Multiply two linear algebraic expressions (Theory) : Additional Activities : Mathematics : C-DAC Online Lab

Multiply two linear algebraic expressions

Objective

To multiply two linear algebraic expressions (polynomials).

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.

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.

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 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").

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

Students should already be familiar with: