8.5 - Multiplying a Polynomial by a Polynomial

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Introduction to Polynomial Multiplication

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Teacher
Teacher

Welcome everyone! Today, we're diving into multiplying polynomials. Can anyone tell me what a polynomial is?

Student 1
Student 1

Isn't a polynomial an expression made up of variables and coefficients?

Teacher
Teacher

Exactly, Student_1! Now, when we multiply polynomials, we often start with binomials. Let's look at how we can multiply them using the distributive law. Can anyone remind me what that law states?

Student 2
Student 2

It means we can distribute each term over the others!

Teacher
Teacher

Well done! For example, in multiplying the binomial (2a + 3b) with (3a + 4b), we would do this: multiply each term in the first binomial by each term in the second one.

Student 3
Student 3

So we would do 2a times 3a, then 2a times 4b, and so on?

Teacher
Teacher

Exactly right! Each multiplication gives us a term, and we then combine like terms.

Student 4
Student 4

What happens if there are no like terms?

Teacher
Teacher

Good question! If there are no like terms, we just list all the terms in our final answer. Now let’s do a quick example together.

Multiplying Binomials Example

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Teacher
Teacher

Let's multiply (x - 4) and (2x + 3). Who wants to lead the way?

Student 1
Student 1

I’ll try! First, I do x times 2x, which is 2x².

Teacher
Teacher

Great start! Now what’s next?

Student 2
Student 2

Then I do x times 3, which gives me 3x.

Teacher
Teacher

Perfect! Now let’s move on to -4. What do we get?

Student 3
Student 3

That gives -8x and -12, right?

Teacher
Teacher

Correct again! Now let's combine those like terms to finish.

Student 4
Student 4

So, 2x² - 5x - 12 is our final answer?

Teacher
Teacher

Exactly! Great teamwork! Remember to always combine like terms when finishing up.

Multiplying a Binomial with a Trinomial

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Teacher
Teacher

Now, let’s increase the complexity. How do you think we would multiply a binomial with a trinomial? For instance, (a + 7)(a² + 3a + 5).

Student 1
Student 1

Do we still use the distributive law?

Teacher
Teacher

That's right! We distribute each term in the binomial over each term in the trinomial.

Student 2
Student 2

So it’s like multiplying the binomial by each term in the trinomial?

Teacher
Teacher

Exactly! You will end up with three terms from the trinomial multiplied by two from the binomial. This means we’ll have six total terms to combine. Let’s work through it together!

Student 3
Student 3

I see! So, we would get a³, 3a², 5a, 7a², 21a, and 35.

Teacher
Teacher

Great job! Now can anyone tell me how we simplify those terms?

Student 4
Student 4

We combine like terms! So we’d get a³ + 10a² + 26a + 35.

Teacher
Teacher

Absolutely! Remembering to combine like terms is key in these multiplications.

Introduction & Overview

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

Quick Overview

This section covers the methods and processes involved in multiplying polynomials, specifically focusing on multiplying binomials and the distributive law.

Standard

In section 8.5, we learn about multiplying polynomials, particularly binomials. The process involves applying the distributive law, where each term in one polynomial multiplies every term in the other. Examples illustrate this process, emphasizing the importance of combining like terms to simplify the results.

Detailed

Detailed Summary

In this section, we delve into the multiplication of polynomials, particularly focusing on the multiplication of binomials by binomials and binomials by trinomials using the distributive law.

Key Points:

  1. Multiplication of Binomials: When multiplying two binomials, such as (2a + 3b) and (3a + 4b), every term in the first binomial is multiplied by every term in the second binomial. This process results in multiple terms, which may include like terms that can be combined for simplification. For example, the product (2a + 3b)(3a + 4b) yields several products: 6a² + 9ab + 8ab + 12b². Combining like terms here gives us 6a² + 17ab + 12b².
  2. Distributive Law: The distributive law is pivotal in polynomial multiplication. It states that a(b + c) = ab + ac. This law allows us to expand the multiplication process systematically.
  3. Practical Examples: Several examples demonstrate how to apply these principles, including how to simplify expressions and handle multiple terms.

Overall, mastering polynomial multiplication is essential for dealings in algebra, which often encounters expressions necessitating expansion, such as in area computation and in various algebraic identities.

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Audio Book

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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,

(3a + 4b) × (2a + 3b) = 3a × (2a + 3b) + 4b × (2a + 3b)
= (3a × 2a) + (3a × 3b) + (4b × 2a) + (4b × 3b)
= 6a² + 9ab + 8ba + 12b²
= 6a² + 17ab + 12b² (Since ba = ab).

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

To multiply two binomials, we apply the distributive property. This means that every term in the first binomial multiplies every term in the second binomial. In our example, we multiplied each term in (3a + 4b) by each term in (2a + 3b). This first gives us four products, and then we combine any like terms to simplify the expression. The key point here is to remember to distribute each term properly.

Examples & Analogies

Imagine you have two bags of fruits: one bag contains 2 apples and 3 oranges, while the other bag contains 3 apples and 4 oranges. If you combine the fruits from both bags, you would not just count the apples and oranges separately; instead, you would see how many apples you have in total and how many oranges, just like combining terms in our multiplication.

Examples of Binomial Multiplication

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Example 8: Multiply

(i) (x – 4) and (2x + 3)
Solution:
(i) (x – 4) × (2x + 3) = x × (2x + 3) – 4 × (2x + 3)
= (x × 2x) + (x × 3) – (4 × 2x) – (4 × 3) = 2x² + 3x – 8x – 12
= 2x² – 5x – 12 (Adding like terms)

(ii) (x – y) and (3x + 5y)
= x × (3x + 5y) – y × (3x + 5y)
= (x × 3x) + (x × 5y) – (y × 3x) – (y × 5y)
= 3x² + 5xy – 3yx – 5y² = 3x² + 2xy – 5y² (Adding like terms)

Detailed Explanation

The examples demonstrate how to multiply binomials by applying the distributive property step-by-step. For the first example, we take each term of (x – 4) and multiply it by each term of (2x + 3), which results in several products, some of which are like terms. After combining like terms, we arrive at the final polynomial. The same method applies to the second example, reinforcing the understanding of this multiplication process.

Examples & Analogies

Think of it like packing boxes. You have a box labeled 'x - 4' for small items and another box labeled '2x + 3' for larger items. When packing, you want to see how many combinations of small and large items you can fit into a bigger box. Each combination gives you a product, similar to how we multiply terms in algebra.

Multiplying Binomials by Trinomials

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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. Consider (a + 7) × (a² + 3a + 5) = a × (a² + 3a + 5) + 7 × (a² + 3a + 5)
= a³ + 3a² + 5a + 7a² + 21a + 35
= a³ + (3a² + 7a²) + (5a + 21a) + 35
= a³ + 10a² + 26a + 35.

Detailed Explanation

When multiplying a binomial and a trinomial, you multiply each term in the binomial by each term in the trinomial, resulting in multiple terms. In our example, we work through the multiplication systematically, combining like terms afterward. This emphasizes the importance of organizing the operation and being attentive to like terms.

Examples & Analogies

Visualize mixing paint. You have two colors represented by the binomial and three shades by the trinomial. When you mix these, you get a range of new shades (terms) from each combination. Some shades may look similar (like terms) and can be blended together, which represents simplifying the polynomial after multiplication.

Simplifying Products

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Example 10: Simplify (a + b) (2a – 3b + c) – (2a – 3b) c.
Solution:
We have (a + b) (2a – 3b + c) = a (2a – 3b + c) + b (2a – 3b + c)
= 2a² – 3ab + ac + 2ab – 3b² + bc
= 2a² – ab – 3b² + bc + ac (Note, –3ab and 2ab are like terms)
and (2a – 3b) c = 2ac – 3bc.
Therefore,
(a + b) (2a – 3b + c) – (2a – 3b) c = 2a² – ab – 3b² + bc + ac – (2ac – 3bc)
= 2a² – ab – 3b² + (bc + 3bc) + (ac – 2ac)
= 2a² – 3b² – ab + 4bc – ac.

Detailed Explanation

This example illustrates how to simplify expressions involving polynomial products by organizing steps and carefully combining like terms. First, we distribute both terms in the binomial to each term in the trinomial, followed by managing the additional polynomial and resulting expressions. The focus should be on accuracy and clarity in combining like terms before arriving at the final simplified result.

Examples & Analogies

Picture a recipe where you first combine flavors (the binomial) with multiple ingredients (the trinomial). After mixing, you may need to adjust the flavors by adding or subtracting certain ingredients to reach a delicious outcome! Just like cooking, careful combination and adjustments lead to a refined result.

Definitions & Key Concepts

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

Key Concepts

  • Polynomial: A mathematical expression comprising variables and coefficients.

  • Binomial: A polynomial with two terms.

  • Trinomial: A polynomial with three terms.

  • Distributive Law: A fundamental property used to simplify multiplication.

  • Combining Like Terms: The process of combining expressions with the same variables and exponents.

Examples & Real-Life Applications

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

Examples

  • (x - 4)(2x + 3) = 2x² - 5x - 12

  • (a + 7)(a² + 3a + 5) = a³ + 10a² + 26a + 35

Memory Aids

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

🎵 Rhymes Time

  • To multiply two binomials, you expand with glee, every term you see.

📖 Fascinating Stories

  • Imagine multiplying the prices of apples and oranges; combine them for a fruit basket total.

🧠 Other Memory Gems

  • Remember 'FOIL' when multiplying binomials: First, Outside, Inside, Last.

🎯 Super Acronyms

BIP - Binomial Interaction Principle

  • remember to combine like terms!

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Polynomial

    Definition:

    An algebraic expression composed of variables and coefficients, involving only non-negative integer exponents.

  • Term: Binomial

    Definition:

    A polynomial with exactly two terms.

  • Term: Trinomial

    Definition:

    A polynomial with exactly three terms.

  • Term: Distributive Law

    Definition:

    A property that allows for the multiplication of a sum by distributing the multiplier across each addend.

  • Term: Like Terms

    Definition:

    Terms that contain the same variables raised to the same power.