Explain the long division method in polynomials.

The long division method in polynomials is used to divide one polynomial by another.

To perform long division with polynomials, we first write the dividend (the polynomial being divided) and the divisor (the polynomial dividing the dividend) in descending order of degree. We then divide the highest degree term of the dividend by the highest degree term of the divisor, and write the result as the first term of the quotient. We then multiply the divisor by this term, and subtract the result from the dividend. This gives us a new polynomial, which we repeat the process with, dividing the highest degree term by the highest degree term of the divisor, and so on, until we have no terms left in the dividend with a degree greater than or equal to the degree of the divisor.

For example, let's divide the polynomial x^3 + 2x^2 - 3x - 4 by the polynomial x - 2. We first write the dividend and divisor in descending order of degree:

x^3 + 2x^2 - 3x - 4 ÷ x - 2

We then divide the highest degree term of the dividend (x^3) by the highest degree term of the divisor (x), giving us x^2. We write this as the first term of the quotient:

x^2

We then multiply the divisor (x - 2) by this term (x^2), giving us x^3 - 2x^2. We subtract this from the dividend, giving us a new polynomial:

x^3 + 2x^2 - 3x - 4 - (x^3 - 2x^2) = 4x^2 - 3x - 4

We then repeat the process with this new polynomial, dividing the highest degree term (4x^2) by the highest degree term of the divisor (x), giving us 4x. We write this as the second term of the quotient:

x^2 + 4x

We then multiply the divisor (x - 2) by this term (4x), giving us 4x^2 - 8x. We subtract this from the new polynomial, giving us a remainder of 5x - 4:

4x^2 - 3x - 4 - (4x^2 - 8x) = 5x - 4

Since the degree of the remainder (1