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Polynomial Functions

Dividing Polynomials

As has been seen, polynomials can be added, subtracted and multiplied. As it turns out, they can also be divided. For this, the methods of long division or synthetic division can be used.

How to Interpret Long Division

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


Terminology for Polynomial Long Division

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


Polynomial Long Division

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.


Sort the terms and write as long division

To begin, sort the terms in the polynomial by their degrees, in descending order, and write the division.


Divide the first term in the numerator by the first term in the denominator
Divide the first term in the numerator by the first term in the denominator. In this case, should be divided by The result, is written over the numerator.


Multiply the quotient in Step by the denominator

The quotient from the previous step is now multiplied by the denominator. This gives


Subtract the product in Step from the numerator
The product from the previous step is subtracted from the numerator.
The simplification in the last step is clarified below.

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.


Repeat Steps to 2 until the numerator has a lower degree than the denominator
Steps are repeated until the polynomial in the numerator has a lower degree than the polynomial in the denominator.
Now, the numerator has a degree of and the denominator has a degree of Thus, the degree of the numerator is lower than that of the denominator, meaning that the division is complete. When long division is performed on paper, it looks something like this.
Polynomial division.svg

The quotient of is the sum the expression above the division symbol and the remainder divided by the denominator.


Synthetic Division

When dividing polynomials by binomials of the form where is the independent variable and is a real number, there is an alternative to polynomial long division. Synthetic division uses and the coefficients of the numerator to find the quotient. The result of the example division above can be determined using this process.


Set up the division using and the coefficients of the numerator

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


Bring down the first coefficient
First, bring down the left-most coefficient. Here, that is so is written below the horizontal line.
Bring down


Multiply and the new number below the line
The newest number under the line is now multiplied by Here that means The result is placed under the second term in the numerator.


Add the numbers in the column above the line
Now, the numbers in the column created by multiplying are added.


Repeat Steps - through the last column
These steps are repeated through the last column.
Now that the division is complete, the numbers below the horizontal line are the coefficients of the quotient.


Write the quotient

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.


Remainder Theorem

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


To show why the remainder can be given by the division can be rewritten by multiplying both sides by the binomial
By evaluating  when the expression can be simplified to
Therefore, the remainder is
The Remainder Theorem can be used to calculate the value of a polynomial function for a number By dividing the polynomial with the binomial the reminder will be equal to the function value If the remainder is zero, the binomial is a factor of the polynomial

Use synthetic division to evaluate if

Show Solution
To evaluate the polynomial, we could substitute into the rule and simplify. However, the Remainder Theorem tells us that is the remainder when dividing by Thus, by determining the remainder of will give As instructed, we'll use synthetic division. The coefficients are and
Bring down
The remainder of the division is which means that
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