Algebra and Algebraic Expressions
Core concepts
Master polynomial arithmetic, factoring patterns (difference of squares, sum/difference of cubes, perfect-square trinomials), rational expressions and their domains, exponent laws (including fractional and negative exponents), radicals, and logarithm rules. Know the binomial theorem and Pascal's triangle for expansion coefficients.
Traps
- Cancelling across a sum: you may cancel factors, never addends.
- Domain restrictions: denominators nonzero, even roots nonnegative, log arguments positive.
- Sign errors in factoring: (a − b)² = a² − 2ab + b², not a² − b².
- Extraneous roots introduced by squaring or clearing denominators.
- Logarithm bases: log(x²) = 2 log|x|, not 2 log x without care.
Rule of thumb
Before simplifying a rational expression, factor numerator and denominator completely; then cancel only common factors and record excluded values.
Worked example
Simplify (x² − 9)/(x² + x − 12). Factor: numerator (x − 3)(x + 3); denominator (x + 4)(x − 3). Cancel (x − 3), giving (x + 3)/(x + 4), with x ≠ 3 and x ≠ −4. Note that x = 3 is excluded even though it cancels - a classic trap.