Multiplying a binomial by a trinomial

8.5.2 Multiplying a binomial by a trinomial

Description

Quick Overview

This section discusses the procedure for multiplying a binomial by a trinomial, illustrating how to apply the distributive law.

Standard

In this section, we learn how to multiply a binomial by a trinomial using the distributive law. The process involves multiplying each term in the binomial by each term in the trinomial, leading to a collection of terms that may be combined if they are like terms.

Detailed

Multiplying a Binomial by a Trinomial

Multiplying a binomial by a trinomial involves applying the distributive law of multiplication. When we have a binomial such as
(b + c) and a trinomial like (a^2 + ba + c), we multiply each term from the binomial with each term of the trinomial. For instance, if we consider (a + 7) multiplied by (a^2 + 3a + 5), the process involves:

  1. Distributing each term of the binomial:
  2. Compute a * (a^2 + 3a + 5) which produces terms a^3, 3a^2, and 5a.
  3. Then compute 7 * (a^2 + 3a + 5) resulting in 7a^2, 21a, and 35.
  4. Combining all the terms: After distributing, we collect and simplify the terms:
  5. Grouping like terms from both distributions gives us: a^3 + 10a^2 + 26a + 35.

This technique allows for systematic organization of products and is essential for simplifying higher algebra expressions robustly.

Key Concepts

  • Distributive Law: Each term in a binomial must multiply each term in a trinomial.

  • Combining Like Terms: After multiplication, group similar terms to simplify the final expression.

Memory Aids

🎵 Rhymes Time

  • Binomial pairs just two, Trinomials are three, together they make a great product for all to see.

📖 Fascinating Stories

  • Once upon a time, in a garden of math, there were two flowers named Binomial and Trinomial. Together, they formed beautiful products that bloomed numerically!

🧠 Other Memory Gems

  • BTTP (Binomial and Trinomial Terms Product) - Remember to Multiply each term in Binomial with each term in Trinomial!

🎯 Super Acronyms

CRM (Combine, Reduce, Multiply) - Always remember to Combine like terms, Reduce where possible, and Multiply each term properly.

Examples

  • Example 1: (x + 3)(x^2 + 2x + 1) = x^3 + 2x^2 + x + 3x^2 + 6x + 3 = x^3 + 3x^2 + 7x + 3, combining like terms.

  • Example 2: (a + 4)(a^2 + 2a + 2) = a^3 + 2a^2 + 2a + 4a^2 + 8a + 8 = a^3 + 6a^2 + 10a + 8.

Glossary of Terms

  • Term: Binomial

    Definition:

    An algebraic expression containing two terms.

  • Term: Trinomial

    Definition:

    An algebraic expression containing three terms.

  • Term: Distributive Law

    Definition:

    A property that states a(b + c) = ab + ac.

  • Term: Like Terms

    Definition:

    Terms in an algebraic expression that have the same variable parts.