Algebraic Expressions and Identities - Addition, Subtraction, Multiplication and Division of Algebraic Expressions

Algebraic expressions are formed by combining constants and variables using operations like addition, subtraction, multiplication, and division. A term is a product of factors; for instance, in $5xy$, $5$ is the numerical coefficient and $x, y$ are literal factors. Think of an expression like a multi-level structure where terms are distinct rooms separated by plus or minus signs.

Like terms are terms that have the exact same literal (variable) factors with the same exponents, such as $4x^2y$ and $-7x^2y$. Only like terms can be added or subtracted. You can visualize this by imagining like terms as identical geometric shapes (e.g., all $x^2$ terms are large squares); you can only combine squares with squares and triangles with triangles.

To add or subtract algebraic expressions, group the like terms together. During subtraction, the sign of every term in the expression being subtracted (the subtrahend) must be changed (e.g., $+2x$ becomes $-2x$). Visually, this is similar to aligning columns in a ledger to ensure you only calculate totals for the same category of items.

The Distributive Law is the foundation of multiplication: $a(b + c) = ab + ac$. When multiplying a monomial by a polynomial, each term of the polynomial is multiplied by the monomial. This can be visualized as finding the area of a large rectangle that has been partitioned into smaller sections.

Multiplication of two binomials $(a+b)(c+d)$ results in $ac + ad + bc + bd$. This is often called the FOIL method (First, Outer, Inner, Last). Imagine a square with side lengths split into $a+b$ and $c+d$; the total area is the sum of the four internal rectangles created by these segments.

When multiplying variables, use the law of exponents: $a^m \times a^n = a^{m+n}$. Conversely, when dividing variables, subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$. If you visualize $x^3$ as $x \cdot x \cdot x$, dividing by $x$ is simply 'canceling out' one of those factors from the group.

Division of a polynomial by a monomial is performed by dividing each term of the polynomial by the monomial individually. For example, $\frac{10x^2 + 5x}{5x} = \frac{10x^2}{5x} + \frac{5x}{5x}$. If you visualize a group of objects being shared equally, you must ensure every part of the original group is divided by the divisor.

Standard identities are equations that are true for any value of the variables involved. For example, $(a+b)^2 = a^2 + 2ab + b^2$. Visually, $(a+b)^2$ represents a large square with side $a+b$, which is composed of one square of area $a^2$, one square of area $b^2$, and two rectangles each of area $ab$.