In long division notation, the numerator is placed underneath the division symbol, and the denominator is placed to the left. The expression below shows the division When the division is complete, there is a number above the division sign. Sometimes, two values will not divide evenly. As a result, the quotient contains a remainder. Here, there is a remainder of which is expressed as divided by or The quotient can be written as
The concepts of numerator, denominator, quotient, and remainder are also used when working with polynomial long division. If, for example, is divided by the division is written as follows. When the division is finished, the quotient is and the remainder is . Thus, the result is written as
To divide expressions where both the numerator and denominator are polynomials, polynomial long division can be used. The process is similar to when dividing real numbers. Consider the following division.
To begin, sort the terms in the polynomial by their degrees, in descending order, and write the division.
The quotient from the previous step is now multiplied by the denominator. This gives
The expression calculated from the long division can now be interpreted as Since the numerator in the remaining fraction has the same degree as the denominator, perform Steps - again.
The quotient of is the sum the expression above the division symbol and the remainder divided by the denominator.
The division symbol for synthetic division is L-shaped. The number is written to the left. Here it's Next, write the coefficients of the numerator to the right. The numerator is so the coefficients are and Note there is no -term so that coefficient is
Keeping in mind that the quotient is one degree lower than the numerator, the quotient can be written using the results of the synthetic division above. Since the last number below the horizontal line was not there is a remainder. The remainder is written as the quotient of the remainder and the denominator.
If a polynomial is divided by a binomial in the form , where is a real number, the result will be the sum of another polynomial and a fraction where the numerator is the remainder, Then, according to the Remainder Theorem, the remainder can be determined by In other words, the remainder is given by the function value of when
Use synthetic division to evaluate if