Algebraic Expressions - Like and Unlike Terms

Algebraic Terms and Expressions: An algebraic expression is a combination of variables and constants connected by arithmetic operators. For example, in the expression $5x^2 - 3x + 8$, the parts $5x^2$, $-3x$, and $8$ are called terms. Visually, you can imagine terms as the individual building blocks or 'packets' that are added or subtracted to form a complete structure.

Factors of a Term: Each term is a product of its factors. For example, the term $7xy$ is the product of $7$, $x$, and $y$. Visually, this is often represented using a 'Factor Tree', where the term sits at the top and its individual numerical and literal factors branch out below it.

Numerical Coefficients: The numerical factor in a term is known as its coefficient. In the term $-9a^2b$, the coefficient is $-9$. Visually, the coefficient acts like a scale or a multiplier, telling you 'how many' of that specific variable combination you have.

Like Terms: Terms that have the same algebraic factors (the same variables with the same exponents) are called Like Terms. For instance, $4mn$ and $-10mn$ are like terms. Visually, imagine sorting a collection of objects; like terms are items of the exact same shape and size, regardless of their color (coefficient).

Unlike Terms: Terms that have different algebraic factors are called Unlike Terms. This occurs if the variables are different (e.g., $3x$ and $3y$) or if the powers of the same variables differ (e.g., $5x^2$ and $5x$). Visually, these are like 'apples' and 'oranges'—because they are fundamentally different, they cannot be merged into a single item.

Identifying Like Terms: To identify like terms, ignore the numerical coefficient and focus only on the literal (variable) part. For example, in the group $2ab, 5ba, 7a^2$, the terms $2ab$ and $5ba$ are like terms because the order of multiplication does not change the factors. Visually, you are looking for a perfect match in the variable 'labels'.

Combining Like Terms: Simplification involves grouping like terms together and adding or subtracting their coefficients. Visually, this is similar to gathering all identical blocks in a pile to see the total height; the nature of the block (variable) stays the same, but the total count (coefficient) changes.