Algebraic Operations
Algebraic Operations
This domain covers polynomial arithmetic, factoring, exponent rules, rational expressions, radicals, and expression simplification. The exam checks whether you can rewrite expressions correctly, combine fractions, and state the restrictions that keep an expression defined.
Core rules
- Polynomial addition and subtraction: align like terms. When subtracting, distribute the minus sign to every term of the second polynomial.
- Multiplication: use the distributive property or FOIL. Remember (a + b)² = a² + 2ab + b², not a² + b².
- Factoring: remove the greatest common factor first, then use difference of squares a² − b² = (a − b)(a + b), sum or difference of cubes, and trinomial factoring.
- Exponent rules: a^m·a^n = a^(m+n); a^m/a^n = a^(m−n); (a^m)^n = a^(mn); a^(−n) = 1/a^n; a^(1/n) = ⁿ√a.
- Rational expressions: factor and cancel factors, not individual terms. State excluded values where any original denominator equals zero.
- Complex fractions: simplify the numerator and denominator separately, or multiply by the least common denominator of all small fractions.
- Radicals: √(ab) = √a·√b. Rationalize a denominator by multiplying by the conjugate or the needed radical.
Worked example
Simplify [(x² − 9)/(x² + 5x + 6)] ÷ [(x − 3)/(x + 2)].
Factor: x² − 9 = (x − 3)(x + 3), and x² + 5x + 6 = (x + 2)(x + 3). Rewrite division as multiplication by the reciprocal:
[(x − 3)(x + 3)]/[(x + 2)(x + 3)] · (x + 2)/(x − 3).
Cancel (x + 3), (x + 2), and (x − 3), leaving 1. Restrictions are x ≠ −3, −2, 3 because those values make an original denominator zero.
Traps that cost marks
- Canceling terms across a sum, such as turning (x + 3)/x into 3; only factors cancel.
- Sign errors when subtracting polynomials, especially with negative coefficients.
- Applying an exponent to every term incorrectly: (2x)³ = 8x³, but (x + y)² ≠ x² + y².
- Dropping domain restrictions after simplifying a rational expression.
- Adding rational expressions without a common denominator, or adding denominators instead of finding an LCD.
- Rationalizing by multiplying only the numerator, which changes the value.