Multiply two linear algebraic expressions

Objective

To multiply two linear algebraic expressions (polynomials).

Algebraic expressions

A mathematical expression that uses any or all of the four basic operations (+, -, x, and ) to connect variables and constants is known as an algebraic expression.

 

  • Variable: The unknown quantity used in algebraic expressions such as x, y, z, a, b, etc.
  • Constant: A constant in an expression is a number or value that never changes. For example: 1, 2, 3, 4, or even 0.3 or ¾.
  • Coefficient: A number or quantity associated with a variable is called the coefficient of that variable.
  • Term: A term is any number or variable separated by operators.

Consider an expression like: 5n + 3

Fig. (1) Expression

Terms of algebraic expressions

The terms of an algebraic expression are the parts of an expression that are separated by the addition (+) and subtraction (-) operations.

  • Monomial: An expression with one term. For example: 4x, 5y, -6k, -z.
  • Binomial: An expression with two terms. For example: 4x+4, 5y+2, 6z+7, -2x+3, 3y-8.
  • Trinomial: An expression with three terms. For example: 4x²+4x+1, -5y²+2y+6.

Linear algebraic expression

A linear algebraic expression or linear expression is a mathematical expression that consists of constants, variables, and coefficients, combined using addition and subtraction operations, where the variables are raised to the power of 1.
Here are some examples of linear expressions:

  • 9y + 5 ⇒ In this expression, "9" and "5" are constants, and "y" is the variable raised to the power of 1.
  • 4a + 7b - 5c ⇒ This expression involves multiple variables (a, b, and c) with coefficients.
  • 3x ⇒ Even when there's only one variable involved, it's still considered a linear expression.

Like terms

Like terms are terms that have the same variables raised to the same powers. The coefficients may be different, but the variables and their exponents must match.
Like terms can be combined or added together.
Example: 4x + 5x
In this example, "4x" and "5x" are like terms because they have the same variable ("x").

Unlike terms

Unlike terms cannot be combined or added together.
Unlike terms are terms in an algebraic expression that have different variables or the same variables with different exponents.
Example: 4x and 3y
In this example, "4x" and "3y" are unlike terms because they have different variables ("x" and "y").

Multiplication of algebraic expressions

To multiply a polynomial by another polynomial (say, a binomial by another binomial (FOIL method) or a binomial by a trinomial), we apply the distributive law of multiplication. In distributive property, each term of one polynomial has to be multiplied by each term of the other polynomial, and all the like terms are grouped together through addition or subtraction.
The word FOIL is an acronym for the four terms of the product:
First( the first terms of each binomial are multiplied together),
Outer( the outside terms are multiplied—that is, the first term of the first binomial and the second term of the second),
Inner( the inside terms are multiplied—that is, the second term of the first binomial and first term of the second),
Last( the last terms of each binomial are multiplied).
The general form is:
(a + b) (c + d) = ac + ad + bc + bd

Prerequisite knowledge

Students should already be familiar with

  • Rules of multiplication
  • Area of a rectangle and a square
  • Multiplying binomials
  • Distributive property of multiplication
  • Combining like terms