Polynomial Long Division
Polynomial long division is a mathematical technique used to divide polynomials using long division with variables. This method is essential for simplifying complex polynomial expressions and finding quotients and remainders.
Definition: Polynomial long division is a process of dividing one polynomial dividend by another polynomial divisor to obtain a quotient and potentially a remainder.
The process of polynomial long division closely resembles the long division method used for numbers. Here's how it works:
- Arrange the polynomials in descending order of degree.
- Divide the first term of the dividend by the first term of the divisor.
- Multiply the result by the divisor and subtract from the dividend.
- Bring down the next term and repeat the process until the degree of the remainder is less than the degree of the divisor.
Example: Let's consider dividing x² - 2x - 24 by x + 4
The solution is presented step-by-step:
- Set up the division: x2−2x−24 ÷ x+4
- Divide x² by x to get x
- Multiply x+4 by x: x² + 4x
- Subtract: x² - 2x - 24 - x2+4x = -6x - 24
- Bring down all terms
- Divide -6x by x to get -6
- Multiply x+4 by -6: -6x - 24
- Subtract: -6x - 24 - −6x−24 = 0
The final result is: x² - 2x - 24 = x+4x−6 + 0
Highlight: The quotient is x−6, and the remainder is 0.
Other examples provided in the image demonstrate similar processes for different polynomial divisions, including cases with remainders.
Vocabulary:
- Dividend: The polynomial being divided
- Divisor: The polynomial we're dividing by
- Quotient: The result of the division
- Remainder: What's left over after division
This method can be applied to various polynomial division problems, including those with higher degrees and more complex terms. It's a fundamental skill in algebra and forms the basis for more advanced mathematical concepts.
