Polynomial long division is a method used in algebra to divide one polynomial by another. It is similar to the long division process used in arithmetic but involves dividing polynomial terms. This technique is useful for simplifying complex polynomial expressions or finding remainders in polynomial division.
Steps for Polynomial Long Division
- Arrange both the dividend (the polynomial to be divided) and the divisor (the polynomial by which we divide) in descending order of their degrees.
- Divide the first term of the dividend by the first term of the divisor. Place the result above the dividend.
- Multiply the entire divisor by this result and subtract it from the dividend. Bring down the next term of the dividend.
- Repeat this process until all terms of the dividend have been brought down and divided.
- The expression above the dividend line is the quotient, and any remainder is left over the divisor.
Example of Polynomial Long Division
Example: Divide 2x^3 – 9x^2 + 21x – 18 by x – 3.
1. Divide 2x^3 by x to get 2x^2. Place 2x^2 above the dividend.
2. Multiply (x – 3) by 2x^2 to get 2x^3 – 6x^2 and subtract this from the dividend.
3. Bring down the next term to get a new dividend of -3x^2 + 21x. Divide -3x^2 by x to get -3x and repeat the process.
4. Continue this process until all terms are brought down and divided.
5. The quotient and remainder are then determined from the final division step.
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