8.5.2 - Multiplying a binomial by a trinomial

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Introduction to Multiplying Binomials and Trinomials

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Teacher
Teacher

Today, we're going to learn how to multiply a binomial by a trinomial. Can anyone tell me what a binomial is?

Student 1
Student 1

A binomial has two terms, like (x + 2).

Teacher
Teacher

Exactly! And what about a trinomial?

Student 2
Student 2

A trinomial has three terms, like (x^2 + x + 1).

Teacher
Teacher

Great! Now, when we multiply them, we’ll use the distributive law, which states that each term in the first expression multiplies each term in the second. Let's take a look at an example to see how it works.

Student 3
Student 3

What’s an example?

Teacher
Teacher

Let's consider (a + 7) and (a^2 + 3a + 5).

Student 4
Student 4

How do we start with that?

Teacher
Teacher

First, we distribute 'a' to each term of the trinomial. Who wants to help me do that?

Student 1
Student 1

So it will be a^3 + 3a^2 + 5a?

Teacher
Teacher

Correct! Now we distribute '7'.

Student 2
Student 2

That's 7a^2 + 21a + 35!

Teacher
Teacher

Excellent! Now we combine the like terms. What do we have?

Student 3
Student 3

It simplifies to a^3 + 10a^2 + 26a + 35!

Teacher
Teacher

Exactly! Remember, it’s crucial to combine like terms.

Practical Application of Multiplication

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Teacher
Teacher

Now that we know how to multiply, let's discuss why this is important. Can anyone share an example of where you might use this in real life?

Student 4
Student 4

When calculating the area of a space that has a length and width expressed as functions.

Teacher
Teacher

Precisely! In areas, we can use the length and width expressions, just like binomials and trinomials.

Student 1
Student 1

Can you show us a real example?

Teacher
Teacher

Sure! Let’s say the dimensions of a rectangle are given by (l + 2) and (w^2 + 3w + 4). The area will be the product of these two expressions, which we can find using the same multiplication technique.

Student 2
Student 2

So we multiply l by everything in the trinomial and then 2 too?

Teacher
Teacher

Exactly! Just remember to collect like terms at the end.

Complex Examples and Clarifications

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Teacher
Teacher

Let’s tackle a more complex example: (x - 3)(2x + 4 + x^2). What do we start with?

Student 3
Student 3

We can distribute x first over the trinomial.

Teacher
Teacher

Correct! After distributing, what do we expect to see?

Student 4
Student 4

Terms like 2x^2 + 4x - 6x - 12.

Teacher
Teacher

Great observations! Now let’s combine those terms.

Student 1
Student 1

So that gives us 2x^2 - 2x - 12?

Teacher
Teacher

Exactly! This shows the importance of careful organization when multiplying.

Student 2
Student 2

I see how that simplifies the process!

Teacher
Teacher

Keep practicing, and you'll get even better!

Introduction & Overview

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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.

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Audio Book

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Understanding Multiplication of a Binomial and a Trinomial

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In this multiplication, we shall have to multiply each of the three terms in the trinomial by each of the two terms in the binomial. We shall get in all 3 × 2 = 6 terms, which may reduce to 5 or less, if the term by term multiplication results in like terms.

Detailed Explanation

When multiplying a binomial and a trinomial, you have to multiply every term in the trinomial by every term in the binomial. For a binomial like (a + 7) and a trinomial like (a^2 + 3a + 5), you first take the first term of the binomial (which is 'a') and multiply it by each term in the trinomial (a^2, 3a, and 5). Then, you do the same with the second term of the binomial (which is '7'). The purpose of this process is to ensure that every combination of terms gets considered, leading to an accurate result. In this case, you end up multiplying 2 terms from the binomial with 3 terms from the trinomial, resulting in a total of 6 products: a × a^2, a × 3a, a × 5, 7 × a^2, 7 × 3a, and 7 × 5.

Examples & Analogies

Think of preparing a meal. If you're making two types of sandwiches (the binomial), say ham and cheese (first type) and peanut butter and jelly (second type), and you want to add three kinds of toppings (the trinomial)—lettuce, tomato, and pickles—to every sandwich. For every type of sandwich, you can add each topping. This means that if you have 2 types of sandwiches and 3 toppings, you'll have a total of 6 different combinations of sandwiches with toppings.

Applying the Distributive Law

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Consider (a+7) × (a^2 +3a +5) =a × (a^2 + 3a + 5) + 7 × (a^2 + 3a + 5)

Detailed Explanation

When you use the distributive law, you distribute each term in the binomial (a and 7) to each term in the trinomial (a^2, 3a, and 5). For instance, when you multiply 'a' with 'a^2', you get 'a^3'. When you multiply 'a' with '3a', you get '3a^2'. Continuing the multiplication, you include '7 × a^2', '7 × 3a', and '7 × 5'. This creates a long expression of terms that may need simplification at the end.

Examples & Analogies

Imagine you're organizing a sports tournament. You have two types of games to organize (say soccer and basketball), and for each game, there are three kinds of awards to distribute (gold, silver, and bronze). Using the distributive law, you reason that each game (type) will have each award category attached to it. So, you will calculate the total number of awards given out across all game types.

Combining Like Terms

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Observe, every term in one binomial multiplies every term in the other binomial. = 6a^2 + 9ab + 8ba + 12b^2 = 6a^2 + 17ab + 12b^2 (Since ba = ab)

Detailed Explanation

After you have carried out the multiplication and obtained several products, the next step is to combine like terms to simplify the expression. For example, the products '9ab' and '8ba' are like terms. Since 'ba' and 'ab' are the same, you can combine them into '17ab'. The aim is to express the results in their simplest form.

Examples & Analogies

Think of sorting items in your room. If you've mixed different types of toys and you want to organize them, you first gather all the cars, action figures, and dolls. After that, instead of keeping them mixed, you would combine all the similar toys in one box. This makes it easier to find and manage your toys later.

Final Expression Result

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Thus we get 3 terms as a final result after combination, instead of the 6 initially obtained.

Detailed Explanation

In the multiplication process, although mathematically, we might anticipate having a certain number of distinct products (like 6 terms from 2 terms and 3 terms), the reality is many of those products will share similar variables. After combining like terms, we may end up with fewer distinct terms than we started with.

Examples & Analogies

Continuing the earlier example of organizing toys, this would be similar to how the total number of boxes might be reduced after combining the toys. Instead of six separate boxes for each different toy, you might find you can simply combine them into three or four boxes by grouping similar toys together.

Definitions & Key Concepts

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

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.

Examples & Real-Life Applications

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

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.

Memory Aids

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

🎵 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.

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for 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.