Algebraic Expressions - Terms, Factors and Coefficients of an Expression

Algebraic Expressions are mathematical phrases formed by combining variables (letters like $x, y$) and constants (fixed numbers like $5, -10$) using arithmetic operations. Think of an expression as a 'construction' where variables and constants are the building blocks connected by $+$, $-$, $\times$, or $\div$.

Terms are the individual components of an expression that are separated by plus ($+$) or minus ($-$) signs. For example, in the expression $4x^2 - 3xy + 7$, there are three terms: $4x^2$, $-3xy$, and $7$. Visually, you can imagine an expression as a 'Term Tree' where the expression is the trunk and each term is a major branch.

Factors are the pieces that are multiplied together to create a single term. For the term $5xy$, the factors are $5$, $x$, and $y$. In a tree diagram, if a term is a branch, the factors are the smaller twigs or 'leaves' growing out of that branch. Factors cannot be further broken down by multiplication.

The Numerical Coefficient (or simply coefficient) is the number that multiplies the variable part of a term. In the term $-7ab$, the coefficient is $-7$. If a term appears as just $x^2$, its coefficient is understood to be $1$. If it is $-x$, the coefficient is $-1$.

Constants are terms that consist only of a number and do not have any variables attached to them. In the expression $x + 5$, the number $5$ is the constant. Unlike variables, the value of a constant never changes, representing a fixed point on a number line.

Like Terms are terms that share the exact same algebraic (variable) factors. For instance, $2xy$ and $-5xy$ are like terms because their variable part $xy$ is identical. Visually, like terms are like objects of the same shape and size that can be stacked or grouped together.

Unlike Terms are terms that have different algebraic factors. For example, $3x$ and $3x^2$ are unlike terms because the power of $x$ is different. Similarly, $4x$ and $4y$ are unlike terms. These cannot be combined into a single term, much like trying to add apples and oranges.