Maths: Why a Cancelled Factor Still Excludes a Value

Rational functions often hide a removable discontinuity — a single point where the expression breaks down even though the graph itself looks perfectly smooth. Factorising the numerator of A(x) = (x² + 2x − 15)/(x − 3) reveals a common factor of (x − 3), which cancels to leave the simpler linear form x + 5. The catch is that the cancelled factor does not vanish from the story: the original denominator still equals zero at x = 3, so that value must be excluded from the domain. This matters because mathematics rarely stays abstract. When a function models a real rectangle, an excluded x-value signals something physically impossible — a side length that cannot exist, leaving the area undefined. Recognising the link between algebraic restrictions and real-world constraints is the heart of this topic. From there, solving an area equation becomes a matter of setting the simplified expression equal to the target value, then checking each solution against the domain restriction to decide which one genuinely holds.