Multiplying a Polynomial by a Polynomial

8.5 Multiplying a Polynomial by a Polynomial

Description

Quick Overview

This section covers the methods and processes involved in multiplying polynomials, specifically focusing on multiplying binomials and the distributive law.

Standard

In section 8.5, we learn about multiplying polynomials, particularly binomials. The process involves applying the distributive law, where each term in one polynomial multiplies every term in the other. Examples illustrate this process, emphasizing the importance of combining like terms to simplify the results.

Detailed

Detailed Summary

In this section, we delve into the multiplication of polynomials, particularly focusing on the multiplication of binomials by binomials and binomials by trinomials using the distributive law.

Key Points:

  1. Multiplication of Binomials: When multiplying two binomials, such as (2a + 3b) and (3a + 4b), every term in the first binomial is multiplied by every term in the second binomial. This process results in multiple terms, which may include like terms that can be combined for simplification. For example, the product (2a + 3b)(3a + 4b) yields several products: 6a² + 9ab + 8ab + 12b². Combining like terms here gives us 6a² + 17ab + 12b².
  2. Distributive Law: The distributive law is pivotal in polynomial multiplication. It states that a(b + c) = ab + ac. This law allows us to expand the multiplication process systematically.
  3. Practical Examples: Several examples demonstrate how to apply these principles, including how to simplify expressions and handle multiple terms.

Overall, mastering polynomial multiplication is essential for dealings in algebra, which often encounters expressions necessitating expansion, such as in area computation and in various algebraic identities.

Key Concepts

  • Polynomial: A mathematical expression comprising variables and coefficients.

  • Binomial: A polynomial with two terms.

  • Trinomial: A polynomial with three terms.

  • Distributive Law: A fundamental property used to simplify multiplication.

  • Combining Like Terms: The process of combining expressions with the same variables and exponents.

Memory Aids

🎵 Rhymes Time

  • To multiply two binomials, you expand with glee, every term you see.

📖 Fascinating Stories

  • Imagine multiplying the prices of apples and oranges; combine them for a fruit basket total.

🧠 Other Memory Gems

  • Remember 'FOIL' when multiplying binomials: First, Outside, Inside, Last.

🎯 Super Acronyms

BIP - Binomial Interaction Principle

  • remember to combine like terms!

Examples

  • (x - 4)(2x + 3) = 2x² - 5x - 12

  • (a + 7)(a² + 3a + 5) = a³ + 10a² + 26a + 35

Glossary of Terms

  • Term: Polynomial

    Definition:

    An algebraic expression composed of variables and coefficients, involving only non-negative integer exponents.

  • Term: Binomial

    Definition:

    A polynomial with exactly two terms.

  • Term: Trinomial

    Definition:

    A polynomial with exactly three terms.

  • Term: Distributive Law

    Definition:

    A property that allows for the multiplication of a sum by distributing the multiplier across each addend.

  • Term: Like Terms

    Definition:

    Terms that contain the same variables raised to the same power.