1. Hawkes Learning Systems: College Algebra
Section 5.2: Polynomial Division and the Division Algorithm

2. Objectives The Division Algorithm and the Remainder Theorem.
Polynomial long division and synthetic division.
Constructing polynomials with given zeros.

3. The Division Algorithm
Let and be polynomials such that and with the degree of less than or equal to the degree of Then there are unique polynomials and , called the quotient and the remainder, respectively, such that
dividend
quotient divisor remainder
The degree of the remainder, , is less than the degree of the divisor, , or else the remainder is 0, in which case we say divides evenly into the polynomial If the remainder is 0, the two polynomials and are factors of

4. The Division Algorithm
If we divide every term in by the polynomial , we obtain the form: This fact may be stated, “If one polynomial is divided by another of smaller degree, the result is a polynomial plus, possibly, a ratio of two polynomials, the numerator of which has a smaller degree than the denominator.”

5. Zeros and Linear Factors
The number k is a zero of a polynomial if and only if the linear polynomial is a factor of p. In this case for some quotient polynomial q. This also means that k is a solution of the polynomial equation , and if p is a polynomial with real coefficients and if k is a real number, then k is an x-intercept of p.

6. The Remainder Theorem
If the polynomial is divided by , the remainder is That is,

7. Polynomial Long Division
Polynomial long division is the analog of numerical long division, and provides the means for dividing any polynomial by another of equal or smaller degree.

8. Example 1: Polynomial Long Division
Divide the polynomial by the polynomial
The first step is to arrange the dividend and the divisor in descending order. The first term of the quotient is then the first term of the dividend divided by the first term of the divisor, giving us
Then, multiply by to obtain
Subtract from the dividend.

9. Example 1: Polynomial Long Division (Cont.)
To determine the second term of the quotient, we repeat the previous process. We will continue to repeat this process until we are no longer able to divide the dividend by the divisor.

10. Polynomial Long Division
Caution! Although polynomial long division is a straightforward process, one common error is to forget to distribute the minus sign in each step as one polynomial is subtracted from the one above it. A good way to avoid this error is to put parentheses around the polynomial being subtracted, as in Example 1.

11. Synthetic Division
Synthetic division is a shortened version of polynomial long division, and can be used when the divisor is of the form for some constant . Synthetic division does not do anything that long division can’t do (and in fact is only applicable in certain circumstances), but the speed of synthetic division is often convenient. Instead of various powers of the variable, synthetic division uses a tabular arrangement to keep track of the coefficients of the dividend and, ultimately, the coefficients of the quotient and remainder.

12. Synthetic Division
Compare the division of by below, using long division on the left and synthetic division on the right.
Note: the numbers in blue are the coefficients of the dividend and the numbers in pink are the coefficients of the quotient and remainder.

13. Synthetic Division
Step 1: Write and the coefficients of the dividend. Copy the leading coefficient of the dividend in the first slot below the horizontal line. Step 2: Multiply this number by and write the result directly below the second coefficient of the dividend. Step 3: Add the two numbers in that column and write the result in the second slot below the horizontal line. Step 4: Repeat the process until the last column is completed and the last number written down is the remainder.
Continued on the next slide…

14. Synthetic Division
The other numbers in the bottom row constitute the coefficients of the quotient, which will be a polynomial of one degree less than the dividend. For example,

15. Synthetic Division
Because synthetic division is much faster than long division, it is very useful in determining if is a factor of a given polynomial. By the remainder theorem, synthetic division also provides a quick means of determining for a given polynomial since is the remainder when is divided by .

16. Example 2: Synthetic Division
Determine if the given k is a zero. If not, determine
Note: It is essential that we place a number of 0’s in certain slots in the first row, as these serve as placeholders for the missing terms of the dividend (namely ).
The fact that the last number is non-zero means is not a zero of p. We can conclude that

17. Example 3: Synthetic Division
Determine if the given k is a zero. If not, determine
In this case, the remainder is 0, and hence 4 is a zero of the polynomial p. Since the remainder is 0, we now know of two factors of , as illustrated above.

18. Polynomial Long Division and Synthetic Division
When graphing polynomials, we will be concerned with those that have only real coefficients, but complex zeros and coefficients may still arise in intermediate stages of the graphing process. In solving polynomial equations, we have already seen (in the case of quadratic equations) that complex numbers may be the only solutions. For these reasons, it is important for us to be able to handle complex numbers as they arise.

19. Example 4: Polynomial Long Division
Divide by using polynomial long division.
Note: The term exists as a placeholder for the missing x term in

20. Example 5: Synthetic Division
Divide by using synthetic division.

21. Constructing Polynomials with Given Zeros
We now know the connection between zeros and factors: k is a zero of the polynomial if and only if is a factor of We can make use of this fact to construct polynomials that have certain desired properties, as illustrated in the following examples.

22. Example 6: Polynomials with Given Zeros
Construct a polynomial that has the given properties: third degree, zeros of and goes to as
First, note that must be factors of the polynomial we are about to construct, since these factors give rise to the desired zeros. Since a cubic with a positive leading coefficient goes to as , we must multiply the three linear factors by a negative constant to achieve the desired behavior.

23. Example 7: Polynomials with Given Zeros
Construct a polynomial that has the given properties: fourth degree, zeros of and a y-intercept of
Because the polynomial has a y-intercept of 108, we know that
p(0) = 108.
Remember, the linear factors can multiplied by any non-zero constant a without affecting the zeros.