Algebraic Expressions and Operations
Algebraic Expressions and Operations
This domain tests whether you can move fluently between verbal descriptions, expressions, and simplified or factored forms. Items typically ask you to simplify polynomial, radical, or rational expressions, apply exponent rules, evaluate an expression for given values, or write an expression that models a context such as perimeter, area, cost, or consecutive integers.
Core rules
- Combine like terms only when the variable parts match exactly: 3x² and 5x² combine; 3x² and 5x do not.
- Product rule aᵐ·aⁿ = aᵐ⁺ⁿ; quotient aᵐ/aⁿ = aᵐ⁻ⁿ; power (aᵐ)ⁿ = aᵐⁿ; negative exponent a⁻ⁿ = 1/aⁿ; zero exponent a⁰ = 1 for a ≠ 0. Rational exponents: a^(m/n) = ⁿ√(aᵐ).
- Special products: (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², (a + b)(a − b) = a² − b².
- Factoring order: greatest common factor first, then difference of squares, then trinomials (find two numbers whose product is c and whose sum is b when a = 1), then grouping.
- A rational expression is undefined wherever its denominator equals zero, so state restrictions before simplifying.
Worked example
Simplify (3x²y⁻¹)²(2x⁻³y⁴).
Square each factor first: 3²(x²)²(y⁻¹)² = 9x⁴y⁻². Then multiply: 9x⁴y⁻² · 2x⁻³y⁴ = 18x^(4−3)y^(−2+4) = 18xy².
Traps that cost marks
- −2² = −4, not 4, while (−2)² = 4. The exponent binds only to the base directly beneath it.
- (x + 3)² is not x² + 9; the middle term 6x is required.
- Terms do not cancel across addition: (x + 5)/x is not 5.
- When subtracting polynomials, distribute the minus sign to every term of the second polynomial.
- Factor out the GCF before applying other patterns, or the answer will be incomplete.