Long division with rational functions is similar to traditional long division with numbers but involves polynomials. It's used to divide a polynomial (the numerator) by another polynomial (the denominator). This process helps simplify complex rational expressions.
Understanding the Process
Here's a step-by-step guide to performing long division with rational functions:
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Set Up the Division: Arrange the polynomials in the long division format, with the dividend (the numerator) inside the division symbol and the divisor (the denominator) outside. Ensure the polynomials are in descending order of exponents and include placeholders (0x) for any missing terms.
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Divide the Leading Terms: Divide the leading term of the dividend by the leading term of the divisor. This will be the first term of the quotient. For example, as explained in the reference video, to divide x² by x, we would multiply by x. We can see this in the video at [1:17].
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Multiply: Multiply the term found in step 2 by the entire divisor. Place this result under the dividend, aligning like terms.
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Subtract: Subtract the result from step 3 from the dividend. Change the signs of each term of the subtracted polynomial, and combine the like terms.
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Bring Down: Bring down the next term from the original dividend.
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Repeat: Repeat steps 2 through 5 until the degree of the remaining polynomial is less than the degree of the divisor.
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Remainder: The remaining polynomial at the end is the remainder, and the final answer can be expressed as:
Quotient + (Remainder/Divisor)
Example
Let's illustrate with a simplified example (not from the provided video). Suppose we want to divide (x² + 3x + 5) by (x + 1):
| Steps | Calculation | Result |
|---|---|---|
| 1 | Set up: (x² + 3x + 5) ÷ (x + 1) | x + 1 |
| 2 | x² / x = x | x |
| 3 | x * (x + 1) = x² + x | x² + x |
| 4 | (x² + 3x) - (x² + x) = 2x | 2x + 5 |
| 5 | Bring down the +5 | 2x + 5 |
| 6 | 2x / x = 2 | + 2 |
| 7 | 2 * (x + 1) = 2x + 2 | 2x + 2 |
| 8 | (2x + 5) - (2x + 2) = 3 | 3 |
Therefore, the answer is x + 2 + 3/(x+1)
Key Considerations
- Placeholders: Use placeholders (e.g.,
0x²) for any missing terms in your dividend or divisor to maintain proper alignment and calculation. - Sign Changes: Carefully change the signs during subtraction to avoid errors.
- Checking Your Work: To verify your result, you can multiply the quotient by the divisor and add the remainder. The result should equal the original dividend.
Practical Insights
- Long division is most helpful when the degree of the numerator is greater than or equal to the degree of the denominator.
- It can simplify complex rational expressions, making them easier to integrate or analyze in calculus.
- The process may seem daunting initially, but with practice and careful attention to detail, it becomes straightforward.
Summary
Long division with rational functions involves dividing polynomials using a systematic process that mirrors numerical long division. By focusing on the leading terms, subtracting carefully, and bringing down terms systematically, one can simplify rational expressions. The YouTube video mentioned in the reference further illustrates the initial steps in polynomial long division, for example it illustrates why you multiply x by x to get x^2 and placing that multiplication result above the x^2 value [1:17].
