8.4 - Multiplying a Monomial by a Polynomial

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Introduction to Monomials and Polynomials

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Teacher
Teacher

Today we're going to discuss how to multiply a monomial by a polynomial. Remember, a monomial is an expression with just one term. Can anyone give me an example of a monomial?

Student 1
Student 1

4x is a monomial!

Teacher
Teacher

That's correct! Now, a polynomial can have one or more terms. Who can give me an example?

Student 2
Student 2

How about x² - 3x + 5?

Teacher
Teacher

Great job! Now, we will focus on multiplying a monomial with a polynomial, such as the binomial. What do we remember about the distributive law?

Student 3
Student 3

It means we multiply each term in the parentheses separately!

Teacher
Teacher

Exactly! Let's remember to distribute with an acronym: D for Distribute, E for Every term. Now, let’s see an example.

Multiplying a Monomial by a Binomial

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Teacher
Teacher

Let’s compute 3x × (5y + 2). Using the distributive law: what will be our first step?

Student 4
Student 4

We multiply 3x by both 5y and 2!

Teacher
Teacher

That's right! So we get: 15xy + 6x. Can anyone simplify: 7x multiplied by (2a - 4)?

Student 1
Student 1

That would be 14ax - 28!

Teacher
Teacher

Fantastic! Let's summarize: Remember to always apply our D.E. method. Distribute across every term.

Multiplying a Monomial by a Trinomial

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Teacher
Teacher

Next, let’s multiply a monomial by a trinomial. For instance, 3p × (4p² + 5p + 7). What’s our first step?

Student 2
Student 2

We’ll multiply 3p by each term in the trinomial!

Teacher
Teacher

Correct! So, we get 12p³ + 15p² + 21p. Why do we have three terms?

Student 3
Student 3

Because there are three terms in the trinomial!

Teacher
Teacher

Exactly! Great job! Remember to look out for combining like terms when multiplying!

Application and Practice

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Teacher
Teacher

To apply these skills, let’s think of practical examples. If I say the length of a rectangle is (3x) and its width is (4y + 2), what is the area?

Student 4
Student 4

We would multiply 3x by (4y + 2) to get 12xy + 6x!

Teacher
Teacher

Great! Let's practice multiplying some more. What is the product of 5(a + 3)?

Student 1
Student 1

It's 5a + 15.

Teacher
Teacher

Fantastic! I want everyone to practice these problems and remember our D.E. method.

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

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 × (5y + 2) = (3x × 5y) + (3x × 2) = 15xy + 6x.

The section further explores multiplying a monomial by a trinomial through an example of 3p × (4p² + 5p + 7), resulting in:
- (3p × 4p²) + (3p × 5p) + (3p × 7) = 12p³ + 15p² + 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.

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Audio Book

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Definition of Polynomials

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An expression that contains two terms is called a binomial. An expression containing three terms is a trinomial and so on. In general, an expression containing one or more terms with non-zero coefficients (with variables having non-negative integers as exponents) is called a polynomial.

Detailed Explanation

In mathematics, polynomials are expressions made up of variables and coefficients. A single term is called a monomial, two terms make a binomial, and three terms create a trinomial. For instance, '3x + 4' is a binomial, while '2x^2 + 3x + 5' is a trinomial.

Examples & Analogies

Think of polynomials like a recipe: a monomial is a single ingredient, a binomial combines two ingredients, and a trinomial adds a third ingredient. Just as you can combine various ingredients to create different dishes, you can combine terms in mathematics to create different expressions.

Multiplying a Monomial by a Binomial

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Let us multiply the monomial 3x by the binomial 5y + 2, i.e., find 3x × (5y + 2) = ? Recall that 3x and (5y + 2) represent numbers. Therefore, using the distributive law, 3x × (5y + 2) = (3x × 5y) + (3x × 2) = 15xy + 6x.

Detailed Explanation

Multiplying a monomial by a binomial involves distributing the monomial across each term in the binomial. In our example, we take 3x and multiply it by both 5y and 2 separately. The results of these multiplications (15xy and 6x) are then added together to get the final product of 15xy + 6x.

Examples & Analogies

Imagine you are packaging boxes of fruit. If you have 3 boxes (the monomial) and each box has 5 apples and 2 oranges (the binomial), you would calculate the total number of fruits you have by multiplying the number of boxes by the amount of each type of fruit.

Multiplying a Monomial by a Trinomial

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Consider 3p × (4p² + 5p + 7). As in the earlier case, we use the distributive law; 3p × (4p² + 5p + 7) = (3p × 4p²) + (3p × 5p) + (3p × 7) = 12p³ + 15p² + 21p.

Detailed Explanation

When multiplying a monomial by a trinomial, we apply the same distributive law by multiplying the monomial by each term of the trinomial separately. Here, we take 3p and multiply it by 4p², 5p, and 7. This gives us three products, which we then add together to yield the final answer of 12p³ + 15p² + 21p.

Examples & Analogies

Consider a gardener with a set of plants. If the gardener has 3 pots of plants, and each pot has 4 potted flowers, 5 potted herbs, and 7 potted vegetables (this is the trinomial), the gardener can find out the total number of each type by multiplying the number of pots by each type of plant separately.

Using Distributive Law in Examples

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For example: (–3x) × (–5y + 2) = (–3x) × (–5y) + (–3x) × (2) = 15xy – 6x.

Detailed Explanation

Using the distributive law can also apply to negative numbers. Here, when we multiply –3x by –5y, the product is positive, resulting in 15xy. When we multiply –3x by 2, we get –6x. So, the entire expression simplifies to 15xy – 6x.

Examples & Analogies

If you owe money (represented by –3x) but receive payments (represented by a positive outcome of the binomial) for two different types of loans, you can calculate new totals. The distribution ensures you accurately account for each type of transaction.

Terminology: Binomials and Trinomials

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What about a binomial × monomial? For example, (5y + 2) × 3x = ? We may use commutative law as: 7 × 3 = 3 × 7

Detailed Explanation

The commutative law states that the order in which two numbers are multiplied does not change the product. So whether we calculate (5y + 2) × 3x or 3x × (5y + 2), we will arrive at the same result, 15xy + 6x. This highlights the flexibility in multiplication – we can rearrange terms as needed.

Examples & Analogies

Imagine a light-weight box filled with 5 bags of flour and two bags of sugar. Whether we multiply the amount of flour first and then add sugar, or combine both types simultaneously, will yield the same total weight of the goods you're carrying.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

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.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • 3x × (5y + 2) = 15xy + 6x.

  • 3p × (4p² + 5p + 7) = 12p³ + 15p² + 21p.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

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

📖 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.

🧠 Other Memory Gems

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

🎯 Super Acronyms

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

Flash Cards

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Glossary of Terms

Review the Definitions for 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.