Algebraic Expressions, Functions, and Systems
Algebraic Expressions
Simplify, expand, factor
An algebraic expression contains no equals sign, so the goal is rewriting rather than solving. Combine like terms only when the variable parts match exactly: 5x and −2x combine, but 5x and 5x² do not.
Exponent rules carry much of this domain. When multiplying powers with the same base, add exponents; when dividing, subtract; when raising a power to a power, multiply. A negative exponent means reciprocal, and any nonzero base to the zero power equals 1. Exponents distribute over products and quotients but never over sums, so (x + 3)² is not x² + 9.
Polynomial arithmetic: to subtract polynomials, change the sign of every term in the subtracted polynomial. To multiply binomials, distribute each term of the first across the second.
Factoring always begins with the greatest common factor. Then look for a difference of squares, a² − b² = (a + b)(a − b), or a trinomial whose two constants multiply to c and add to b. Factor and cancel only factors, never individual terms across a plus sign.
Worked example: 2x(3x − 4) − (x² − 5x + 1) becomes 6x² − 8x − x² + 5x − 1, which simplifies to 5x² − 3x − 1.
Rule of thumb: when evaluating an expression for a given value, substitute inside parentheses, then follow order of operations.
Functions and Systems
Function notation, slope, and solution counts
A function pairs each input with exactly one output. The notation f(3) means substitute 3 for x, while f(a + 1) means replace every x with the entire quantity (a + 1) - a common trap is treating it as f(a) + 1. Domain restrictions come from denominators that would be zero and even roots of negatives.
Slope measures rate of change: m = (y₂ − y₁)/(x₂ − x₁). Slope-intercept form y = mx + b shows slope and y-intercept directly. Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals, so their product is −1.
Systems of two equations can be solved by substitution when one variable is already isolated, or by elimination when coefficients can be matched and the equations added or subtracted. A system has one solution when the lines intersect, none when they are parallel (same slope, different intercept), and infinitely many when the equations describe the same line.
Traps: sign errors when subtracting entire equations, multiplying only one term when scaling an equation, and confusing x- and y-intercepts, which are found by setting the other variable to zero.
Worked example: solve 2x + 3y = 12 and x − y = 1. The second gives x = y + 1. Substituting: 2(y + 1) + 3y = 12, so 5y + 2 = 12 and y = 2, giving x = 3. Check: 2(3) + 3(2) = 12.