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Introduction

This tutorial will guide you through the process of polynomial division using the stacked method, a technique commonly used in algebra to find the quotient and remainder when dividing polynomials. Mastering this method is essential for solving various mathematical problems and is particularly useful in higher-level math courses.

Step 1: Understanding the Terms

Before diving into polynomial division, familiarize yourself with key terms:

• Polynomial: An expression consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents.
• Dividend: The polynomial you want to divide.
• Divisor: The polynomial you are dividing by.
• Quotient: The result of the division.
• Remainder: The amount left over after division.

Step 2: Set Up the Problem

1. Identify your dividend and divisor.
2. Write the dividend under the long division symbol.
3. Place the divisor to the left outside of the long division symbol.

Example:

• Dividend: ( x^3 + 2x^2 - 5x + 6 )
• Divisor: ( x - 1 )

Step 3: Divide the Leading Terms

1. Divide the leading term of the dividend by the leading term of the divisor.
2. Write the result above the division line as the first term of the quotient.

Example:

• ( \frac{x^3}{x} = x^2 )

Step 4: Multiply and Subtract

1. Multiply the entire divisor by the term you just found in the quotient.
2. Write this result below the dividend.
3. Subtract this result from the dividend to find the new polynomial.

Example Calculation

• Multiply: ( x^2 (x - 1) = x^3 - x^2 )
• Subtract from dividend:

   x^3 + 2x^2 - 5x + 6
 - (x^3 - x^2)
 ------------------------
          3x^2 - 5x + 6

Step 5: Repeat the Process

1. Take the new polynomial obtained from subtraction as your new dividend.
2. Repeat steps 3 and 4 until the degree of the new polynomial is less than the degree of the divisor.

Continuing the Example

• New dividend: ( 3x^2 - 5x + 6 )
• Divide leading terms: ( \frac{3x^2}{x} = 3x )
• Multiply and subtract again.

Step 6: Find the Remainder

Once you can no longer divide (the degree of the new polynomial is less than the divisor), what remains is your remainder.

In Our Example

• Final subtraction might yield a remainder, say ( R ).
• Write this as ( R = \text{(the remaining polynomial)} ).

Conclusion

In this guide, you learned how to divide polynomials using the stacked method, including identifying key terms, setting up the problem, and performing the division step-by-step. Practice with different polynomials to solidify your understanding and improve your proficiency in polynomial division. For further learning, consider exploring more advanced topics like synthetic division or factoring polynomials.