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Polynomial long division is a method used to divide polynomials, similar to long division with numbers. It allows us to divide one polynomial by another polynomial to find the quotient and remainder.

Step-by-Step Process

To perform polynomial long division, follow these steps:

1. Arrange the terms of the dividend and divisor in descending order of their degrees.
2. Divide the leading term of the dividend by the leading term of the divisor to obtain the first term of the quotient.
3. Multiply the entire divisor by the first term of the quotient, and subtract this product from the dividend.
4. Bring down the next term of the dividend, and repeat steps 2 and 3 until all terms have been processed.
5. The resulting expression is the quotient, and any remaining terms constitute the remainder.

Example

Let's perform polynomial long division for the division $$\left(2x^3+3x^2-4x+1\right)÷\left(x-1\right)$$:

1. Dividend: $$2x^3+3x^2-4x+1$$
2. Divisor: $$x-1$$
3. Quotient: $$2x^2+5x+1$$
4. Remainder: $$0$$

Therefore, $$\left(2x^3+3x^2-4x+1\right)÷\left(x-1\right)=2x^2+5x+1$$.

Polynomial long division is a powerful tool for simplifying polynomial expressions and solving equations.