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Polynomial long division

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Intros
Lessons
  1. Review: Long division with numbers
    • long division with polynomials works in the same way as long division with numbers
    • exercise:
    658 ÷ 20
    i) Operate long division
    ii) Identify:
    • Dividend:
    • Division:
    • Quotient:
    • Remainder:
    iii) Write the division statement in 2 ways.
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Examples
Lessons
  1. Use long division to divide polynomials
    Apply long division, and then use the result to write the division statement in 2 ways.
    1. (8x3−14x2+7x−1)÷(2x−5)\left( {8{x^3} - 14{x^2} + 7x - 1} \right) \div \left( {2x - 5} \right)
    2. (−x2+6x−2)÷(x)\left( { - {x^2} + 6x - 2} \right) \div \left( x \right)
  2. Watch out for "Missing Terms"! → Use "Placeholder"
    Operate long division.
    1. (29x3−12x4−x)÷(−3x+5)\left( {29{x^3} - 12{x^4} - x} \right) \div \left( { - 3x + 5} \right)
    2. (x3+5x−11)÷(x−2)\left( {{x^3} + 5x - 11} \right) \div \left( {x - 2} \right)
    3. (4x3+6x2−9x+5)÷(2x2−1)\left( {4{x^3} + 6{x^2} - 9x + 5} \right) \div \left( {2{x^2} - 1} \right)
  3. Application of a Division Statement
    Given the long division as shown:
    Polynomial long division
    Determine the polynomial P(x)P\left( x \right).
    1. Determine "Factors"
      (6x2+7x−5)÷(3x+5)\left( {6{x^2} + 7x - 5} \right) \div \left( {3x + 5} \right)
      1. Operate long division.
      2. Write the division statement.
      3. Is (3x+5)\left( {3x + 5} \right) a factor of (6x2+7x−5)\left( {6{x^2} + 7x - 5} \right)? Explain.
    Topic Notes
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    It may sound hard, but the idea of polynomial long division is basically the same as any other long divisions. A division statement has 4 elements: dividend, division, quotient, and remainder.

    Introduction: Understanding Polynomial Long Division

    Polynomial long division is a fundamental technique in algebra that extends the familiar concept of long division to polynomials. Just as we divide numbers, we can divide polynomials using a similar step-by-step process. This method is crucial for simplifying complex algebraic expressions, solving equations, and understanding rational functions. Our introduction video will guide you through the long division method for polynomials, making this sometimes challenging topic more accessible. You'll learn how to set up the division, perform each step systematically, and interpret the results. Whether you're new to the concept or looking to refresh your skills, this video will help you master how to divide polynomials with confidence. By understanding polynomial long division, you'll unlock a powerful tool that's essential for advanced mathematics and many real-world applications. Get ready to enhance your algebraic toolkit and tackle more complex problems with ease!

    Review of Regular Long Division

    Understanding Long Division

    Long division is a fundamental mathematical operation that allows us to divide larger numbers by breaking them down into smaller, manageable steps. Let's explore this process using the example 658 ÷ 20, which will help us understand how to solve long division problems effectively.

    Key Terms in Long Division

    Before we dive into the process, let's define some important terms:

    • Dividend: The number being divided (in our example, 658)
    • Divisor: The number we're dividing by (in our example, 20)
    • Quotient: The result of the division
    • Remainder: Any leftover amount after division

    Step-by-Step Long Division Process

    Now, let's break down the long division rules and process for 658 ÷ 20:

    1. Set Up the Problem

    Write the dividend (658) under the division symbol (÷) and the divisor (20) to the left of it.

    2. Divide, Multiply, Subtract, Bring Down

    This is the core of how to use long division:

    • Divide: How many times does 20 go into 65? It goes 3 times.
    • Multiply: 3 x 20 = 60
    • Subtract: 65 - 60 = 5
    • Bring Down: Bring down the next digit (8) next to the 5

    3. Repeat the Process

    Now we have 58. Repeat the divide, multiply, subtract, bring down steps:

    • Divide: 20 goes into 58 two times
    • Multiply: 2 x 20 = 40
    • Subtract: 58 - 40 = 18
    • Bring Down: There are no more digits to bring down

    4. Finalize the Answer

    Since 18 is less than our divisor (20), we can't divide further. This means:

    • Quotient: 32 (3 from the first step, 2 from the second)
    • Remainder: 18

    Similarity to Polynomial Long Division

    The process we've just walked through for regular long division is remarkably similar to polynomial long division. Both methods follow the same basic steps of divide, multiply, subtract, and bring down. The main difference lies in working with variables in polynomial division instead of just numbers.

    Tips for Mastering Long Division

    To become proficient in how to solve long division problems, consider these tips:

    • Practice estimating to check if your answer makes sense
    • Double-check your multiplication at each step
    • Remember that the remainder should always be less than the divisor
    • Use scratch paper to work out each step if needed

    By following these long division rules and practicing regularly, you'll find that even complex division problems become manageable. Remember, the key is to break down the process into smaller steps, just as we did with 658 ÷ 20. With time and practice,

    Writing Division Statements

    Understanding how to solve division problems is a crucial mathematical skill. In this section, we'll explore two methods of writing division statements, using the example 658 ÷ 20. Both approaches are valuable, and mastering them will enhance your problem-solving abilities.

    Method 1: Long Division

    Long division is a traditional method that breaks down the division process into manageable steps:

    1. Write the dividend (658) inside the division bracket and the divisor (20) outside.
    2. Divide 20 into 65 (the first two digits of 658). It goes 3 times with a remainder of 5.
    3. Write 3 above the division bracket and multiply 20 by 3 (60). Subtract 60 from 65.
    4. Bring down the next digit (8) to create 58.
    5. Divide 20 into 58. It goes 2 times with a remainder of 18.
    6. Write 2 next to the 3 above the bracket.
    7. Multiply 20 by 2 (40) and subtract from 58.
    8. The final answer is 32 remainder 18, or 32.9.

    Method 2: Short Division

    Short division is a more compact method, ideal for mental calculations:

    1. Write 658 ÷ 20.
    2. Divide 20 into 65, which goes 3 times with a remainder of 5.
    3. Write 3 as the first digit of the answer.
    4. Carry the 5 over to the next digit (8), making it 58.
    5. Divide 20 into 58, which goes 2 times with a remainder of 18.
    6. Write 2 as the second digit of the answer.
    7. The final result is 32 remainder 18, or 32.9.

    Importance of Both Methods

    Understanding both long and short division methods is essential because:

    • They reinforce conceptual understanding of division.
    • Long division helps with complex problems and decimal answers.
    • Short division is faster for mental math and simpler calculations.
    • Knowing multiple methods increases problem-solving flexibility.

    Practice Examples

    Try these division problems using both methods:

    1. 924 ÷ 28
    2. 1,575 ÷ 45
    3. 3,216 ÷ 64

    By practicing both long and short division methods, you'll become more proficient in solving division problems quickly and accurately. Remember, the key to mastering division statements is consistent practice and understanding the underlying concepts behind each method.

    Introduction to Polynomial Long Division

    As students progress in their mathematical journey, they often encounter a natural transition from regular long division to polynomial long division. This shift builds upon familiar concepts while introducing new complexities. Understanding how to do polynomial long division is crucial for advancing in algebra and calculus.

    Regular long division and polynomial long division share several similarities. Both processes involve dividing a larger quantity by a smaller one, following a step-by-step approach to find a quotient and a remainder. The layout of the problem looks remarkably similar, with the dividend placed under a "division bracket" and the divisor to the left.

    However, there are key differences when learning how to divide a polynomial. Instead of working with simple numbers, polynomial long division deals with algebraic expressions containing variables. This introduces an additional layer of complexity, as one must consider the degrees of the polynomials involved.

    The basic concept of dividing polynomials revolves around breaking down a higher-degree polynomial (the dividend) by a lower-degree polynomial (the divisor). The goal is to express the dividend as the product of the divisor and a quotient, plus a remainder if necessary.

    Let's illustrate this process with a simple example. Suppose we want to divide x² + 3x + 2 by x + 1. To divide using polynomial long division, we follow these steps:

    1. Arrange the polynomials in descending order of degree.
    2. Divide the leading term of the dividend by the leading term of the divisor.
    3. Multiply the result by the divisor and subtract from the dividend.
    4. Bring down the next term and repeat the process until the degree of the remainder is less than the degree of the divisor.

    In this case, the process would look like this:

          x + 2
    ___________
x + 1 ) x² + 3x + 2
        x² + x
        ________
          2x + 2
          2x + 2
          ________
              0

    Understanding the degree of polynomials is crucial in this process. The degree of a polynomial is the highest power of the variable in the expression. In our example, the dividend (x² + 3x + 2) has a degree of 2, while the divisor (x + 1) has a degree of 1. The division process continues until the remainder has a degree less than that of the divisor.

    As you practice how to do polynomial long division, you'll notice that the process becomes more intuitive. It's essential to keep your work organized and pay close attention to signs and coefficients. Remember that the goal is to express the dividend in terms of the divisor, quotient, and remainder, following the form: dividend = (divisor × quotient) + remainder.

    Mastering polynomial long division opens doors to more advanced mathematical concepts. It's a fundamental skill for simplifying rational expressions, finding roots of polynomials, and solving differential equations. By building on the familiar concept of regular long division and applying it to algebraic expressions, students develop a deeper understanding of polynomial manipulation and algebraic reasoning.

    Step-by-Step Guide to Polynomial Long Division

    Polynomial long division is a fundamental skill in algebra that allows you to divide one polynomial by another. If you've ever wondered how to divide polynomials step by step, this comprehensive guide will walk you through the process. We'll use a specific example to illustrate how to do long division with polynomials, breaking down each step clearly and providing tips to avoid common mistakes.

    1. Set Up the Problem

    Let's start with an example: Divide (x³ + 2x² - 5x - 6) by (x + 2)

    To begin, set up the division problem similar to regular long division, with the dividend (x³ + 2x² - 5x - 6) under a division bar and the divisor (x + 2) outside.

    2. Divide the First Term

    Look at the highest degree terms of both polynomials. Divide the first term x³ by x to get x². Write this above the division bar.

    Tip: Always start with the term of the highest degree in the dividend and divisor.

    3. Multiply and Subtract

    Multiply the result (x²) by the divisor (x + 2) to get x³ + 2x². Write this under the dividend and subtract.

    Common mistake: Don't forget to distribute the multiplication to all terms of the divisor.

    4. Bring Down the Next Term

    Bring down the next term of the dividend (2x²) to the result of the subtraction (0 + 2x²).

    5. Repeat Steps 2-4

    Continue the process: - Divide the first term 2x² by x to get 2x. Write this next to x² above the division bar. - Multiply (x + 2) by 2x and subtract from 2x² - 5x. - Bring down the next term (-6).

    6. Final Division

    Divide the remaining polynomial by the divisor: - Divide -9x - 6 by x + 2 to get -9. - Multiply (x + 2) by -9 and subtract from -9x - 6.

    7. Identify the Quotient and Remainder

    The terms above the division bar form the quotient: x² + 2x - 9 The final result of subtraction is the remainder: 12

    8. Express the Final Answer

    Write the answer as: (x³ + 2x² - 5x - 6) ÷ (x + 2) = x² + 2x - 9 with a remainder of 12

    Tips for Mastering Polynomial Long Division

    1. Always arrange the polynomials in descending order of degree before starting. 2. Ensure that you're comfortable with basic polynomial operations before attempting long division. 3. Practice with simpler examples before moving to more complex ones. 4. Double-check your work by multiplying the quotient by the divisor and adding the remainder this should equal the original dividend.

    Common Mistakes to Avoid

    1. Forgetting to bring down terms: Always bring down every term of the dividend. 2. Incorrect signs: Be careful with negative signs during subtraction steps. 3. Incomplete division: Continue the process until the degree of the remainder is less than the degree of the divisor. 4. Misaligning terms: Keep like terms aligned vertically for easier calculation.

    By following this step-by-step guide on how to divide polynomials, you'll be able to tackle even complex polynomial divisions with confidence. Remember, the key to mastering

    Common Challenges and Solutions in Polynomial Long Division

    Polynomial long division is a fundamental skill in algebra, but it can be challenging for many students. Let's explore some common hurdles and practical solutions to help you master this important technique.

    1. Setting up the problem correctly: One of the first challenges students face is properly arranging the polynomial division. To overcome this, always start by writing the dividend (the polynomial being divided) under the division symbol, and the divisor (the polynomial you're dividing by) outside. Ensure that both polynomials are written in descending order of degree.

    Example: When dividing x³ + 2x² - 5x + 3 by x - 2, set it up as:

    x - 2 | x³ + 2x² - 5x + 3

    2. Identifying the first term to divide: Students often struggle with determining which terms to divide first. The key is to focus on the highest degree terms of both polynomials. Divide the leading term of the dividend by the leading term of the divisor.

    Tip: In our example, divide x³ by x, giving you x² as the first term of the quotient.

    3. Multiplying and subtracting in polynomial division: A common mistake is forgetting to distribute the multiplication or making errors in subtraction. To avoid this, take your time and double-check each step. Use a different color or underlining to keep track of the terms you're working with.

    Example: x² * (x - 2) = x³ - 2x²

    Subtract this from the dividend: (x³ + 2x² - 5x + 3) - (x³ - 2x²) = 4x² - 5x + 3

    4. Bringing down terms: Students sometimes forget to bring down terms or bring down the wrong ones. Remember to bring down the next term of the original dividend after each subtraction step. If there are no more terms to bring down, your division is complete.

    5. Dealing with missing terms in polynomials: When a polynomial has missing terms (e.g., x³ + x), students may get confused. The solution is to write the polynomial with zero coefficients for the missing terms (x³ + 0x² + x + 0) to keep the degrees clear.

    6. Handling remainders: Not all polynomial divisions result in zero remainders. If you have a remainder of lower degree than the divisor, express it as a fraction over the divisor in your final answer.

    Example: (x² + 3x + 2) ÷ (x + 1) = x + 2 with a remainder of 0. The final answer is x + 2.

    7. Simplifying complex polynomial problems: When faced with more complex polynomials, break the problem down into smaller steps. Focus on one term at a time and maintain a systematic approach.

    Let's walk through a tricky problem: Divide 2x - 3x³ + 5x - 7 by x² + 2

    Step 1: Set up the division
    x² + 2 | 2x - 3x³ + 0x² + 5x - 7

    Step 2: Divide 2x by x², giving 2x²
    2x² * (x² + 2) = 2x + 4x²
    Subtract: (2x - 3x³ + 0x² + 5x - 7) - (2x + 4x²) = -3x³ - 4x² + 5x - 7

    Step 3: Bring down the next term (already done in this case)
    Divide -3x³ by x², giving -3x
    -3x * (x² + 2) = -3x³ - 6x
    Subtract: (-3x³ - 4x² + 5x - 7) - (-3x³ - 6x) = -4x² + 11x - 7

    Step 4: Multiplying and subtracting in polynomial division again: Divide -4x² by x², giving -4
    -4 * (x² + 2) = -4x² - 8
    Subtract: (-4x² + 11x - 7) - (-4x² - 8) = 11x + 1

    Step 5: Dealing with missing terms in polynomials again: Since there are no more terms to bring down, the remainder is 11x + 1. The final answer is 2x² - 3x - 4 with a remainder of 11x + 1.

    Step 6: Simplifying complex polynomial problems again: Always double-check your work to ensure accuracy.

    Practice Problems and Solutions

    Ready to put your polynomial long division skills to the test? We've prepared a set of practice problems ranging from simple to more complex. Try to solve these on your own before checking the solutions. Remember, practice makes perfect!

    Problem 1: Simple Polynomial Division

    Divide (x² + 3x - 10) by (x + 5)

    Problem 2: Polynomial Division with Remainder

    Divide (2x³ - 7x² + 4x - 15) by (x - 3)

    Problem 3: More Complex Polynomial Division

    Divide (3x - 5x³ + 2x² - x + 7) by (x² + 2)

    Solutions:

    Solution 1:

    Step 1: Set up the division
    Step 2: Divide x² by x to get x
    Step 3: Multiply (x + 5) by x and subtract
    Step 4: Bring down the next term
    Step 5: Divide -2x by x to get -2
    Step 6: Multiply (x + 5) by -2 and subtract
    Result: Quotient = x - 2, Remainder = 0

    Solution 2:

    Step 1: Set up the division
    Step 2: Divide 2x³ by x to get 2x²
    Step 3: Multiply (x - 3) by 2x² and subtract
    Step 4: Bring down the next term
    Step 5: Divide -x² by x to get -x
    Step 6: Multiply (x - 3) by -x and subtract
    Step 7: Bring down the next term
    Step 8: Divide x by x to get 1
    Step 9: Multiply (x - 3) by 1 and subtract
    Result: Quotient = 2x² - x + 1, Remainder = -12

    Solution 3:

    Step 1: Set up the division
    Step 2: Divide 3x by x² to get 3x²
    Step 3: Multiply (x² + 2) by 3x² and subtract
    Step 4: Bring down the next terms
    Step 5: Divide -5x³ by x² to get -5x
    Step 6: Multiply (x² + 2) by -5x and subtract
    Step 7: Bring down the remaining terms
    Step 8: Divide 12x² by x² to get 12
    Step 9: Multiply (x² + 2) by 12 and subtract
    Result: Quotient = 3x² - 5x + 12, Remainder = -25x + 7

    Tips for Checking Your Work:

    • Multiply the quotient by the divisor and add the remainder. This should equal the original polynomial.
    • Ensure the degree of the remainder is less than the degree of the divisor.
    • Check for sign errors, a common mistake in polynomial division.
    • Verify that you've brought down all terms correctly during the division process.
    • Double-check your arithmetic, especially when dealing with negative numbers.

    Remember, polynomial long division practice is crucial for mastering this skill. If you're struggling with any

    Conclusion: Mastering Polynomial Long Division

    In this article, we've explored the essential steps of polynomial long division, a crucial skill in algebra. We've covered how to set up the problem, divide using polynomial long division, and interpret the results. Remember, mastering this technique requires consistent practice and patience. Don't hesitate to rewatch the introduction video for reinforcement of key concepts. The more you practice, the more comfortable you'll become with solving complex polynomial divisions. Challenge yourself with additional practice problems to solidify your understanding. As you progress, consider exploring related topics like synthetic division or polynomial factoring to broaden your mathematical toolkit. By dedicating time to honing your skills in polynomial long division, you'll build a strong foundation for more advanced mathematical concepts. Keep practicing, stay curious, and watch your proficiency grow in this fundamental algebraic technique.

    The more you practice, the more comfortable you'll become with solving complex polynomial divisions. Challenge yourself with additional practice problems to solidify your understanding. As you progress, consider exploring related topics like synthetic division or polynomial factoring to broaden your mathematical toolkit.

    Example:

    Use long division to divide polynomials
    Apply long division, and then use the result to write the division statement in 2 ways. (8x3−14x2+7x−1)÷(2x−5)\left( {8{x^3} - 14{x^2} + 7x - 1} \right) \div \left( {2x - 5} \right)

    Step 1: Set Up the Division

    To begin, set up the polynomial long division. Write the dividend 8x3−14x2+7x−18x^3 - 14x^2 + 7x - 1 under the long division symbol and the divisor 2x−52x - 5 outside the symbol.

    Step 2: Divide the Leading Terms

    Divide the leading term of the dividend 8x38x^3 by the leading term of the divisor 2x2x. This gives 4x24x^2. Write 4x24x^2 above the division symbol.

    Step 3: Multiply and Subtract

    Multiply 4x24x^2 by the entire divisor 2x−52x - 5, resulting in 8x3−20x28x^3 - 20x^2. Subtract this from the original dividend to get a new polynomial: −14x2−(−20x2)=6x2-14x^2 - (-20x^2) = 6x^2. Bring down the next term from the dividend, which is 7x7x, to get 6x2+7x6x^2 + 7x.

    Step 4: Repeat the Process

    Next, divide the leading term of the new polynomial 6x26x^2 by the leading term of the divisor 2x2x, resulting in 3x3x. Write 3x3x above the division symbol. Multiply 3x3x by the divisor 2x−52x - 5 to get 6x2−15x6x^2 - 15x. Subtract this from the current polynomial to get a new polynomial: 7x−(−15x)=22x7x - (-15x) = 22x. Bring down the next term from the dividend, which is −1-1, to get 22x−122x - 1.

    Step 5: Final Division

    Divide the leading term of the new polynomial 22x22x by the leading term of the divisor 2x2x, resulting in 1111. Write 1111 above the division symbol. Multiply 1111 by the divisor 2x−52x - 5 to get 22x−5522x - 55. Subtract this from the current polynomial to get the remainder: −1−(−55)=54-1 - (-55) = 54.

    Step 6: Write the Quotient and Remainder

    The quotient is 4x2+3x+114x^2 + 3x + 11 and the remainder is 5454. Therefore, the division statement can be written in two ways:

    Division Statement Method 1

    The first way to write the division statement is:
    Dividend = Divisor × Quotient + Remainder
    8x3−14x2+7x−1=(2x−5)(4x2+3x+11)+548x^3 - 14x^2 + 7x - 1 = (2x - 5)(4x^2 + 3x + 11) + 54

    Division Statement Method 2

    The second way to write the division statement is:
    Dividend / Divisor = Quotient + Remainder / Divisor
    8x3−14x2+7x−12x−5=4x2+3x+11+542x−5\frac{8x^3 - 14x^2 + 7x - 1}{2x - 5} = 4x^2 + 3x + 11 + \frac{54}{2x - 5}

    Here is the HTML content for the FAQs section on polynomial long division:

    FAQs

    How do you divide polynomials using long division?

    To divide polynomials using long division: 1. Arrange both polynomials in descending order of degree. 2. Divide the first term of the dividend by the first term of the divisor. 3. Multiply the result by the divisor and subtract from the dividend. 4. Bring down the next term and repeat steps 2-3 until the remainder's degree is less than the divisor's.

    What is the rule for polynomial division?

    The key rule for polynomial division is that you can only divide terms with like variables and exponents. Always start by dividing the term with the highest degree in the dividend by the term with the highest degree in the divisor. Continue this process, subtracting at each step, until the remainder has a lower degree than the divisor.

    How do you check the division of a polynomial?

    To check polynomial division: 1. Multiply the quotient by the divisor. 2. Add the remainder (if any) to this product. 3. The result should equal the original dividend. For example, if (ax + b) ÷ (cx + d) = q with remainder r, then (cx + d)q + r should equal ax + b.

    Is long division of polynomials hard?

    Long division of polynomials can be challenging at first, but with practice, it becomes more manageable. The process is similar to arithmetic long division, but with variables. Key challenges include keeping terms organized, handling negative terms, and remembering to bring down all terms. Regular practice and a step-by-step approach can make it easier to master.

    What is the first step in dividing polynomials using long division?

    The first step in dividing polynomials using long division is to arrange both the dividend and divisor in descending order of degree (highest to lowest power of the variable). This organization ensures that you start by dividing the terms with the highest degrees, making the process more straightforward and systematic.

    Prerequisite Topics for Polynomial Long Division

    Understanding polynomial long division is crucial in advanced algebra, but it's essential to grasp several prerequisite topics to fully comprehend this concept. One of the fundamental skills is polynomial synthetic division, which serves as a shortcut method to divide polynomials step by step. This technique is closely related to long division and can help students develop a deeper understanding of polynomial operations.

    Before diving into long division, it's important to be proficient in solving polynomial equations. This skill allows students to manipulate and analyze polynomial expressions, which is crucial when performing long division. Additionally, simplifying rational expressions and understanding restrictions is vital, as polynomial long division often results in rational expressions that need simplification.

    While not directly related to long division, knowledge of integration of rational functions by partial fractions can provide a broader context for the importance of polynomial division in higher-level mathematics. This advanced topic demonstrates how polynomial division is applied in calculus and emphasizes its significance in mathematical problem-solving.

    Students should also be comfortable with adding and subtracting rational expressions, as these operations are often necessary when working with the results of polynomial long division. Furthermore, understanding how to solve polynomials with unknown coefficients can enhance a student's ability to work with more complex polynomial expressions during division.

    Lastly, exploring the applications of polynomial functions provides real-world context for polynomial long division. This knowledge helps students appreciate the practical significance of the technique and motivates them to master it for use in various fields such as physics, engineering, and economics.

    By thoroughly understanding these prerequisite topics, students will be well-prepared to tackle polynomial long division with confidence. Each concept builds upon the others, creating a strong foundation for mastering this important algebraic technique. As students progress through these topics, they'll develop a comprehensive understanding of polynomials and their operations, ultimately leading to success in more advanced mathematical concepts.

    A division statement can be written in 2 ways:
    i) dividend = (divisor) (quotient) + remainder
    ii) dividenddivisor\frac{dividend}{divisor} = quotient + remainderdivisor\frac{remainder}{divisor}
    Basic Concepts
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