Algebra - Algebraic Expressions and Identities

Terms, Factors, and Coefficients: An algebraic expression is made up of terms separated by plus or minus signs. For example, in $5x^2 - 3xy + 7$, the terms are $5x^2$, $-3xy$, and $7$. Each term is a product of factors; for $5x^2$, the factors are $5$, $x$, and $x$. The numerical factor is called the coefficient. Visually, you can represent an expression as a 'factor tree' where the expression branches into terms, and each term branches into its individual numerical and variable factors.

Classification of Polynomials: Expressions are categorized by the number of terms they contain. A Monomial has exactly one term (e.g., $4x^2$), a Binomial has two terms (e.g., $a + b$), and a Trinomial has three terms (e.g., $x^2 + 2x + 1$). Any expression with one or more terms is broadly called a Polynomial. Imagine these as boxes of different sizes: a small box for one item, a medium for two, and a large box for three or more.

Like and Unlike Terms: Like terms are terms that have the exact same variables raised to the same powers, such as $7ab$ and $-2ab$. Unlike terms have different variables or powers, such as $3x$ and $3x^2$. Only like terms can be added or subtracted. Visually, think of sorting fruit: you can add 3 apples to 2 apples to get 5 apples, but you cannot combine 3 apples and 2 oranges into a single type of fruit.

Addition and Subtraction of Expressions: To add or subtract algebraic expressions, collect and combine only the like terms. This process is often visualized by writing expressions in a vertical format, aligning like terms in columns (e.g., $x^2$ terms under $x^2$ terms), similar to how we align units, tens, and hundreds in arithmetic.

Multiplication of Algebraic Expressions: Multiplying a monomial by a polynomial involves the distributive law, where the monomial is multiplied by every term inside the parentheses. Multiplying a binomial by another binomial follows the FOIL method (First, Outer, Inner, Last). This can be visualized using an 'Area Model' where the two expressions represent the length and width of a rectangle, and the product represents the total area of the smaller rectangular segments inside.

Algebraic Identities: Identities are equality relations that hold true for any value of the variables involved. They serve as 'shortcut templates' for expansion and factorization. Instead of multiplying term-by-term, you recognize the pattern and apply the formula. Think of these as blueprints that allow you to see the final structure of an expanded expression without doing all the manual labor of multiplication.