Cancel the shared factor. The factor \(x+3\) appears in both products, so the simplified expression is:
Check: Use \(x=2\), which is allowed. In the original expression, the numerator is \(2^2-9=-5\), and the denominator is \(2^2+3(2)=10\), giving \(-5/10=-1/2\). In the simplified expression, \((2-3)/2=-1/2\). The values agree.
Worked Example: Factoring Quadratics with a Common Factor
Simplify \(\frac{2x^2-10x}{2x^2-14x+20}\), keeping the original domain.
Step 1: Find the restrictions. Factor the original denominator:
The denominator is zero at \(x=5\) and \(x=2\). Therefore the original expression requires \(x\ne5\) and \(x\ne2\).
Step 2: Factor both polynomials. Factor out the common numerical factor in each polynomial, then factor the remaining numerator and quadratic:
Step 3: Cancel shared factors. Both products contain \(2\) and \(x-5\). Cancelling them gives:
Check: Take \(x=1\), an allowed input. The original numerator is \(2-10=-8\), and the denominator is \(2-14+20=8\), so the value is \(-8/8=-1\). The simplified expression gives \(1/(1-2)=-1\), as required.
Worked Example: Cancelling a Repeated Factor
Simplify \(\frac{x^2-4x+4}{x^2-4}\) and state all restrictions.
Step 1: Find the restrictions. Factor the original denominator:
It vanishes at \(x=2\) and \(x=-2\), so those values are excluded.
Step 2: Factor the numerator and denominator. The numerator is a perfect square trinomial:
Step 3: Cancel one shared factor.
Check: At the allowed input \(x=0\), the original numerator is \(4\) and the denominator is \(-4\), so the quotient is \(-1\). The simplified expression gives \((0-2)/(0+2)=-2/2=-1\). The results match.
Cancellation Applies to Factors, Not Terms
A common error is to cancel matching pieces of expressions that are sums or differences rather than products. For example, the \(x\) in \(x+3\) is a term; it is not a factor multiplying the entire numerator. The same applies to \(x+5\) in the denominator.
A Common Mistake: Cancelling Terms
Worked Example: Correcting an Invalid Cancellation
A tempting but incorrect calculation is:
The \(x\)'s cannot be cancelled: each is part of a sum, and neither numerator nor denominator is written as a product with \(x\) as a common factor. For a concrete check, take \(x=1\). The original fraction is \((1+3)/(1+5)=4/6=2/3\), whereas the incorrect result is \(3/5\). Since \(2/3\ne3/5\), the cancellation has changed the value.
The correction is to leave the expression as \(\frac{x+3}{x+5}\) unless a valid factoring step reveals a shared factor. At \(x=1\), this form gives \(2/3\), confirming the calculation.
A separate error is to omit a restriction after cancelling. For instance, after simplifying \(\frac{x^2-9}{x^2+3x}\) to \(\frac{x-3}{x}\), one might list only \(x\ne0\), because the simplified denominator is \(x\). That misses \(x=-3\), which made the original denominator zero. The correct record is \(x\ne0,-3\). Always obtain restrictions before cancellation.
How to Present a Simplified Rational Expression
A clear final answer includes both the simplified quotient and the restrictions inherited from the original denominator. This makes the algebraic result usable without losing track of where the original expression was defined.
| Task | What to do |
|---|---|
| Find restrictions | Set the original denominator equal to zero and exclude its real solutions. |
| Factor | Write the numerator and denominator as products. |
| Cancel | Remove only identical factors multiplying the full numerator and denominator. |
| State the answer | Give the reduced expression and retain every original restriction. |
Factoring is useful because a complicated quotient can conceal a simple shared factor. Keeping the original restrictions alongside the reduced expression is just as important as doing the cancellation correctly.
Check Your Understanding
For each question, remember to use the original denominator when identifying restrictions.
- Simplify \(\frac{x^2-16}{x^2+4x}\), and state all values excluded from its original domain.
- Factor \(3x^2-12x\) and \(3x^2-27\). Which factors are shared?
- Simplify \(\frac{x^2-6x+9}{x^2-9}\), retaining the original restrictions.
- Explain why cancelling the \(x\) terms in \(\frac{x+4}{x+7}\) is invalid.
- In a rational expression, a denominator factor is cancelled. What must still be recorded in the final answer?
