To perform long division with polynomials, follow these clear steps:

Steps for Long Division of Polynomials

1. Arrange terms : Write both the dividend (the polynomial you are dividing) and the divisor (the polynomial you are dividing by) in descending order of their degrees. Include zero coefficients for any missing terms to keep the alignment consistent.
2. Divide leading terms : Divide the first term of the dividend by the first term of the divisor. This gives the first term of the quotient.
3. Multiply and subtract : Multiply the entire divisor by the term just found in the quotient. Write this product under the dividend. Subtract this product from the dividend to get a new polynomial.
4. Bring down next term : If there are more terms in the dividend, bring down the next term to the remainder polynomial.
5. Repeat : Repeat the process - divide the first term of the new polynomial by the first term of the divisor, multiply, subtract, and bring down the next term - until the degree of the remainder is less than the degree of the divisor.
6. Write the answer : The quotient is the polynomial formed by all the terms you found in step 2 each time you divided. The remainder is the last polynomial left over. If the remainder is not zero, write the answer as:

Quotient+RemainderDivisor\text{Quotient}+\frac{\text{Remainder}}{\text{Divisor}}Quotient+DivisorRemainder​

This process is analogous to long division with numbers but applied to polynomial terms

Example Outline

For example, dividing x2+5x+6x^2+5x+6x2+5x+6 by x+2x+2x+2:

• Divide x2x^2x2 by xxx to get xxx.
• Multiply xxx by x+2x+2x+2 to get x2+2xx^2+2xx2+2x.
• Subtract x2+2xx^2+2xx2+2x from x2+5x+6x^2+5x+6x2+5x+6, resulting in 3x+63x+63x+6.
• Bring down terms if needed (here, continue with 3x+63x+63x+6).
• Divide 3x3x3x by xxx to get 333.
• Multiply 333 by x+2x+2x+2 to get 3x+63x+63x+6.
• Subtract 3x+63x+63x+6 from 3x+63x+63x+6, remainder is 0.
• Quotient is x+3x+3x+3, remainder 0.

This example shows the quotient without remainder

. This method works for polynomials with one or more variables and can handle missing terms by inserting zeros as placeholders

. It is a fundamental algorithm for polynomial division and factoring