9. Polynomial and Rational Expression Fluency

Polynomial and Rational Expression Fluency

Mastery algebra begins with structure. Before expanding, look for a common
factor, difference of squares, or trinomial pattern. Distribute every term and
combine only like terms.

For rational expressions, factor before canceling:
\[ \frac{x^2-9}{x^2+x-6} =\frac{(x-3)(x+3)}{(x+3)(x-2)} =\frac{x-3}{x-2}, \]
with \(x\ne-3,2\) from the original denominators. Cancel factors, never terms
joined by addition or subtraction.

For a compound rational expression, factor each part, rewrite division as
multiplication by the reciprocal, then cancel. Keep every restriction from the
original expression even if a factor disappears. Test an allowed numerical
value to confirm two forms are equivalent.

Learning objectives

You will recognize factor structures, simplify rational products without
illegal cancellation, preserve excluded values, and verify equivalence.

Worked example

Simplify
\[ \frac{x^2-16}{x^2-2x-8}\cdot\frac{x-4}{x+4}. \]
Factor every polynomial:
\[ \frac{(x-4)(x+4)}{(x-4)(x+2)} \cdot\frac{x-4}{x+4} =\frac{x-4}{x+2}. \]
The original expression excludes \(x=4,-2,-4\). Even though factors cancel,
those inputs remain invalid because they made an original denominator zero.

To verify, choose an allowed value such as \(x=6\). Both the original and
simplified expressions should produce the same result. A disagreement exposes
a factor, sign, or copying error.

Common traps

Factoring \(x^2-16\) as \((x-4)^2\) is incorrect; it is a difference of squares.
Canceling an \(x\) from \(x+4\) is also invalid because \(x\) is a term, not a
factor of the entire sum.

Mastery practice

Factor three expressions before performing any cancellation. For each final
form, list restrictions from the unsimplified expression and verify one allowed
input numerically.