Do not divide out terms of polynomials
Avoid canceling polynomial terms incorrectly: only factors multiplying the entire numerator and denominator can be divided out — never cancel individual terms within a sum.
Key Concepts
You cannot cancel individual terms that are part of a sum or difference within a polynomial. Cancellation is only allowed for complete factors that are being multiplied. For instance, simplifying $$ \frac{x+1}{x 3} $$ to $$ \frac{1}{ 3} $$ is incorrect because x is a term, not a factor. The terms are bound by addition or subtraction.
Incorrect: $$ \frac{y+8}{y+2} \neq \frac{8}{2} $$ (You cannot cancel the y terms). Correct: $$ \frac{8(y+2)}{3(y+2)} = \frac{8}{3} $$ (The entire factor (y+2) is canceled). Correct process: $$ \frac{x^2 9}{x 3} = \frac{(x 3)(x+3)}{x 3} = x+3 $$.
Terms joined by addition or subtraction, like (x+5), are a package deal; you can't cancel individual parts. Think of them as glued together. You can only cancel an entire package if an identical one appears on the other side of the fraction bar. Breaking up these polynomial terms is a major math foul that must be avoided!
Common Questions
Why can't you cancel terms within a polynomial fraction?
Canceling requires common factors across the entire numerator and denominator, not individual additive terms. In (x+3)/3, the 3 in the numerator is part of a sum, not a factor. Canceling it incorrectly gives x, but the correct simplification is x/3 + 1.
What is the correct way to simplify a rational expression?
Factor the numerator and denominator completely first. Then cancel any common factors that appear in both. For (x^2-9)/(x-3), factor to (x+3)(x-3)/(x-3), then cancel (x-3) to leave (x+3), which is valid because (x-3) is a factor.
What is a common example of this error that students make?
A classic error is simplifying (x+5)/5 to x by canceling the 5. Since 5 is only a term in the sum x+5, not a factor of the entire numerator, this is invalid. The correct form remains (x+5)/5 unless the numerator can be factored to include 5 as a multiplied factor.