Algebraic Expressions - Monomials, Binomials, Trinomials and Polynomials

An algebraic expression is formed from variables and constants. Variables like $x, y, l, m$ represent unknown values that can change, while constants like $4, 100, -7$ have fixed numerical values. Visually, think of a variable as a box that can hold any number of items, while a constant is a fixed stack of items.

Expressions are made up of terms. A term is a product of factors. For example, in the expression $5x + 3$, the terms are $5x$ and $3$. In the term $5x$, the factors are $5$ and $x$. You can visualize this as a 'Tree Diagram' where the expression is the root, terms are the branches, and factors are the leaves.

The numerical factor of a term is called its numerical coefficient or simply the coefficient. In the term $7xy$, $7$ is the coefficient; in $-5ab$, $-5$ is the coefficient. If no number is visible, such as in $x^2y$, the coefficient is understood to be $1$.

Algebraic expressions are classified based on the number of terms they contain. A Monomial has only one term (e.g., $5x^2$); a Binomial has two terms (e.g., $a + b$); a Trinomial has three terms (e.g., $x^2 + x + 1$). Any expression with one or more terms is generally called a Polynomial.

Terms that have the same algebraic factors are called Like Terms (e.g., $2xy$ and $-5xy$). Terms that have different algebraic factors are called Unlike Terms (e.g., $4x$ and $4x^2$). Visually, adding like terms is like grouping fruits of the same kind together; you cannot add 3 apples and 2 oranges to get 5 'apple-oranges'.

To add or subtract algebraic expressions, only like terms can be combined. When adding, we add the numerical coefficients of the like terms. For example, $3x + 4x = (3+4)x = 7x$. In subtraction, the sign of every term in the expression being subtracted is changed before adding it to the first expression.

The value of an expression depends on the values assigned to the variables it contains. For example, if $x = 2$, then the value of $4x - 3$ is $4(2) - 3 = 8 - 3 = 5$. This can be visualized as a function machine where you input a number for $x$ and get a specific result out.