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3.3.3 - Division

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Introduction to Division of Monomials

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Teacher
Teacher

Today, we are going to look at how to divide monomials. When we divide one monomial by another, we can simplify it. Does anyone know how we do that?

Student 1
Student 1

Do we just divide the numbers and then the variables?

Teacher
Teacher

Exactly! We divide the coefficients and then subtract the exponents of the same base. Let's look at the example: How do we divide 6x^3 by 3x?

Student 2
Student 2

So we would do 6 divided by 3, which is 2?

Teacher
Teacher

That's correct! Next, we subtract the exponents, 3 - 1 gives us 2. Therefore, what is the final answer?

Student 3
Student 3

It’s 2x^2!

Teacher
Teacher

Well done! So remember, when dividing monomials, it’s all about dividing the numbers and subtracting the exponents on the variables.

Division of Polynomials by Monomials

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Teacher
Teacher

Now, let's move on to dividing a polynomial by a monomial using the distributive property. Can anyone explain what that means?

Student 4
Student 4

Do we divide each term of the polynomial by the monomial?

Teacher
Teacher

Exactly, we divide each term of the polynomial by the monomial separately. Let’s say we have the polynomial 8x^3 + 4x^2 + 2x and we want to divide this by 2x. What do we do first?

Student 1
Student 1

We divide 8x^3 by 2x first!

Teacher
Teacher

Correct! What's the result of that division?

Student 2
Student 2

It would be 4x^2.

Teacher
Teacher

Good! Now let's divide the next term, 4x^2 by 2x. What do we get?

Student 3
Student 3

That would be 2x.

Teacher
Teacher

Very nice work! Finally, what about the last term, 2x?

Student 4
Student 4

That would be 1!

Teacher
Teacher

Excellent! So when we combine all those together, what's our simplified expression?

Student 1
Student 1

It’s 4x^2 + 2x + 1!

Teacher
Teacher

Great job! This is how we simplify polynomials through division, taking one term at a time.

Introduction & Overview

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Quick Overview

This section discusses the division of algebraic expressions, specifically the division of monomials and polynomials by monosomials using the distributive law.

Standard

The division of algebraic expressions, particularly monomials and polynomials, is essential in simplifying equations. This section outlines the process of dividing these expressions, emphasizing the significance of the distributive law in this context.

Detailed

Division of Algebraic Expressions

In algebra, division plays a critical role in simplifying expressions and solving equations. In this section, we discuss two main types of division: the division of monomials and the division of polynomials by a monomial.

Division of Monomials

Monomials are single-term expressions. When dividing a monomial by another, we can simply divide the coefficients and subtract the exponents of like bases.

Example:

If we divide 6x^3 by 3x, we perform the following operation:

  • Divide the coefficients:

6 ÷ 3 = 2

  • Subtract the exponents of 'x':

3 - 1 = 2

Thus,

$$ \frac{6x^3}{3x} = 2x^2 $$

Division of Polynomials by a Monomial

The division of polynomials by a monomial is performed using the distributive property. Each term of the polynomial is divided by the monomial separately.

Example:
If we have the polynomial 8x^3 + 4x^2 + 2x and divide it by 2x, we perform:

$$ \frac{8x^3}{2x} + \frac{4x^2}{2x} + \frac{2x}{2x} = 4x^2 + 2x + 1 $$

This clear division method is crucial for simplifying expressions and finding solutions to equations in algebra.

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Division of Monomials

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● Division of a monomial by a monomial.

Detailed Explanation

In algebra, a monomial is an expression that contains only one term. When we divide a monomial by another monomial, we divide the coefficients and subtract the exponents of like bases. For example, if we have 6x^3 ÷ 3x^2, we divide the coefficients: 6 ÷ 3 = 2. For the variables with the same base, we subtract the exponents: x^3 ÷ x^2 = x^(3-2) = x^1. Therefore, 6x^3 ÷ 3x^2 = 2x.

Examples & Analogies

Think of monomials like ingredients in a recipe. If you have 6 cups of flour and you want to distribute it evenly among 3 bakers, you divide 6 by 3. Each baker gets 2 cups of flour. Similarly, when you divide monomials, you're distributing the quantities.

Division of Polynomials by Monomials

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● Division of a polynomial by a monomial using distributive law.

Detailed Explanation

When dividing a polynomial by a monomial, we apply the distributive property. This means we will divide each term of the polynomial by the monomial separately. For example, if we have the polynomial 8x^3 + 4x^2 - 12x and we want to divide it by 4x, we perform the division term by term. So, (8x^3 ÷ 4x) + (4x^2 ÷ 4x) - (12x ÷ 4x) = 2x^2 + x - 3. This gives us a new polynomial of lower degree.

Examples & Analogies

Imagine you have a garden with different sections, each containing various types of flowers. If you decide to split the flowers evenly among your friends, you would take each type of flower from each section and divide it by the number of friends. Each friend gets their share of flowers from each section, just like we divide each term of a polynomial.

Definitions & Key Concepts

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Key Concepts

  • Division of Monomials: This is performed by dividing coefficients and subtracting exponents.

  • Distributive Property: This property allows us to divide each term of a polynomial by a monomial separately.

  • Simplifying Expressions: These operations help in breaking down complex algebraic expressions into simpler forms.

Examples & Real-Life Applications

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

Examples

  • Dividing 12x^5 by 4x^2 yields 3x^3 (12 ÷ 4 = 3 and 5 - 2 = 3).

  • Dividing the polynomial 6x^2 + 9x + 3 by 3, results in 2x^2 + 3x + 1.

Memory Aids

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🎵 Rhymes Time

  • To divide monomials, don’t feel lost, divide that number and subtract, it's the perfect cost!

📖 Fascinating Stories

  • Once in Algebra Land, there lived a mathematician named Polly. Polly loved to divide her cookies equally. So, she would take a big plate of cookies, count them, and share them based on her friends. This is like dividing polynomials, sharing with care term by term!

🧠 Other Memory Gems

  • Divide by number, for exponents remember: subtract!

🎯 Super Acronyms

DAS - Divide And Subtract for dividing monomials!

Flash Cards

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Glossary of Terms

Review the Definitions for terms.

  • Term: Division of Monomials

    Definition:

    The process of dividing one monomial by another, involving the division of coefficients and subtracting exponents of like bases.

  • Term: Polynomial

    Definition:

    An algebraic expression consisting of one or more terms, which can include constants, variables, and exponents.

  • Term: Distributive Property

    Definition:

    A mathematical property used to multiply a monomial by each term in a polynomial.