DIVISION OF POLYNOMIALS

DIVIDING A MONOMIAL BY A MONOMIAL
In a similar manner to multiplying two monomials, we divide numbers by numbers and like bases by like bases following the laws for dividing with exponents (taking the top exponent minus the bottom.)

DIVIDING A POLYNOMIAL BY A MONOMIAL
This is called term by term division. We break the polynomial into its terms then divide each term by the monomial.

 

LONG DIVISION OF POLYNOMIALS
 

  1.  Write as a long division problem. Put the divisor on the outside and put the dividend inside filling in “missing” powers of x using a coefficient of 0.
  2. Take the first term of the inside divided by the first term of the outside and put the result on top.
  3. Take the result from the division in part 2 and multiply it by the outside polynomial. Change the signs of every term and put the results under the like terms in the inside polynomial. Add.
  4. The result of the addition is now the new inside polynomial. If this polynomial is of lower degree than the outside polynomial, it is the remainder. If not return to step 1 using the new inside polynomial and repeat the process until the inside polynomial is of lower degree than the outside.
  5. The answer is the polynomial on top plus the remainder.
 
 
SYNTHETIC DIVISION
If the divisor is of the form (x - a)   or (x + a) there is a quicker method of dividing than using long division.
  1. Write the divisor in the form x - a.
  2. Write the value of the number, a in a box. (Notice it will have the opposite sign of the binomial. For (x + 2) we would write -2. For (x - 2) we would write +2.
  3. Next to the box on the same line fill in the coefficients of the variable from the highest power to the zero power (constant term.) Put a coefficient of zero for any missing power.
  4. Leave a blank space (where numbers can be put in later) and draw a line. Write the first coefficient below the line.
  5. Take the number below the line and multiply by the number in the box. Put the result under the next coefficient. Add and put the result below the line.
  6. Take the number you just wrote below the line, multiply it by the box and put it under the next coefficient. Add.
  7. Continue this process to the end of the line.
Example:
  
would look like this as a synthetic division problem
 
These are the coefficients of the answer. The first number is the coefficient of a polynomial one degree lower than the one you started with. The last number is the remainder.
The result of the division above is x3 - x2 - 2x + 7  remainder -8.