8.3.1 - Multiplying two monomials

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Basics of Monomials

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Teacher
Teacher

Today we will start with monomials. A monomial is a single term that contains a coefficient and one or more variables. Can anyone give me an example of a monomial?

Student 1
Student 1

How about `3x`?

Teacher
Teacher

Exactly! `3x` is a monomial. What do you think `0xy` is? Is it a monomial?

Student 2
Student 2

No, because it has a coefficient of zero, so it equals zero!

Teacher
Teacher

Right! Now let's move to multiplying monomials.

Multiplying Coefficients

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Teacher
Teacher

When multiplying two monomials like `3x` and `4y`, what do we do with the coefficients?

Student 3
Student 3

We multiply them together!

Teacher
Teacher

Correct! So how would we decide what the new coefficient would be for `3x * 4y`?

Student 4
Student 4

We would multiply `3` and `4`, giving us `12`.

Teacher
Teacher

Great job! Now if we multiply `3x` by `-2y`, what's our result?

Student 1
Student 1

It would be `-6xy`!

Combining Variables

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Teacher
Teacher

Now let’s look at the variables. When multiplying, what’s the rule for combining the variables?

Student 2
Student 2

We add the exponents of like variables.

Teacher
Teacher

Correct! So if we multiply `x^2` and `x^3`, what do we get?

Student 3
Student 3

`x^{2+3}` which is `x^5`.

Teacher
Teacher

Excellent! Remember the rule: a^m * a^n = a^{m+n}.

Examples in Context

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Teacher
Teacher

Let’s do some examples. What is `5x * 3x^2`?

Student 1
Student 1

It’s `15x^{1+2}` which equals `15x^3`.

Teacher
Teacher

Fantastic! Now what about `5x * -4xy`?

Student 4
Student 4

That’s `-20x^{1+1}y`, which simplifies to `-20x^2y`.

Teacher
Teacher

Exactly! Let’s remember these steps as we tackle more complex multiplication.

Introduction & Overview

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

Quick Overview

This section introduces the multiplication of two monomials, showing how to multiply their coefficients and variables.

Standard

The section covers the process of multiplying two monomials, explaining the importance of multiplication rules for coefficients and variables. It demonstrates various examples to illustrate how to deal with negative signs and the properties of exponents.

Detailed

Multiplying Two Monomials

In algebra, a monomial is a single term consisting of a coefficient and one or more variables raised to non-negative integer powers. When multiplying two monomials, we use the distributive property and the laws of exponents.

The general rule for multiplication of monomials involves multiplying the coefficients (numerical parts) together and then multiplying the variable parts, adding the exponents of any like variables.

  • Basic Examples:
  • For instance, multiplying 3x by 4x^2 results in:

3x * 4x^2 = (3 * 4) (x^1 * x^2) = 12x^{1+2} = 12x^3

  • Involving a Negative Coefficient:
  • Similarly, if one monomial has a negative coefficient, like -2y, multiplying by 5y gives:

-2y * 5y = (-2 * 5)(y^1 * y^1) = -10y^{1+1} = -10y^2

The section also covers how to handle additional variables and constants within monomials. The multiplication operation applies equally regardless of the complexity, whether it's between positive, negative, or zero coefficients and any number of variables.

Overall, understanding how to multiply monomials is fundamental to more complex algebraic expression manipulations such as polynomials.

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Audio Book

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Introduction to Multiplying Monomials

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We begin with 4 × x = x + x + x + x = 4x as seen earlier. Notice that all the three products of monomials, 3xy, 15xy, –15xy, are also monomials.

Detailed Explanation

When we multiply a number by a variable, such as multiplying 4 by x, we're effectively adding x together 4 times. This gives us 4x. The example also highlights that products like 3xy and 15xy are classified as monomials because they consist of a number multiplied by variables.

Examples & Analogies

Think of multiplying 4 by x as having 4 bags, each containing x apples. When we say '4 times x', we're counting all the apples, which gives us 4x apples in total.

Basic Examples of Multiplying Monomials

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(i) x × 3y = x × 3 × y = 3 × x × y = 3xy
(ii) 5x × 3y = 5 × x × 3 × y = 5 × 3 × x × y = 15xy
(iii) 5x × (–3y) = 5 × x × (–3) × y = 5 × (–3) × x × y = –15xy

Detailed Explanation

When multiplying two monomials, we multiply their coefficients (the numbers) and then multiply their variables. For instance, for 5x and 3y, we multiply 5 and 3 to get 15, and then combine the variables x and y to give us 15xy. If one of the coefficients is negative, such as in the case of 5x and –3y, the final result becomes negative, resulting in –15xy.

Examples & Analogies

Imagine 5 boxes, each containing strawberries (x), and you want to combine them with each of the 3 baskets of cherries (y). If one box also contains a negative 3 cherries, then when you combine them, you end up with a negative count of these cherries in your final mix!

Multiplying Monomials with Variable Powers

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(iv) 5x × 4x² = (5 × 4) × (x × x²) = 20 × x³ = 20x³
(v) 5x × (–4xyz) = (5 × –4) × (x × xyz) = –20 × (x × x × yz) = –20x²yz

Detailed Explanation

In these examples, we also deal with powers of variables. When multiplying like variables, we add their exponents. For example, in 5x and 4x², since the x has an exponent of 1, we add it to the exponent of 2 from x² to get x³, giving us a final product of 20x³. Similarly, in the second example, we multiply the coefficients and add the powers of x as well.

Examples & Analogies

Consider you have 5 containers each filled with 1 liter of liquid (x) and another container with 4 liters (x²). When you combine them, you can think of the liters of liquid not just being in separate containers, but scaling up – each container fully filled compounds the total volume!

Collecting Variable Powers

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Observe how we collect the powers of different variables in the algebraic parts of the two monomials. While doing so, we use the rules of exponents and powers.

Detailed Explanation

When multiplying two monomials, it’s crucial to keep track of how we manipulate the exponents of the variables. The rules of exponents tell us that when we multiply powers with the same base, we add the exponents together. So, if we multiply x¹ with x², we end up with x³.

Examples & Analogies

If you view your variable x as a garden where each plant type has an age represented by the exponent, multiplying the plants together means you’re pairing those ages together, allowing for more growth, which is seen in the higher exponent when you plant different mature plants together.

Definitions & Key Concepts

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

Key Concepts

  • Monomial: An algebraic expression with one term, such as 5x.

  • Coefficient: The numerical factor in a term of a polynomial, e.g., in 4xy the coefficient is 4.

  • Multiplication of Monomials: When multiplying, multiply coefficients and add the exponents of like variables.

  • Distributive Property: a(b + c) = ab + ac, useful when multiplying two monomials.

Examples & Real-Life Applications

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

Examples

  • Example: Multiply 3x * 4x^2 = 12x^3.

  • Example: Multiply -2y * 5y = -10y^2.

  • Example: Multiply 5a * -6b = -30ab.

Memory Aids

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

🎵 Rhymes Time

  • Multiply the numbers, combine the facts, add the powers, and see the impacts.

📖 Fascinating Stories

  • Once upon a time, there lived two numbers, 2 and 3, who wanted to join hands. When they met, they found they could create 6 together. They also summoned their friends x and y to multiply and dance in harmony producing new terms.

🧠 Other Memory Gems

  • Remember: M.C.A! Multiply Coefficients, Add exponents.

🎯 Super Acronyms

M.M.E

  • Monomials Multiply Easily!

Flash Cards

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Glossary of Terms

Review the Definitions for terms.

  • Term: Monomial

    Definition:

    An algebraic expression consisting of a single term. Examples include 3x, -4y^2, 5xyz.

  • Term: Coefficient

    Definition:

    The numerical factor in a term of a polynomial or monomial. For example, in 4x, the coefficient is 4.

  • Term: Exponent

    Definition:

    A number indicating how many times to multiply the base. For example, in x^2, the exponent is 2, meaning x is multiplied by itself once.

  • Term: Variable

    Definition:

    A symbol used to represent a quantity that can change, commonly represented by letters like x, y, etc.