Algebra
Algebra
The Algebra domain covers seeing structure in expressions, polynomial arithmetic, creating equations, and reasoning with equations and inequalities. OST items ask you to justify each step, choose efficient methods, and interpret solutions in context.
Core rules
- Combine like terms and apply the distributive property before factoring. Common patterns: a^2 - b^2 = (a - b)(a + b) and x^2 + (a + b)x + ab = (x + a)(x + b).
- For quadratics, factor first; if not factorable, complete the square or use the quadratic formula x = (-b ± √(b^2 - 4ac))/(2a).
- The zero product property: if AB = 0, then A = 0 or B = 0. Always set the equation to zero before factoring.
- For systems, substitution works well when one variable is isolated; elimination works when coefficients are opposites or multiples.
- Multiplying or dividing an inequality by a negative number reverses the inequality sign.
Worked example
Solve x^2 - 5x - 6 = 0. Find two numbers with product -6 and sum -5: -6 and 1. Factor: (x - 6)(x + 1) = 0. By the zero product property, x - 6 = 0 or x + 1 = 0, so x = 6 or x = -1. Check x = 6: 36 - 30 - 6 = 0. Check x = -1: 1 + 5 - 6 = 0.
Traps
Factoring x^2 - 5x - 6 as (x - 3)(x + 2) because the signs are guessed; forgetting that a quadratic can have two, one, or no real solutions; dividing both sides by a variable and losing a solution; and reversing an inequality only when it is convenient rather than whenever multiplying or dividing by a negative.