Multiplying a Monomial by a Polynomial

8.4 Multiplying a Monomial by a Polynomial

Description

Quick Overview

This section focuses on the multiplication of monomials by polynomials, showing how to apply the distributive law effectively.

Standard

In this section, students learn how to multiply a monomial by a polynomial, covering both binomials and trinomials. The distributive law is introduced as a fundamental concept to simplify calculations, and examples illustrate the step-by-step process and applications.

Detailed

Detailed Summary

This section, titled "Multiplying a Monomial by a Polynomial," outlines the method of multiplying monomials with polynomials, highlighting binomials and trinomials as key categories of polynomial expressions.

The multiplication of a monomial by a binomial is illustrated using examples, such as multiplying 3x by the binomial (5y + 2). Using the distributive law, the multiplication is expressed as:

  • Distributive Law: 3x Ɨ (5y + 2) = (3x Ɨ 5y) + (3x Ɨ 2) = 15xy + 6x.

The section further explores multiplying a monomial by a trinomial through an example of 3p Ɨ (4pĀ² + 5p + 7), resulting in:
- (3p Ɨ 4pĀ²) + (3p Ɨ 5p) + (3p Ɨ 7) = 12pĀ³ + 15pĀ² + 21p.

Prominent throughout is the emphasis on the distributive law, a principle that allows the simplification of expressions by distributing the monomial across each term in the polynomial. The section also underscores the importance of recognizing like terms when combining results after multiplication. By the end of this section, students will have established a clear understanding of multiplying monomials by polynomials, thereby strengthening their foundation for more complex algebraic operations.

Key Concepts

  • Distributive Law: A method for multiplying a single term by each term in a polynomial.

  • Monomial: An algebraic expression consisting of exactly one term.

  • Polynomial: An algebraic expression containing one or more terms.

  • Binomial: A polynomial with two terms.

  • Trinomial: A polynomial with three terms.

Memory Aids

šŸŽµ Rhymes Time

  • When you multiply, don't forget the rule, Distribute each term, that's the golden tool!

šŸ“– Fascinating Stories

  • Once upon a time, there was a wise owl named 'Distributive.' He taught the students to share everything equally, so when multiplying, they learned to distribute their values across all terms in a polynomial.

šŸ§  Other Memory Gems

  • D.E. = Distribute Every term! A reminder to always distribute correctly!

šŸŽÆ Super Acronyms

M.P.P. = Monomial x Polynomial = Product in Polynomial form.

Examples

  • 3x Ɨ (5y + 2) = 15xy + 6x.

  • 3p Ɨ (4pĀ² + 5p + 7) = 12pĀ³ + 15pĀ² + 21p.

Glossary of Terms

  • Term: Monomial

    Definition:

    An algebraic expression containing one term.

  • Term: Polynomial

    Definition:

    An algebraic expression containing one or more terms.

  • Term: Binomial

    Definition:

    A polynomial that contains two terms.

  • Term: Trinomial

    Definition:

    A polynomial that contains three terms.

  • Term: Distributive Law

    Definition:

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