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12.3.1 - Division of a monomial by another monomial

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Introduction to Monomial Division

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Teacher
Teacher

Today, we're going to talk about dividing monomials. Can anyone tell me what a monomial is?

Student 1
Student 1

Is it a mathematical expression with just one term?

Teacher
Teacher

Exactly! A monomial is a single term, like 3x or 5y². Now, when we divide one monomial by another, we can simplify it by factoring.

Student 2
Student 2

How do you factor them exactly?

Teacher
Teacher

Great question! We express them in their irreducible factors. For example, 6x³ can be expressed as 2 × 3 × x × x × x. This helps us see common factors clearly.

Student 3
Student 3

What about dividing things like –20x⁴ by 10x²?

Teacher
Teacher

Let's think of how we can break those down. Remember, we can write –20x⁴ as –2 × 2 × 5 × x × x × x × x, then we can find our common factors!

Teacher
Teacher

So, to summarize: When dividing monomials, express them in factor form and then simplify by canceling out common factors.

Performing Divisions - Examples

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Teacher
Teacher

Let’s look at a practical example. How would we divide 24xy²z³ by 6yz²?

Student 2
Student 2

We can start by breaking both down into prime factors!

Teacher
Teacher

Right! Dividing gives us 4x when we cancel out the common factors. What did we cancel?

Student 1
Student 1

The 6 from the denominator cancels with the 24 in the numerator, along with the y's and z's.

Teacher
Teacher

Perfect! Let's try another: if we take 7x²y²z² divided by 14xyz, what do we get?

Student 4
Student 4

We can simplify it to 1/2xy!

Teacher
Teacher

Exactly! Can someone remind us about the steps involved in cancelling?

Student 3
Student 3

Factor, identify common factors, and cancel them out!

Teacher
Teacher

Great job! Remember, always look for the irreducible factors first.

Understanding with Real-Life Applications

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Teacher
Teacher

Now that we've covered the theory and calculations, how does this relate to real-world scenarios?

Student 1
Student 1

Like dividing total costs between items?

Teacher
Teacher

Exactly! If you know the price of items in terms of monomials, you can divide to find out how much one item costs.

Student 2
Student 2

So, if we used an example of 60 apples divided among 12 baskets... it could be expressed as a monomial!

Teacher
Teacher

Yes! That’s a great example. It's akin to simplifying – 60 is 5 × 12, simplifying gives us how many apples per basket.

Student 4
Student 4

This makes sense! So we can apply math to everyday situations routinely.

Teacher
Teacher

Exactly! Let’s always try to connect our learning with the real world.

Introduction & Overview

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Quick Overview

This section explains how to divide a monomial by another monomial using factorization.

Standard

The section covers the process of dividing monomials by expressing them in factor form. It introduces examples to illustrate the cancellation of common factors and demonstrates division in a systematic manner.

Detailed

In this section, we begin by exploring the division of a monomial by another monomial, which is a fundamental operation in algebra. We learn that division is the inverse operation of multiplication, and this relationship extends to algebraic expressions. By expressing both the dividend and the divisor in their irreducible factor forms, we can easily perform the division by canceling out common factors. For example, in the case of 6x³ divided by 2x, the process involves re-arranging the expression to allow for this cancellation, ultimately leading to a simplified expression. The section provides examples and explanations that illustrate how to carry out these operations effectively, laying the groundwork for further exploration of polynomial division.

Example: Do the following divisions.
(i) \(-30x^5 \div 10x^3\)
(ii) \(12a^3b^2c \div 4abc^3\)

Solution:
(i) \(-30x^5 \div 10x^3 = -3 \times 3 \times 2 \div 1 \times 1 \times 2 \times x^{5-3} = -3x^2\)

(ii) \(12a^3b^2c \div 4abc^3 = 3 \times 1 \times b^{2-1} \times a^{3-1} \times c^{1-3} = 3ab \div c^2\)

Therefore,
\(-30x^5 \div 10x^3 = -3x^2\) and \(12a^3b^2c \div 4abc^3 = 3abc^{-2}\)

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Audio Book

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Introduction to Division of Monomials

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Consider 6x³ ÷ 2x
We may write 2x and 6x³ in irreducible factor forms,
2x = 2 × x
6x³ = 2 × 3 × x × x × x
Now we group factors of 6x³ to separate 2x,
6x³ = 2 × x × (3 × x × x) = (2x) × (3x²)

Detailed Explanation

When dividing monomials, we can break each expression down into its irreducible factors. For example, 6x³ can be expressed as the product of its factors: 2, 3, and x repeated three times. When we divide 6x³ by 2x, we identify common factors in the numerator (6x³) and the denominator (2x). In this case, dividing by 2 and x gives us the result of 3x².

Examples & Analogies

Think of it like dividing a bag of 6 apples (where each apple is represented as x) among 2 friends. If you take 2 apples out (2x), each friend will get the remaining apples equally divided. So, 6 apples shared among 2 friends results in 3 apples each, represented as 3 times the number of apples each friend takes.

Cancellation of Common Factors

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Therefore, 6x³ ÷ 2x = 3x².
A shorter way to depict cancellation of common factors is as we do in division of numbers: 77 ÷ 7 = 11.

Detailed Explanation

In algebra, we can simplify calculations by eliminating common factors directly. Just like in basic arithmetic (where dividing 77 by 7 simplifies to 11 by canceling out 7), we can do the same in algebra. When we simplify 6x³ by 2x, both have the factor 2, allowing us to cancel that out and similarly with x, leaving us with 3 and x².

Examples & Analogies

Imagine you have 6 candies to share. If each candy can be seen as both a whole (2 candies) and a component (the flavor), dividing just removes overlapping aspects until you're left with the most basic formula for each person, which in the end gives you a clean count of what remains.

Examples of Dividing Monomials

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Example 13: Do the following divisions.
(i) –20x⁴ ÷ 10x²
(ii) 7x²y²z² ÷ 14xyz

Detailed Explanation

Let’s look at the examples of dividing monomials. In the first example, to divide -20x⁴ by 10x², we express -20x⁴ as -2 × 2 × 5 × x × x × x × x and 10x² as 2 × 5 × x × x. Here we can cancel the common factors (2 and 5) and two x's, allowing us to arrive at the final answer of -2x². Similarly, for the second example, we apply the same principle by canceling common factors from both numerator and denominator to simplify the expression more efficiently.

Examples & Analogies

Imagine -20 represents an obligation (maybe you owe someone 20 dollars worth of hours), while dividing by 10 tells you how many hours you're accountable for per task. The result gives you the number of tasks left after accounting for the hours already completed — in this case, how many full segments of work you owe after sharing responsibilities.

Definitions & Key Concepts

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

Key Concepts

  • Monomial Division: The operation of dividing one monomial by another.

  • Factorization: Breaking down expressions into irreducible parts to facilitate division.

  • Common Factors: Factors that are shared by both the dividend and divisor, which can be canceled.

Examples & Real-Life Applications

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

Examples

  • Example: 6x³ ÷ 2x = (2 × 3 × x × x × x) ÷ (2 × x) = 3x².

  • Example: –20x⁴ ÷ 10x² = (–2 × 2 × 5 × x × x × x × x) ÷ (2 × 5 × x × x) = –2x².

Memory Aids

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

🎵 Rhymes Time

  • To divide a monomial, take a factor ride, cancel what's the same, and let simplification coincide.

📖 Fascinating Stories

  • Imagine two kids sharing apples; they each take from the same bunch, the one sharing takes letters off, while the other munches.

🧠 Other Memory Gems

  • D.C.C - Divide, Check Common factors, Cancel.

🎯 Super Acronyms

DMC - Divide, Multiply factors, Cancel common terms.

Flash Cards

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Glossary of Terms

Review the Definitions for terms.

  • Term: Monomial

    Definition:

    An algebraic expression that consists of one term, such as 3x or 5y².

  • Term: Factorization

    Definition:

    The process of breaking down an expression into a product of its factors.

  • Term: Dividend

    Definition:

    The number or expression that is being divided.

  • Term: Divisor

    Definition:

    The number or expression by which the dividend is divided.

  • Term: Irreducible Factors

    Definition:

    Factors that cannot be factored further into simpler components.