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3.2.1: Resources and Key Concepts
Opening the book…
Resources
Key Concepts
Definitions
Algebraic Expression: A mathematical phrase that is a combination of numbers, variables, and mathematical operations, but does not contain an equals sign.
Evaluate (an algebraic expression): The process of substituting a given value for the variable(s) in an algebraic expression and simplifying the result.
Theorems
Difference of Squares: For any algebraic expressions \(F\) and \(L\), the formula for factoring a difference of squares is \(F^2 - L^2 = (F - L)(F + L)\).
Sums and Differences of Cubes: For any algebraic expressions \(F\) and \(L\), the formulas for factoring are:
Sum of Cubes: \(F^3 + L^3 = (F + L)(F^2 - FL + L^2)\)
Do NOT "Split" Arguments of Functions: A function cannot be distributed across terms inside its argument. For example, \(\sin(A + B)\) is not equal to \(\sin(A) + \sin(B)\). The expression inside the parentheses must be evaluated first.
Incorrect Notation for Negative Exponents: The shorthand notation \(\cos^2(\theta)\) for \((\cos(\theta))^2\) does not apply to negative exponents. Specifically, \([\cos(\theta)]^{-1}\) is not the same as \(\cos^{-1}(\theta)\), which represents the inverse cosine function.
- Factoring a Sum of Squares: A sum of two squares, such as \(F^2 + L^2\), is not factorable over the real numbers. Attempting to factor it as \((F+L)(F+L)\) or in any other way is incorrect.