Factorization and Reasoning Problems

Vocabulary and concepts regarding algebraic factorization methods (common factor, difference of squares, trinomials) and steps for solving mathematical reasoning problems.

Common Factor

A technique used when all terms in an expression have a common factor that can be extracted, such as $6x + 12 = 6(x + 2)$.

Difference of Squares Formula

The algebraic identity represented by the formula $a^2 - b^2 = (a + b)(a - b)$.

Difference of Squares Recognition

Identifying an expression that has two terms, a subtraction operation, and where both terms are perfect squares.

Perfect Square Trinomial (PST) Formulas

The identities characterized by $a^2 + 2ab + b^2 = (a + b)^2$ and $a^2 - 2ab + b^2 = (a - b)^2$.

Perfect Square Trinomial (PST) Recognition

A trinomial where the first and last terms are perfect squares and the central term is double the product of their square roots.

Factoring $x^2 + bx + c$

A method involving finding two numbers that sum to $b$ and multiply to $c$, for example $x^2 + 7x + 12 = (x + 3)(x + 4)$.

Factoring $ax^2 + bx + c$

The process of multiplying $a \times c$, finding two suitable numbers, splitting the central term, and grouping, e.g., $2x^2 + 7x + 3 = (x + 3)(2x + 1)$.

First Step in Reasoning Problems

The process of identifying the provided data and determining what the problem is asking for.

Variable Representation

Using a symbol, generally $x$, to represent an unknown number in a problem.

Double of a Number

An algebraic representation written as $2x$.

Half of a Number

An algebraic representation written as $\frac{x}{2}$.

Three Less Than a Number

An algebraic representation written as $x - 3$.

Steps for Solving Reasoning Problems

The sequential process of reading, defining a variable, setting up an equation, solving it, and verifying the answer.