What is the quotient in polynomial form?

The quotient in polynomial division is the result you get when you divide one polynomial by another. Polynomials are expressions that consist of variables and coefficients, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

To find the quotient when dividing polynomials, you can use long division or synthetic division (if dividing by a binomial of the form x - c). The process is similar to long division with numbers. Here's a general step-by-step procedure using long division:

1. Write the dividend (the polynomial you're dividing) and the divisor (the polynomial you're dividing by) in descending order of their degrees.

2. Divide the first term of the dividend by the first term of the divisor. This gives you the first term of the quotient.

3. Multiply the entire divisor by the first term of the quotient, and subtract this from the dividend.

4. Bring down the next term of the dividend if there is one. This forms a new, smaller polynomial dividend.

5. Repeat the process: divide the first term of the new dividend by the first term of the divisor, multiply the divisor by the result, subtract, and bring down the next term.

6. Continue this process until the degree of the remaining dividend (referred to as the remainder) is less than the degree of the divisor.

7. The quotient is the polynomial formed by the terms you've obtained during the division process. If there's a remainder, it's written as a fraction over the original divisor.

Here's a simple example:

Divide the polynomial 2x^3 + 3x^2 - x + 5 by the polynomial x - 2.

1. Divide the first term of the dividend (2x^3) by the first term of the divisor (x). You get 2x^2. This is the first term of the quotient.

2. Multiply the divisor (x - 2) by the first term of the quotient (2x^2), which gives you 2x^3 - 4x^2.

3. Subtract this from the dividend: (2x^3 + 3x^2) - (2x^3 - 4x^2) = 3x^2 + 4x^2 = 7x^2.

4. Drag the next term down: 7x^2 - x.

5. Divide the first term of the new dividend (7x^2) by the first term of the divisor (x). You get 7x.

6. Multiply the divisor by the new term of the quotient (7x), and subtract this from the new dividend.

.and so on, until you can no longer continue the division process.

At the end, you'll have a polynomial quotient and, possibly, a remainder.