8.5.1 - Multiplying a binomial by a binomial

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Understanding Binomials

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Teacher
Teacher

Let's start by discussing what binomials are. A binomial is a polynomial with exactly two terms, such as (2a + 3b). Can anyone give me another example?

Student 1
Student 1

How about (x + 5)?

Teacher
Teacher

Great! Now, when we multiply two binomials, we use the distributive property. Does anyone remember what that means?

Student 2
Student 2

Is it like multiplying each term of one by every term of the other?

Teacher
Teacher

Exactly! This process is sometimes called the FOIL method, which stands for First, Outside, Inside, Last. It helps us remember how to multiply!

Student 3
Student 3

So if I have (a + 2) and (3 + b), I would multiply a by 3 and then by b, right?

Teacher
Teacher

Yes! Let's summarize. Binomials are two-term polynomials, and to multiply them, we apply the distributive property or FOIL.

Step-by-Step Multiplication

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Teacher
Teacher

Now, let's multiply two specific binomials: (2a + 3b) and (3a + 4b). Can someone remind me the first step?

Student 4
Student 4

We first distribute 3a to both terms in (2a + 3b), right?

Teacher
Teacher

Exactly! So we have 3a × 2a and 3a × 3b. Can anyone calculate those?

Student 1
Student 1

That gives us 6a² + 9ab.

Teacher
Teacher

Perfect! Now what do we do next?

Student 2
Student 2

We need to distribute 4b to (2a + 3b) as well.

Teacher
Teacher

Correct! 4b × 2a gives us 8ab and 4b × 3b gives us 12b². What do we do with all these results?

Student 3
Student 3

Combine like terms!

Teacher
Teacher

Right. So we end up with 6a² + 9ab + 8ab + 12b², which simplifies to 6a² + 17ab + 12b².

Student 4
Student 4

I see how combining like terms helps simplify it!

Applying the Concept

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Teacher
Teacher

Now that we understand how to multiply binomials, can anyone think of a real-world scenario where this might apply?

Student 2
Student 2

What about calculating area? Like, if we have a rectangle whose sides are binomials?

Teacher
Teacher

Precisely! The area of a rectangle with lengths defined by binomials requires multiplying those binomials together.

Student 1
Student 1

So if I had a rectangle with sides (x + 2) and (x + 3), I could find its area?

Teacher
Teacher

Exactly! You would calculate (x + 2)(x + 3) using the steps we discussed.

Student 3
Student 3

And then combine like terms at the end!

Teacher
Teacher

Great summary! So remember, binomial multiplication has real-world applications like area calculation.

Introduction & Overview

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Quick Overview

This section teaches the process of multiplying binomials using the distributive property.

Standard

The section explains how to multiply two binomials by using the distributive law. It covers the steps required, provides examples, and emphasizes the importance of combining like terms in the resulting polynomial.

Detailed

In this section, we delve into the multiplication of binomials, a foundational operation in algebra. To multiply two binomials, such as (2a + 3b) and (3a + 4b), we apply the distributive law which states that we multiply each term in the first binomial by every term in the second binomial. The mathematical expression is set up as:

(3a + 4b) × (2a + 3b) = 3a × (2a + 3b) + 4b × (2a + 3b)

This results in:

= (3a × 2a) + (3a × 3b) + (4b × 2a) + (4b × 3b) = 6a² + 9ab + 8ba + 12b²

After identifying like terms (where ba = ab) and combining them, we conclude with:

= 6a² + 17ab + 12b²

The importance of this section lies in understanding how to systematically multiply polynomials of varying degrees and combine like terms, which reinforces key algebraic skills.

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Audio Book

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Introduction to Multiplying Binomials

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Let us multiply one binomial (2a + 3b) by another binomial, say (3a + 4b). We do this step-by-step, as we did in earlier cases, following the distributive law of multiplication.

Detailed Explanation

This chunk introduces the concept of multiplying two binomials by stating an example. We use the distributive law, which states that to multiply a sum by a number, you multiply each addend by the number separately and then sum the results. It sets the stage for the detailed multiplication process that follows.

Examples & Analogies

Imagine you are distributing candies among friends. If you have 2 friends and you give 3 candies to each friend, that’s similar to multiplying the number of friends (2) by the number of candies each (3). You can think of each friend as a binomial in a different candy distribution situation.

Applying Distributive Law

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(3a + 4b) × (2a + 3b) = 3a × (2a + 3b) + 4b × (2a + 3b) = (3a × 2a) + (3a × 3b) + (4b × 2a) + (4b × 3b)

Detailed Explanation

This excerpt shows how to apply the distributive law step by step. We first take each term from the first binomial and multiply it by each term in the second binomial. This results in four products, as every term in one binomial must be multiplied by every term in the other binomial.

Examples & Analogies

Think of this as producing different types of fruit baskets. If you have a basket with 3 apples and 4 oranges (the first binomial), and you make two types of baskets, one with 2 apples and the other with 3 oranges (the second binomial), you are essentially combining all possible varieties of baskets to see how many you can create.

Combining Like Terms

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Observe, every term in one binomial multiplies every term in the other binomial. = 6a² + 9ab + 8ba + 12b² = 6a² + 17ab + 12b² (Since ba = ab)

Detailed Explanation

After calculating each multiplication, we gather all the results into one expression. Here, we notice that the terms 9ab and 8ba are like terms, so we combine them to simplify the expression as much as possible. This is an essential step in polynomial multiplication because it allows us to present our answer in its simplest form.

Examples & Analogies

If you think of the terms as components of a recipe, like 6 cups of flour, 9 cups of sugar, and 12 cups of milk, combining similar types helps in giving a clearer idea of what you have. Just like in cooking, you want to know how many total cups of ingredients you have rather than listing them individually.

Final Step and Example Multiplication

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When we carry out term by term multiplication, we expect 2 × 2 = 4 terms to be present. But two of these are like terms, which are combined, and hence we get 3 terms.

Detailed Explanation

This final step concludes the multiplication process by noting that although we expect to get four terms from multiplying two binomials, combining like terms often reduces those into fewer terms. This illustrates the efficiency of algebra in working with expressions by reducing complexity.

Examples & Analogies

Imagine you are packing boxes for a move. You have 4 boxes, and after many moves, you realize you have packed similar items together. Instead of listing out 4 boxes of different items, you combine them into 2 main boxes. This simplification makes it easier to understand how many items of each type you truly have when you unpack.

Definitions & Key Concepts

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

Key Concepts

  • Distributive Property: Allows multiplication of terms in polynomials.

  • FOIL Method: A technique for multiplying two binomials by considering Four parts (First, Outside, Inside, Last).

  • Combining Like Terms: Important for simplifying results after multiplication.

Examples & Real-Life Applications

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

Examples

  • Example 1: Multiply (2x + 3)(x - 5) = 2x^2 - 10x + 3x - 15 = 2x^2 - 7x - 15.

  • Example 2: Multiply (a + 4)(a - 1) = a^2 - 1a + 4a - 4 = a^2 + 3a - 4.

Memory Aids

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

🎵 Rhymes Time

  • To multiply binomials, don’t be shy, | Use FOIL to simplify, | First, Outside, Inside, Last is the way, | Combine your terms for a brighter day!

🎯 Super Acronyms

FOIL

  • First
  • Outside
  • Inside
  • Last.

📖 Fascinating Stories

  • Imagine two neighbors, Alex and Bella, each planting double rows of flowers labeled with terms. When they make their flowerbeds, each plant type looks to reach the garden sun, just like terms in multiplication combine for the perfect view.

🧠 Other Memory Gems

  • Use F for the first factor, O for outside, I for inside, and L for last!

Flash Cards

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Glossary of Terms

Review the Definitions for terms.

  • Term: Binomial

    Definition:

    A polynomial with exactly two terms.

  • Term: Distributive Law

    Definition:

    A property that allows us to multiply a single term by terms inside parentheses.

  • Term: Like Terms

    Definition:

    Terms in an expression that have the same variable raised to the same power.

  • Term: Polynomial

    Definition:

    An algebraic expression consisting of one or more terms.