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12.4 - Division of Algebraic Expressions Continued

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Introduction to Dividing Polynomials

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Teacher
Teacher

Today, we are focusing on how to divide polynomials. Can anyone tell me what a polynomial is?

Student 1
Student 1

A polynomial is an expression made up of variables and coefficients.

Teacher
Teacher

Excellent! Now, when we divide polynomials, we are looking for common factors. For example, with (7x² + 14x) divided by (x + 2), we can factor the numerator. Can anyone guess what factors we have there?

Student 2
Student 2

We can factor it to 7x(x + 2)!

Teacher
Teacher

That's right! Now we can divide out the (x + 2). What are we left with?

Student 3
Student 3

Just 7x!

Teacher
Teacher

Great job! Remember, when we factor, we simplify our expressions!

Using Factorization for Simplification

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Teacher
Teacher

Next, let's explore a more complex example: dividing 44(x⁴ - 5x³ - 24x²) by 11x(x - 8). How would we start?

Student 4
Student 4

We should factor the numerator first?

Teacher
Teacher

Exactly! We factor out common terms. Can anyone identify the common factor?

Student 1
Student 1

2 times 11 times x²!

Teacher
Teacher

Correct! After factoring it becomes 2 × 2 × 11 × x² (x + 3)(x - 8). What happens when we perform the division now?

Student 2
Student 2

We can cancel the 11 and x.

Teacher
Teacher

Good! That leaves us with 2 × 2 × x(x + 3). Remember to always look for factors first!

Practical Application of Dividing Polynomials

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Teacher
Teacher

Can anyone think of when we might need to divide polynomials outside of math class?

Student 3
Student 3

Maybe in engineering when dealing with polynomial equations in physics?

Teacher
Teacher

Exactly! Now let’s take a look at another example: z(5z² - 80) divided by 5z(z + 4). What is our first step?

Student 4
Student 4

We can factor out the z and apply the difference of squares!

Teacher
Teacher

Yes! Well done! This gives us (z - 4) as a result. Keep practicing these methods at home!

Introduction & Overview

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

Quick Overview

This section discusses the division of polynomials by polynomials and focuses on factorization techniques for algebraic expressions.

Standard

The section elaborates on how to divide algebraic expressions, specifically focusing on checking and matching factors when dividing polynomials by polynomials. It includes several examples to demonstrate the process, emphasizing the importance of factorization in simplification.

Detailed

Division of Algebraic Expressions Continued

This section delves deeper into the concept of dividing algebraic expressions, specifically targeting the division of polynomials by other polynomials. The process begins by factorizing the numerator and the denominator to identify common factors that can be canceled out. The section provides several illustrative examples to clarify how this method is applied effectively.

For instance, to divide the expression (7x² + 14x) by (x + 2), the teacher guides students to first factorize the numerator into 7x(x + 2). This allows for the subsequent simplification by canceling the common factor (x + 2), leading to the result of 7x. Further, examples demonstrate how to handle more complex polynomial divisions, ensuring that students understand the foundational logic behind factorization and division. The exercises challenge students to practice these concepts, reinforcing their learning through application.

Example 16:

Divide \( 36(x^4 - 2x^3 - 8x^2) \) by \( 6(x - 4) \).

Solution:

Factoring \( 36(x^4 - 2x^3 - 8x^2) \), we get:
\[ 36(x^4 - 2x^3 - 8x^2) = 36x^2(x^2 - 2x - 8) \]
(Expanding and factoring out the common factor)

We can further factor \( x^2 - 2x - 8 \):
\[ x^2 - 2x - 8 = (x - 4)(x + 2) \]
So,
\[ 36(x^4 - 2x^3 - 8x^2) = 36x^2(x - 4)(x + 2) \]
Then, we have:
\[
\frac{36(x^4 - 2x^3 - 8x^2)}{6(x - 4)} = \frac{36x^2(x - 4)(x + 2)}{6(x - 4)}
\]
We cancel the factors \( 6 \) and \( x - 4 \):
\[
= 6x^2(x + 2)
\]
Therefore, \( 36(x^4 - 2x^3 - 8x^2) = 6(x - 4) \) results in \( 6x^2(x + 2) \).

Similar Question:

Divide \( 54(x^4 - 3x^3 - 12x^2) \) by \( 9(x - 6) \).

Solution:

Factoring \( 54(x^4 - 3x^3 - 12x^2) \), we get:
\[ 54(x^4 - 3x^3 - 12x^2) = 54x^2(x^2 - 3x - 12) \]
(Expanding and factoring out the common factor)

We can further factor \( x^2 - 3x - 12 \):
\[ x^2 - 3x - 12 = (x - 6)(x + 2) \]
Hence, \( 54(x^4 - 3x^3 - 12x^2) \) simplifies to \( 54x^2(x - 6)(x + 2) \).
Then, we have:
\[
\frac{54(x^4 - 3x^3 - 12x^2)}{9(x - 6)} = \frac{54x^2(x - 6)(x + 2)}{9(x - 6)}
\]
We cancel the factors \( 9 \) and \( x - 6 \):
\[
= 6x^2(x + 2)
\]
Therefore, \( 54(x^4 - 3x^3 - 12x^2) = 9(x - 6) \) results in \( 6x^2(x + 2) \).

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Audio Book

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Dividing a Polynomial by a Binomial

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Consider (7x2 + 14x) ÷ (x + 2)

We shall factorise (7x2 + 14x) first to check and match factors with the denominator:

7x2 + 14x = (7 × x × x) + (2 × 7 × x)

= 7 × x × (x + 2)

Now (7x2 + 14x) ÷ (x + 2) = 7x.

Detailed Explanation

In this example, we start with the polynomial 7x² + 14x. Before performing the division, we factor the polynomial. This means we rewrite it in a form that reveals its components. We notice that both terms have a common factor of 7x. Thus, we can express it as 7x(x + 2). Now, we perform the division by the binomial (x + 2). Since (x + 2) is a factor of the polynomial, we can cancel it, simplifying our expression down to just 7x. This shows us that dividing a polynomial by one of its factors gives a simpler polynomial as a result.

Examples & Analogies

Imagine you have a box of chocolates that comes in two sizes: small and large. In this code, you can represent the total number of chocolates in terms of their size. If you know that a certain number of chocolates can be combined into larger boxes, you can easily find out how many boxes you are left with after distributing them into the larger sections. Just like here, when you divide, you're essentially reorganizing items into simpler groups.

Example of Polynomial Division

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Example 15: Divide 44(x4 – 5x3 – 24x2) by 11x (x – 8)

Solution: Factorising 44(x4 – 5x3 – 24x2), we get

44(x4 – 5x3 – 24x2) = 2 × 2 × 11 × x2(x2 – 5x – 24)
= 2 × 2 × 11 × x2(x – 8)(x + 3) (after factoring further)

Now dividing:

= 4x²(x + 3).

Detailed Explanation

Here, we approach a slightly more complex polynomial division. First, we start with the polynomial 44(x⁴ – 5x³ – 24x²). Our goal is to factor the expression to identify parts that we can cancel out during division. We factor by taking out the common factors (2 × 2 × 11 × x²) and then factoring the quadratic part using methods appropriate for quadratics. After factoring, we have 4x²(x + 3) remaining after canceling the common terms with the divisor 11x(x – 8). This example emphasizes how breaking down polynomials before division can simplify the process and yield clearer results.

Examples & Analogies

Think of dividing a fruit basket shared among friends. You might have various types of fruits mixed together, and you need to separate them into types (apples, oranges, bananas) before dividing. By organizing (factoring) your basket first, it becomes easier to see how many pieces of each fruit each person can receive. Similarly, in polynomial division, recognizing and factoring out common terms simplifies the division process.

Dividing Using Identities

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Example 16: Divide z(5z2 – 80) by 5z(z + 4)

Solution: Dividend = z(5z2 – 80)

=z[(5 × z2) – (5 × 16)]

=5z(z + 4)(z – 4) using the identity (a² – b²) = (a + b)(a – b).

Thus, z(5z² – 80) ÷ 5z(z + 4) = (z – 4).

Detailed Explanation

In this example, we encounter a polynomial expression that is suitable for factoring based on known identities, like the difference of squares. We identify a pattern in z(5z² - 80) which allows us to express it as z times a factored form (common algebraic identity). Then we can proceed to cancel out the common factors with the divisor 5z(z + 4). This ultimately reduces our result to just (z - 4). This technique demonstrates how recognizing identities can significantly streamline polynomial division.

Examples & Analogies

Imagine you have a recipe for a large cake (our polynomial) that includes layers of different flavors. If you know the total size that's meant for, say, 5 guests, and you want to see how many servings you could get out of it based on the individual flavors, factorizing helps you split (or divide) that cake effectively. By using what you know about each flavor (identity), you can quickly find you're left with smaller manageable portions.

Definitions & Key Concepts

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

Key Concepts

  • Division of Polynomials: The process of dividing a polynomial by another polynomial using factorization.

  • Common Factors: Identifying and canceling common factors to simplify the algebraic expression.

  • Polynomial Factorization: Breaking down polynomials into simpler components.

Examples & Real-Life Applications

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

Examples

  • Example of dividing (7x² + 14x) by (x + 2), where we factor to find 7x.

  • Dividing 44(x⁴ - 5x³ - 24x²) by 11x(x - 8) using factorization to simplify.

Memory Aids

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

🎵 Rhymes Time

  • To divide polynomials with ease, find the factors and take what you please!

📖 Fascinating Stories

  • Imagine you're splitting a pizza. The numerator is the whole pizza, and the denominator tells you how many slices you can make. Factor out those slices for an even distribution!

🧠 Other Memory Gems

  • F.A.C.T. - Factor And Cancel Together when dividing.

🎯 Super Acronyms

D.I.V. - Divide, Identify Factors, Validate the result.

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Polynomial

    Definition:

    An algebraic expression consisting of terms with non-negative integer exponents.

  • Term: Factorization

    Definition:

    The process of breaking down an expression into its multiplicative components.

  • Term: Common Factor

    Definition:

    A factor that is shared among two or more terms.