Key Points

An algebraic expression is formed from variables and constants. The parts of an expression separated by '+' or '-' signs are called terms. For example, in $4xy + 7$, the terms are $4xy$ and $7$.

Expressions are classified by the number of terms. A monomial has one term (e.g., $3x^2$), a binomial has two terms (e.g., $x+3$), and a trinomial has three terms. An expression with one or more terms is a polynomial.

Like terms have the same variables raised to the same powers, such as $7x^2$ and $3x^2$. Unlike terms have different variables or different powers, like $7x^2$ and $4xy$. Only like terms can be added or subtracted.

To add expressions, group the like terms together and add their coefficients. For example, to add $7x^2 - 4x + 5$ and $9x - 10$, we combine like terms to get $7x^2 + (-4x + 9x) + (5 - 10) = 7x^2 + 5x - 5$.

To subtract an expression, add its additive inverse. This means changing the sign of every term in the expression being subtracted and then adding. For example, subtracting $5x^2 - 3$ is the same as adding $-5x^2 + 3$.

To multiply two monomials, multiply their numerical coefficients and add the powers of their variables. For example, $(5x) imes (4x^2) = (5 imes 4) imes (x imes x^2) = 20x^3$.

Use the distributive property to multiply each term of the polynomial by the monomial. For example, $3p imes (4p^2 + 5p + 7) = (3p imes 4p^2) + (3p imes 5p) + (3p imes 7) = 12p^3 + 15p^2 + 21p$.

To multiply two binomials, multiply each term of the first binomial by each term of the second binomial. For $(a+b)(c+d)$, the product is $a(c+d) + b(c+d) = ac + ad + bc + bd$. Remember to combine any like terms in the final result.

The square of a sum is given by the identity $(a+b)^2 = a^2 + 2ab + b^2$. This is used for quick expansion of expressions like $(2x+5)^2$.

The square of a difference is given by the identity $(a-b)^2 = a^2 - 2ab + b^2$. This is useful for expanding expressions like $(y-8)^2$.

The product of a sum and a difference is given by the identity $(a+b)(a-b) = a^2 - b^2$. This is a powerful tool for factorization and simplification.

The product of two binomials of the form $(x+a)$ and $(x+b)$ is given by the identity $(x+a)(x+b) = x^2 + (a+b)x + ab$.

An identity is an equality that is true for all values of its variables. For example, $(a+1)^2 = a^2 + 2a + 1$ is an identity because it holds true for any value of $a$.

The volume of a rectangular box is the product of its length, breadth, and height. If the dimensions are algebraic expressions, you multiply them. For example, volume of a box with dimensions $5a, 3a^2, 7a^4$ is $5a imes 3a^2 imes 7a^4 = 105a^7$.