Multiply two linear algebraic expressions

Objective

To multiply two linear algebraic expressions (polynomials)

Algebraic expressions

An algebraic expression is an expression in which variables and constants are connected by any or all of the four fundamental operations (+, – , × and ÷).

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

Terms of algebraic expressions

The parts of an expression separated by the mathematical operations, addition (+) and subtraction (–) are called the terms of algebraic expression.

  • 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

Terms that have the same variables are called like terms and the variables must also have the same power in the like terms. For example: 2x² + 5x², 6y - 2y

Unlike terms

Terms which do not have the same variable or those which have the same variable but unequal powers are called unlike terms. For example: 3y + 5x, 7x² + 3x, 4z - 3x

Multiplication of algebraic expressions

To multiply a polynomial by a 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 (“first” terms of each binomial are multiplied together),
Outer (“outside” terms are multiplied—that is, the first term of the first binomial and the second term of the second),
Inner (“inside” terms are multiplied—second term of the first binomial and first term of the second),
Last (“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
  • Concept of area of rectangle and square
  • Multiplying binomials
  • Distributive property of multiplication
  • Combining like terms