Hawkes Learning Systems: College Algebra

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Hawkes Learning Systems:College Algebra

Section 5.2: Polynomial Division and the Division Algorithm




Objectives

o The Division Algorithm and the Remainder Theorem.

o Polynomial long division and synthetic division.

o Constructing polynomials with given zeros.




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

p x d x 0d x

q x r x

.d xq xp xx r

dividend quotient divisor remainder

p x

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 .

d x

r x d x

d x p x

p x d x q x




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.”

. p x r x

q xd x d x

p x q x d x r x d x




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.

p xx k

p x q x x k

0p x




The Remainder Theorem

If the polynomial is divided by , the remainder is . That is,

p x x k p k

.p x q x x k p k




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.




Example 1: Polynomial Long Division

Divide the polynomial by the polynomial

5 4 3 29 10 18 28 3x x x x x 29 .1x x

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

2 5 4 3 29 1 9 10 18 28 3x x x x x x x

3x

3.x

Then, multiply by to obtain .Subtract from the dividend.

3x 29 1x x 5 4 39x x x

49x

59x 4x 3x

319x 228 3x x

5 4 39x x x




Example 1: Polynomial Long Division (Cont.)

2 5 4 3 29 1 9 10 18 28 3x x x x x x x

3xTo 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.

2x

3 218 29 3x x x

2x

3

227 3 3x x 227 3 3x x

0

59x 4x 3x49x 319x 228 3x x 49x 3x 2x

318x 22x 2x




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.




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.

x k k




Synthetic Division

Compare the division of by below, using long division on the left and synthetic division on the right.

3 22 8 9 7x x x 2x

3 22 8 9 72x x x x 2

22 4 1x x

3 242x x 24 9 7x x

2 84x x

7x 2x

5

2 8 9 7 4 8 2

2 4 1 5

Note: the numbers in blue are the coefficients of the dividend and the numbers in pink are the coefficients of the quotient and remainder.




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.

k

k

Continued on the next slide…




2

2

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,

2 2 8 9 7 4

44

2 8

1

2 2

51

2




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 .

x k

p k p k

p x x k p x




The fact that the last number is non-zero means is not a zero of p. We can conclude that

Example 2: Synthetic Division

Determine if the given k is a zero. If not, determine . p k

8 7 4 22 10 25 3; 5p x x x x x x k

5 2 10 0 0 1 0 25 1 3

2100

00

00

01

55

250

01

58

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 ). 6 5 3 x x x , ,and

5 8.5p





Determine if the given k is a zero. If not, determine . p k

3 22 25 12; 4p x x x x k

4 2 1 25 12

287

283

120

22 7 3 4p x x x x

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. p




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.




Example 4: Polynomial Long Division

Divide by using polynomial long division.

2 1p x x d x x i

2 0 1x i x x x

2x ix

1ix

i

1ix 0

Note: The term exists as a placeholder for the missing x term in .

0x

2 1x

2 1x i x i x





Divide by using synthetic division.

22 3 5 3 9p x x i x i 3d x x i

3 2 3 5 3 9i i i

2

6i

3 i

3 9i

0

22 3 3 2 3 5 3 9x i x i x i x i




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.

p xx k .p x




Example 6: Polynomials with Given Zeros

Construct a polynomial that has the given properties: third degree, zeros of and goes to as .x 4, 6, 1

4 6 1p x x x x

3 23 22 24p x x x x

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.

, , a4 6 1nd x x x

x




Example 7: Polynomials with Given Zeros

Construct a polynomial that has the given properties: fourth degree, zeros of and a y-intercept of .

1, 3, 3, and 6,108 1 3 3 6p x a x x x x

0 1 3 3 6p a

108 54a2 a

2 1 3 3 6p x x x x x

4 3 22 14 6 126 108p x x x x x

Remember, the linear factors can multiplied by any non-zero constant a without affecting the zeros.

Because the polynomial has a y-intercept of 108, we know that p(0) = 108.

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Hawkes Learning Systems: College Algebra

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