Section 6. Long Division of Polynomials INTRODUCTION In Section 6.1 we learned to simplify a rational epression by factoring. For eample, + 3 10 = ( + 5)( ) ( ) = ( + 5) 1 = + 5. However, if we try to simplify either far, because... 15 + 36 5 or + 9 + 6 in this same manner, we won t get very 15 + 36 5 = ( 3)( 1) ( 5) and there are no common factors to divide out; and the numerator in + 9 + 6 is not factorable. Still, we can simplify them in a different sort of way... using long division, and that is what this section is all about. Before we can begin dividing polynomials, though, we need to be sure we have the proper preparation. What follows are some basic ideas with which you are already familiar, but they will help create the foundation used to develop our topic. THE PREPARATION: TYPES OF FRACTIONS Numerically, any fraction in which the value of the numerator is less than the value of the denominator is called a proper fraction, such as 3 8, 1 and 5. In algebra, a proper fraction is one in which the degree of the numerator is less than the degree of the denominator. Eamples of proper algebraic fractions include: 3 8 degree is 1 degree is, 10 3 3 4 degree is 3 degree is 4 and 5 degree is 0 + 3 degree is 1 In 10 3 3 4, even though the numerator has a greater numerical value, it is the degree of the term that we look at. Because the denominator, 3 4, has a degree of 4, and 10 3 has a degree of only 3, 103 3 4 is a proper fraction. Long Division of Polynomials page 6. - 1
Conversely, a numerical improper fraction has a numerator that is greater than or equal to the value of the 8 denominator, such as 3, 164 5, 1 and 5 5. In algebra, this means that the degree of the numerator is greater than or equal to the degree of the denominator. Eamples of improper algebraic fractions include: 3 8 degree is degree is 1, 1 3 5 3 degree is 3 degree is 3 and + 3 10 + 4 degree is degree is 1 Some improper fractions can be simplified directly. For eample, 1 =, 5 5 = 1, 3 3 1 = 4 and 15 3 5 3 = 3. Other improper fractions, especially those that are numerical only, can be rewritten as mied numbers: 8 3 = 3 and 164 5 = 3 4 5. The process for rewriting improper fractions as mied numbers is a called long division: Eample 1: Divide using long division. Write the answer as a mied number. a) 8 3 b) 164 5 Procedure: The numerator is called the dividend and the denominator is called the divisor. a) 8 3 = 8 3: b) 164 5 = 164 5 5 3 1 6 4 1 5 1 4 1 0 4 4 5 or 3 + 4 5 Long Division of Polynomials page 6. -
Eercise 1 Divide using long division. Write the answer as a mied number. a) 91 4 b) 45 6 c) 1,38 11 Just as 3, also written + 3, is a mied number, + 3 + 5 working with mied epressions shortly. Here is more preparation. is a mied epression. We will be THE PREPARATION: SUBTRACTING POLYNOMIALS You are already familiar with how to subtract polynomials: you must distribute the negative sign and add. For eample, subtract: ( + 8) (4 ) ( + 8) + - 1(4 ) Subtracting means adding the opposite. = + 8 4 + Distribute the - 1 to both terms; = 3 + 10 Combine like terms. If we were to set the same problem up vertically, the result will be the same if done correctly. This second method allows you to distribute the negative and rewrite it as a sum; then add: + 8 (4 ) + 8 + - 4 + 3 + 10 Notice that subtraction has been changed to add the opposite. (This may look like you have to write the problem twice. Actually, on your paper, this could all be done in one setting.) Long Division of Polynomials page 6. - 3
Eample : Subtract. a) 3 + 5 b) - 8 + 3 c) + 1 d) 4 3 (3 + ) (- 8 + ) ( 4) (4 8) Procedure: Answer: Keep in mind that Subtracting means adding the opposite. Distribute the negative sign through to the second quantity. (Notice that, within each of these, the binomials have the same first term.) Eercise Subtract. a) 4 + 8 b) - 3 + 5 c) 6 + 1 d) - 5 16 (4 + 6) (- 3 + 9) (6 9) (- 5 10) THE PREPARATION: MULTIPLICATION AND THE DISTRIBUTIVE PROPERTY Multiply 8 13. Here are four ways to approach the same problem. They all involve distribution. a) b) c) d) 8 13 8 (10 + 3) 4 + 80 104 4 + 8 0 1 0 4 8 (10 + 3) 8 0 + 4 = 1 0 4 For our work in dividing polynomials, the fourth way will be most valuable. Eample 3: Multiply, by distributing, the following. Use the method shown in (d) above. a) 3 b) - c) 6 ( + 5) (3 4) ( 1) Procedure: Multiply mentally. The terms will be unlike terms, unable to be combined. a) 3 b) - c) 6 ( + 5) (3 4) ( 1) 3 + 15-6 + 8 1 6 Long Division of Polynomials page 6. - 4
Eercise 3 Multiply, by distributing, the following. Use the method shown in (d) above. a) 5 b) - 3 c) d) - 4 ( + 4) ( 1) (3 5) (5 + ) THE PREPARATION: MISSING TERMS IN A POLYNOMIAL Some polynomial, when written in descending order, have all possible degrees represented. For eample, the polynomial 3 + 6 4 + 9 has all of the possible degrees represented: there are terms with degree 3,, 1 and 0 represented, nothing is missing. Other polynomials have some missing terms. For eample, 4 + 5 4 + 9 is missing an 3 term, and 5 + 5 3 4 + 9 is missing two terms: an 4 term and an term. Each of the missing terms could be put into place using a coefficient of 0 (zero): 4 + 5 4 + 9 5 + 5 3 4 + 9 = 4 + 0 3 + 5 4 + 9 = 5 + 0 4 + 5 3 4 + 0 + 9 Eample 4: Rewrite each polynomial in descending order with a complete set of terms. a) 5 3 6 b) 3 3 Procedure: Identify the missing terms and write them in with a coefficient of 0. a) 5 3 6 b) 3 3 = 5 3 + 0 6 = 3 3 + 0 + 0 Eercise 4 Rewrite each polynomial in descending order with a complete set of terms. a) 3 + 9 b) 1 4 3 6 c) 8 3 Here is a recap of the preparation we ve seen so far in this section: Long Division of Polynomials page 6. - 5
TYPES OF FRACTIONS: We looked at proper fractions, improper fractions and mied numbers SUBTRACTING POLYNOMIALS: Remember to remove the parentheses and add the opposite MULTIPLICATION AND THE DISTRIBUTIVE PROPERTY We were able to multiply a little differently. MISSING TERMS IN A POLYNOMIAL: DIVIDING POLYNOMIALS We learned to fill in all of the terms, in descending order, of a polynomial. To best understand how to divide polynomials we need first look at dividing numbers using long division. What you ll eventually see is how place values (ones, tens and hundreds) play an important role. Consider 31 13 using long division: 13 3 1 : Divide 13 into 3; it goes in times. Multiply 13. 13 3 1 6 Subtract 6 from 3 and bring down the 1. 13 3 1 6! 1 1 1 Take a look back at the first step of dividing 13 into 31. When we begin the process, the first thing we do is divide 13 into 3. This is like the improper fraction 3 13. When dividing, the whole number is and there is a remainder, found by subtracting. Long Division of Polynomials page 6. - 6
We re going to take a different look at the same problem, 31 13. This time we re going to epand both numbers and treat 31 as 300 + 0 + 1 and 13 as 10 + 3. (When written this way, the numbers start to look a little like a trinomial and a binomial.) Throughout this net eercise, you ll see many of the preparation ideas presented earlier in this section put to good use. Divide: (300 + 0 + 1) (10 + 3). each: 300 10 = 30 10 + 3 300 + 0 + 1 1 Divide 10 into 300. This time, instead of dividing 13 into 3, we ll divide the first term of 300 10 Place 30 above 0 (they re both tens ) 30 = 30 10 + 3 300 + 0 + 1 3 Net, multiply 30 by (10 + 3) 300 + 90 Net, we subtract (300 + 90) from what s directly above it. Subtract by adding the opposite. 30 10 + 3 300 + 0 + 1 ( 300 + 90) Add the opposite. Notice that 300 becomes - 300 and + 90 becomes - 90. 30 10 + 3 300 + 0 + 1 + (- 300 + - 90)! - 0 + 1 Repeat the process, this time dividing the new first term, - 0, by 10: - 0 10 = -. (Put this in the ones place.) We ll multiply and subtract (by adding the opposite). 30! 10 + 3 300 + 0 + 1 + (- 300 + - 90) - 0 + 1 (- 0 6) Then, multiply - times (10 + 3) 30! 10 + 3 300 + 0 + 1 + (- 300 + - 90) - 0 + 1 + + + (- 0 6) Add the opposite. - 0 becomes + 0 and - 6 becomes + 6 30! 10 + 3 300 + 0 + 1 + (- 300 + - 90) - 0 + 1 + (- + 0 + 6) + 13 Show the remainder above, in the answer. remainder The answer, 30 + form. 13 looks different from 8 13, but they are the same (30 = 8), just in a different Long Division of Polynomials page 6. -
We will now use the same procedure using algebra: dividing a polynomial by a binomial. The key is dividing first term by first term. Eample 5: Divide 3 + + 1 + 3 ; also written as (3 + + 1) ( + 3). Procedure: Use the method outlined on the previous page. This is the same as the numerical eample (300 + 0 + 1) (10 + 3) with replacing 10. (300 = 3 100 = 3 10 which becomes 3 ) + 3 3 + + 1 1 Divide into 3. 3 Place 3 above (they re like terms) = 3 + 3 Net, multiply 3 3 by ( + 3) 3 3 + + 1 3 + 9 3 + 3 3 + + 1 (3 + 9) Add the opposite. Notice that 3 becomes - 3 and + 9 becomes - 9. 3 + 3 3 + + 1 + (-3 + - 9) - + 1 3 + 3 3 + + 1 + (-3 + - 9) - + 1 (- 6) Multiply - times ( + 3) 3 + 3 3 + + 1 + (-3 + - 9) - + 1 + (- + + 6) Add the opposite. - becomes + and - 6 becomes + 6 3 + + 3 3 + + 1 + (-3 + - 9) - + 1 + (- + + 6) Show the remainder above, in the answer. + 3 remainder Conclusion: (3 + + 1) ( + 3) = 3 + + 3 This very last step is what your work would normally look like. This shows every step compiled into one long-division problem. 3 + + 3 3 + + 1 + (-3 + - 9) - + 1 + (- + + 6) + 3 Long Division of Polynomials page 6. - 8
Eample 6: Divide ( 3 6 + + 1) ( 4) Procedure: Use the same procedure as in Eample 5. This time the division process is shown complete as you might write it. It s recommended that you do the step of changing the subtraction to add the opposite. 4 A B A B 1 + 3 6 + + 1 3 ( 4 ) - + (- + 8) - + 1 C (- + 4) - 3 D C D - 3 4 A B Divide into 3 to get Multiply ( 4) to get 3 4 and subtract. You may now change it to add the opposite. Divide into - to get - Multiply - ( 4) to get - + 8 and subtract. You may now change it to add the opposite. C Divide into - to get - 1. Multiply and subtract (as above) D - 3 is the remainder and shows up in the answer as part of the fraction. Because the remainder is negative, the answer could also be written with the reminader fraction subtracted. 3 1 4 Eercise 5 Divide each using long division. a) 5 + 3 b) 4y + 18y 3 y + 5 Long Division of Polynomials page 6. - 9
c) 3m 3 m + m 5 m d) p 3 + 6p + 9p + 4 p + 4 Before dividing, each polynomial must be put in descending order. Furthermore, if either polynomial has any missing terms, they must be written so that all terms are represented. Eample : Divide. Be sure to write the polynomial in 5 3 1 + 8 descending order and include any missing terms. + Procedure: The fraction can be written as (5 3 1 + 8) ( + ) The dividend, 5 3 1 + 8, is missing a term. Before we can divide we must write it as 5 3 + 0 1 + 8. So, 5 3 1 + 8 + = 5 10 + 3 + + Long Division of Polynomials page 6. - 10
Eample 8: Divide. Be sure to write the polynomial in 5 5 + 3 descending order and include any missing terms. 3 Procedure: The fraction can also be written as (5 5 + 3 ) ( 3) The dividend, 5 5 + 3, is not in descending order; it is also missing a term. Before we can divide we must write it as 3 + 5 + 0 5. 3 + 4 + 6 + 3 + 5 + 0 5 3 ( 3 ) A B A B 8 + 0 (8 1) 1 5 C (1 18) - D C D - 3 A Divide into 3 to get Multiply ( 3) to get 3 3 and subtract. You may now change it to add the opposite. B Divide into 8 to get 4 Multiply 4 ( 3) to get 8 1 and subtract. You may now change it to add the opposite. C Divide into 1 to get 6. Multiply and subtract (as above) D - is the remainder and shows up in the answer as part of the fraction. Because the remainder is negative, the answer could also be written with the reminader fraction subtracted. + 4 + 6 3 So, 5 5 + 3 3 = + 4 + 6 3 Eercise 6 Divide each using long division. Be sure to write each polynomial in descending order and to include any missing terms. a) 5 8 + 3 b) 4 3 + 10 8 6 + 3 Long Division of Polynomials page 6. - 11
c) 3 + + 10 + 4 d) 10 + 6 3 + 1 3 1 Answers to each Eercise Section 6. Eercise 1: a) 3 4 b) 9 1 6 c) 15 11 Eercise : a) b) - 4 c) 10 d) - 6 Eercise 3: a) 10 + 0 b) - 3 + 3 c) 6 10 d) - 0 8 Eercise 4: a) 3 + + 0 9 b) - 4 3 + 0 6 + 1 c) - 3 + 0 + 0 + 8 Eercise 5: a) + 1 + c) 3m m + or 3m m Eercise 6: a) 3 + c) + 4 10 3 b) 4y + y + 5-5 m d) p + p + 1 5 m 1 b) 4 6 + 4 d) + 3 3 1 Long Division of Polynomials page 6. - 1
Section 6. Focus Eercises 1. Divide each using long division. a) y + 10y + 18 y + 5 b) 3 4 8 c) (p 3 + p 5p 1) (p + 3) d) (3m 3 9m 14m + 8) (m 4). Divide each using long division. Be sure to write each polynomial in descending order. a) (5 9 + 4 ) ( 1) b) (3 + 4) ( + ) c) (6 + 3 8 5 ) ( 4) d) (14 + 5 3 4 6) ( + 3) Long Division of Polynomials page 6. - 13
e) (5 3 + 1 6) (5 + ) f) ( + 10 3 + 1) ( 1) 3. Divide each using long division. Be sure to write each polynomial in descending order; also, be sure to include any missing terms. a) ( 3 + 9 + 1) ( + 5) b) (1 11 + 3 3 9) ( 4) c) (16 9 + 4 ) ( ) d) (4 3 5) ( + 1) Long Division of Polynomials page 6. - 14