When you have a product, each numerical or literal (variable) value within that product is called a factor.
Example: In the product $5xy$, the expression can be broken down into $5 \times x \times y$.
Here, $5, x,$ and $y$ are all individual factors of $5xy$.
In a product containing two or more factors, any one factor is called the co-efficient of the product of the remaining factors.
Example 1: In $5x$, the number $5$ is the co-efficient of $x$.
Example 2: In $-6xy$, the number $-6$ is the co-efficient of $xy$.
To master algebra, you need to understand how expressions are built.
Term: A term can be a constant alone, a variable alone, or a combination of both linked by multiplication or division.
Examples: $6$, $x$, $6x$, and $\frac{6}{x}$ are all terms.
Factors: In a product, each numerical or literal (variable) value is a factor.
Example: In the product $5xy$, the factors are $5$, $x$, and $y$.
Co-efficient: When you have a product of multiple factors, any one factor is the co-efficient of the rest.
In $5x$, $5$ is the co-efficient of $x$.
In $-6xy$, $-6$ is the co-efficient of $xy$.
To simplify expressions, you must be able to distinguish between like and unlike terms.
Like Terms (Similar Terms): These are terms that contain the exact same literal (variable) factors.
Examples: ($a$ and $2a$) or ($2a$ and $3a$).
Unlike Terms: These are terms that do not share the same literal factors.
Examples: ($2a$ and $3b$) or ($2$ and $a$).
You can simplify an expression by adding the co-efficients of like terms together.
$3a + 2a = (3+2)a = 5a$
$5a + 6a + 7a = (5+6+7)a = 18a$
Mixed Terms: If an expression has different variables, group the like terms first:
$3a + 5a + 3b + 4b = 8a + 7b$
$3a + 2 + 5a + 7 = 8a + 9$
Just like you can’t add apples and oranges, you can only combine certain parts of an algebraic expression.
Like Terms: Terms that contain the exact same literal (variable) factors.
Example: $2a$ and $3a$ are like terms.
Unlike Terms: Terms that have different variables or are a mix of variables and constants.
Example: $2a$ and $3b$ are unlike terms, as are $2$ and $a$.
The golden rule of algebra is that you can only add or subtract like terms.
To add like terms, simply add their numerical co-efficients.
$3a + 2a = 5a$
$3a + 5a + 3b + 4b = 8a + 7b$ (Notice we kept the $a$ and $b$ groups separate!)
The same logic applies to subtraction.
$5a – 3a = 2a$
$5a – 9a = -4a$
$3x – 12x = -9x$
An Algebraic Expression is the combination of terms created using fundamental operations like addition, subtraction, multiplication, and division.
Examples: $6 + x$, $3x + 6$, or $3x – 5$.
A Monomial is an expression containing only one term where the powers of the variables are non-negative integers (whole numbers).
Is a Monomial: $4x$, $3x^2y$.
Is NOT a Monomial: $\frac{4}{x}$ (because the exponent is $-1$) or $\sqrt{x}$ (because the exponent is $1/2$).
An Algebraic Expression is the combination of terms created using operations like addition and subtraction. These can be categorized by the number of terms they contain:
Monomial: An expression containing only one term where the powers of the variables are whole numbers.
Examples: $4x$, $2xy$, $3x^2y$.
Binomial: An expression containing two unlike terms.
Examples: $x+y$, $4p+2z$.
Trinomial: An expression with three unlike terms.
Examples: $a+4b+2z$.
Multinomial: A general term for an expression with two or more terms.
Polynomial: An expression with one or more terms where variable exponents are non-negative integers.
Degree of a Monomial: The sum of the powers of all variables involved. For example, the degree of $7x^3yz^2$ is $3+1+2 = 6$.
Degree of a Polynomial: The greatest degree among all its individual terms.
Substitution: The method of replacing variables with specific numerical values to find a result. If $x = -1$, then $2x$ becomes $2(-1) = -2$.
There are two primary ways to add complex expressions:
Horizontal Method: Writing the terms in a single line and grouping like terms (e.g., $(3a + 2) + (5a + 7) = 8a + 9$).
Vertical Method: Aligning like terms in columns before adding.