Algebra
Algebra
Algebra II emphasizes structure: rewriting expressions, factoring completely, and solving equations with justification. Be fluent with polynomial operations, special products, factoring by grouping, difference of squares, and sum/difference of cubes. The Remainder Theorem says the remainder when p(x) is divided by x-c is p(c); the Factor Theorem says x-c is a factor exactly when p(c) = 0. For rational expressions, factor first, record excluded values, then cancel. When solving rational or radical equations, multiply or raise to a power, but always test candidates in the original equation because extraneous roots can appear. For exponential equations, express both sides with a common base; for logarithmic equations, check that arguments remain positive. Systems mixing a line and a parabola can have 0, 1, or 2 real solutions, found by substitution.
Common traps: sign errors when subtracting polynomials, canceling terms rather than factors, forgetting that dividing by a variable expression can lose solutions, and reversing an inequality when multiplying or dividing by a negative. On the Regents, solve often expects exact simplified form, not a decimal approximation.
Worked example: Solve 2^(x+1) = 8^(x-1). Since 8 = 2^3, rewrite as 2^(x+1) = 2^(3x-3). Equate exponents: x+1 = 3x-3, so 4 = 2x and x = 2. Check: 2^3 = 8 and 8^1 = 8.