8.4.2 - Multiplying a monomial by a trinomial

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Introduction to Monomials and Trinomials

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Teacher
Teacher

Good morning, class! Today we will learn about multiplying a monomial by a trinomial. But first, let’s clarify what monomials and trinomials are. A monomial is an expression that contains only one term, like `3x`. Can anyone provide an example of a trinomial?

Student 1
Student 1

How about `4x² + 5x + 7`?

Teacher
Teacher

Exactly! Great example, Student_1. Now, when we multiply a monomial with a trinomial, we expand it. This means we multiply the monomial by each term in the trinomial. Remember the acronym 'DPA' for Distribute, Product, and Add. Who can elaborate on how we apply 'DPA'?

Student 2
Student 2

We would first distribute the monomial to each term in the trinomial!

Teacher
Teacher

Correct! And what do we do next?

Student 3
Student 3

Then we find the products of those multiplications!

Teacher
Teacher

Exactly! And finally, we add the results together.

Teacher
Teacher

Let’s summarize: Multiplying a monomial by a trinomial involves distributing, calculating products, and adding them. Remembering 'DPA' can help us keep track of this process.

Applying the Distributive Law

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Teacher
Teacher

Now that we've reviewed the definitions, let’s put this into practice with an example! Consider `3p × (4p² + 5p + 7)`. What is the first step according to the method we've learned?

Student 4
Student 4

We need to distribute `3p` to each term in the trinomial!

Teacher
Teacher

Perfect, Student_4! So, what do we get when we multiply?

Student 1
Student 1

We get `12p³`, `15p²`, and `21p`.

Teacher
Teacher

Excellent job! Now, how do we combine these terms?

Student 2
Student 2

We add them together!

Teacher
Teacher

Exactly! The result is `12p³ + 15p² + 21p`. Let’s note that this is a polynomial as well. Recap: distribute, multiply, and add!

Visualizing the Process

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Teacher
Teacher

Let’s draw this out quickly! Imagine the monomial `3p` as a box and the trinomial as another box split into three parts. If we distribute `3p`, what will we visualize?

Student 3
Student 3

We would see three different sections, each getting multiplied by `3p`!

Teacher
Teacher

That's right! This visual representation helps in understanding the distributive law. Can someone summarize what was drawn?

Student 4
Student 4

The `3p` multiplies with each part of the trinomial, showing how each term interacts separately!

Teacher
Teacher

Excellent explanation! Visualization helps grasp these concepts more intuitively. Remember: you can always sketch a diagram when you’re stuck!

Consistent Practice

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Teacher
Teacher

Let’s discuss practice! It’s crucial for mastery. What kinds of problems can we create using our knowledge of multiplying monomials and trinomials?

Student 1
Student 1

We could use problems like `x × (2x² + 3x + 4)`!

Student 2
Student 2

Or maybe `5y × (y² + 2y + 1)` as practice too!

Teacher
Teacher

Fantastic suggestions! Remember, the key is to apply the 'DPA' method each time. Can anyone present a challenge to solve in pairs?

Student 3
Student 3

How about `6x × (x² + 2 + 3x)`?

Teacher
Teacher

Great choice! Work together and utilize what we've learned. In summary, consistent practice reinforces our skills in multiplying and simplifying these expressions.

Introduction & Overview

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

Quick Overview

In this section, we learn to multiply a monomial by a trinomial using the distributive law, simplifying the process by breaking it down into manageable parts.

Standard

The section provides a clear explanation of how to multiply a monomial by a trinomial using the distributive law, demonstrating the process through various examples and emphasizing the significance of each term's multiplication and simplification.

Detailed

Multiplying a Monomial by a Trinomial

In algebra, multiplication involves not only multiplying numbers but also extending the method to algebraic expressions. A monomial is an expression containing a single term, while a trinomial contains three terms.

To multiply a monomial by a trinomial, we utilize the distributive law, which allows us to multiply each term in the trinomial by the monomial separately. For example, if we take a monomial such as 3p and multiply it by the trinomial 4p² + 5p + 7, we can break this down into:

$$
3p \times (4p² + 5p + 7) = (3p \times 4p²) + (3p \times 5p) + (3p \times 7)
$$

This results in:

  • 12p³ from multiplying 3p and 4p²
  • 15p² from multiplying 3p and 5p
  • 21p from multiplying 3p and 7

The final outcome of this multiplication yields a polynomial: 12p³ + 15p² + 21p. This systematic approach not only simplifies the process of multiplying complex expressions but also sets the foundation for polynomial algebra, making further expressions easier to handle. In summary, mastering this technique is vital for solving more advanced algebraic expressions.

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Audio Book

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Introduction to Multiplying Monomials by Trinomials

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

Detailed Explanation

In this section, we are learning how to multiply a monomial by a trinomial. A monomial is a single term like '3p', while a trinomial has three terms like '(4p^2 + 5p + 7)'. To perform this multiplication, we will use the distributive law, which tells us that we need to multiply the monomial by each term in the trinomial individually.

Examples & Analogies

Think of the monomial as the price of one item and the trinomial as a total cost that consists of different components. For example, if the item costs '3p' and the components of the total cost are '4p^2' (for shipping), '5p' (for handling), and '7' (for taxes), we would calculate the total cost by multiplying the price and adding up all the individual costs.

Applying the Distributive Law

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3p × (4p2 + 5p + 7) =(3p × 4p2) + (3p × 5p) + (3p × 7)

Detailed Explanation

Following the distributive law, we multiply '3p' by each term in the trinomial separately. The first multiplication is 3p × 4p² which gives us 12p³. The second is 3p × 5p which results in 15p². The last multiplication is 3p × 7 resulting in 21p. By performing these calculations step by step, we can simplify our expression.

Examples & Analogies

Picture a worker (3p) who does three different tasks: transporting items (4p²), sorting them (5p), and delivering them (7). The total contribution of the worker for each task shows you how to multiply the worker's unit rate (monomial) by the total tasks (trinomial) he performs.

Combining the Products

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=12p³ + 15p² + 21p

Detailed Explanation

After performing the individual multiplications, we sum up all the products to get the final result: 12p³ + 15p² + 21p. Each term represents a different part of the overall multiplication, and it's crucial to keep them separate because they cannot be simplified further since they are not like terms.

Examples & Analogies

Imagine you have completed three different projects in a week, and you want to know your total contribution. Each project had a different scale of work. Just like you combine your contributions from these projections (12p³ for large tasks, 15p² for medium tasks, and 21p for small tasks), here we combine all the product terms from our multiplication.

Revisiting the Distributive Law

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Multiply each term of the trinomial by the monomial and add products.

Detailed Explanation

This emphasizes the importance of the distributive law in multiplication, where a single term (monomial) is distributed to each term of the polynomial (trinomial). It helps in achieving clarity in multiplication and ensures no steps are skipped in computation.

Examples & Analogies

Think of distributing snacks among a group of friends. If each friend (a term in the trinomial) receives an equal share of snacks (the monomial), it's essential to ensure every friend gets their fair share rather than skipping anyone. Similarly, by applying the distributive law, we ensure every term in the trinomial receives the monomial's respect smoothly.

Definitions & Key Concepts

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

Key Concepts

  • Distributive Law: The technique used to multiply each term in a polynomial by a monomial.

  • Simplifying Expression: The process of combining like terms to present the multiplication result in a simplified format.

  • Term-by-Term Multiplication: The method of multiplying each term of a trinomial by a monomial to obtain the final polynomial.

Examples & Real-Life Applications

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

Examples

  • Multiply 3p by 4p² + 5p + 7: Result is 12p³ + 15p² + 21p.

  • Multiply x by 2x² + 3x + 5: Result is 2x³ + 3x² + 5x.

Memory Aids

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

🎵 Rhymes Time

  • When multiplying, take a cue, Distribute first, it's easy to do!

📖 Fascinating Stories

  • Imagine a shop where a single item price is multiplied by several customers’ carts—a clear view of distribution!

🧠 Other Memory Gems

  • 'DPA' stands for Distribute, Product, Add—keep multiplying until you’re glad!

🎯 Super Acronyms

DPA = Distribute, then Product, and finally Add.

Flash Cards

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Glossary of Terms

Review the Definitions for terms.

  • Term: Monomial

    Definition:

    An algebraic expression that contains only one term.

  • Term: Trinomial

    Definition:

    An algebraic expression that consists of three terms.

  • Term: Distributive Law

    Definition:

    A property of multiplication over addition or subtraction, allowing the multiplication of a single term by each term in a sum or difference.

  • Term: Polynomial

    Definition:

    An algebraic expression formed from one or more monomials.