Multiplying a binomial by a binomial

8.5.1 Multiplying a binomial by a binomial

Description

Quick Overview

This section teaches the process of multiplying binomials using the distributive property.

Standard

The section explains how to multiply two binomials by using the distributive law. It covers the steps required, provides examples, and emphasizes the importance of combining like terms in the resulting polynomial.

Detailed

In this section, we delve into the multiplication of binomials, a foundational operation in algebra. To multiply two binomials, such as (2a + 3b) and (3a + 4b), we apply the distributive law which states that we multiply each term in the first binomial by every term in the second binomial. The mathematical expression is set up as:

(3a + 4b) × (2a + 3b) = 3a × (2a + 3b) + 4b × (2a + 3b)

This results in:

= (3a × 2a) + (3a × 3b) + (4b × 2a) + (4b × 3b) = 6a² + 9ab + 8ba + 12b²

After identifying like terms (where ba = ab) and combining them, we conclude with:

= 6a² + 17ab + 12b²

The importance of this section lies in understanding how to systematically multiply polynomials of varying degrees and combine like terms, which reinforces key algebraic skills.

Key Concepts

  • Distributive Property: Allows multiplication of terms in polynomials.

  • FOIL Method: A technique for multiplying two binomials by considering Four parts (First, Outside, Inside, Last).

  • Combining Like Terms: Important for simplifying results after multiplication.

Memory Aids

🎵 Rhymes Time

  • To multiply binomials, don’t be shy, | Use FOIL to simplify, | First, Outside, Inside, Last is the way, | Combine your terms for a brighter day!

🎯 Super Acronyms

FOIL

  • First
  • Outside
  • Inside
  • Last.

📖 Fascinating Stories

  • Imagine two neighbors, Alex and Bella, each planting double rows of flowers labeled with terms. When they make their flowerbeds, each plant type looks to reach the garden sun, just like terms in multiplication combine for the perfect view.

🧠 Other Memory Gems

  • Use F for the first factor, O for outside, I for inside, and L for last!

Examples

  • Example 1: Multiply (2x + 3)(x - 5) = 2x^2 - 10x + 3x - 15 = 2x^2 - 7x - 15.

  • Example 2: Multiply (a + 4)(a - 1) = a^2 - 1a + 4a - 4 = a^2 + 3a - 4.

Glossary of Terms

  • Term: Binomial

    Definition:

    A polynomial with exactly two terms.

  • Term: Distributive Law

    Definition:

    A property that allows us to multiply a single term by terms inside parentheses.

  • Term: Like Terms

    Definition:

    Terms in an expression that have the same variable raised to the same power.

  • Term: Polynomial

    Definition:

    An algebraic expression consisting of one or more terms.