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:

1. Arrange the 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 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:

1. Set up the division: (x² - 2x - 24) ÷ (x + 4)
2. Divide x² by x to get x
3. Multiply (x + 4) by x: x² + 4x
4. Subtract: x² - 2x - 24 - (x² + 4x) = -6x - 24
5. Bring down all terms
6. Divide -6x by x to get -6
7. Multiply (x + 4) by -6: -6x - 24
8. Subtract: -6x - 24 - (-6x - 24) = 0

The final result is: x² - 2x - 24 = (x + 4)(x - 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.