Chapter Six - Polynomial Division, and Graphing of Rational Functions - This has to be one of the most challenging, and the most fun, chapters of the book

Oct 16, 2026

Chapter Six explains polynomial division and the graphing of rational functions by building on the inverse relationship between multiplication and division. It first revisits long division for numbers, emphasizing place value and how “guess, multiply, subtract, and bring down” works to find quotients and remainders. It then duplicates those steps for polynomials using distributive-property expansion of binomials and systematic subtraction. The chapter shows how aligned powers of x guide the division process.

LanguageEnglish

CategoryExam

FormatPDF

Pages37

Reading Time1 h 33 min

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Polynomial Division, and Graphing of Rational Functions

How We Do Long Division

Polynomial Division

Chapter Six Polynomial Division, and Graphing of Rational Functions This has to be one of the most challenging, and the most fun, chapters of the book. I hope you dig deep and enjoy it a lot! The material presented here pulls together a lot of mathematics into one topic, as you will see. How We Do Long Division Before I get into division of polynomials, I want to go back to simple long division of numbers, which some of you may never have seen, or if you have, it was many years ago. At the outset of this book, I claimed that much of mathematics is undoing an operation, that is, applying the inverse operation to make the problem simpler and arrive at an answer. Recall how multiplication works, that sixteen times seventeen means that I am adding the number 16 to itself 17 times. I could write 16 × 17 = 16 + 16 + 16 + 16 + 16 + 16 + 16 + 16 + 16 + 16 + 16 + 16 + 16 + 16 + 16 + 16 + 16 which is a long and tedious task. But multiplication allows me to quickly determine the answer. There are several techniques that essentially revolve around the distributive property, and what I mean by that is that 16 × 17 = (10 + 6)(10 + 7) and, like we did with the polynomial multiplication earlier, I group these products together, like this: 16 × 17 = (10 + 6)(10 + 7) = 100 + 60 + 70 + 42 = 272 The whole reason I’m pointing this out before we move to the inverse operation, division, is that my quick method of multiplying 16 × 17 uses place value to breakup the 16 and 17. We break it up in powers of ten; for example, one thousand six hundred and twenty-nine is (1 × 1000) + (6 × 100) + (2 × 10) + (9 × 1) or, (1 × 103) + (6 × 102) + (2 × 101) + (9 × 100 ) which I hope is ringing a bell with you, recalling our discussion on polynomials earlier. Now, remember that division is the inverse of multiplication. Suppose we wanted to divide 272 by 16. I will write it like this: 16 goes into 272: (1 × 101) + (6 × 100 ) goes into (2 × 102) + (7 × 101) + (2 × 100 ) My first question to myself is to look at the 1 × 101 and ask myself how many of these are in the first term of the second number, 2 × 102 . Another way to ask the question is, what do I have to multiply 1 × 10 by in order to get 2 × 102 ? The answer is 2 × 10, because 1 × 10 guess in doing long division. times 2 × 10 is 2 × 102. So I make that my first The only problem with my guess is that it is too big. I find that out when I multiply 2 × 10 times the first number, (1 × 10 + 6 × 1), 2 × 10 simply cannot work because I get a number bigger than the 272 I want, namely 320. So, I reduce my first guess to 1 × 10. When I do that I get 1 × 10 times (1 × 10 + 6 × 1), or 160. 160 is obviously less than 272, but I’m on my way. I subtract 160 from 272 to get a remainder. Here, my remainder is 112. I go back to my original question, how many 16s are there in 112; another way to ask it is, what do I multiply 16 by to get 112? The answer is 7. When I add 7 to the 1 × 10 from my previous work, I get 10 + 7 = 17. Now, this is a rotten explanation for what’s going on here, because I’m belaboring every step of the process, and it gets very dull. If you learned long division, you probably learned it like this: Handwritten, of course, but you see how I “guessed” the 2 first, then scrapped it for the 1. Notice several things here. The “2” and later the “ 1” went over the “7” of the number 272. That’s because I’m lining up powers of 10. This is important when we talk polynomial division, as you will see. Notice also the understood subtraction that happens twice, the first when I subtract 16 from 27 to get 11, and the second time when I subtract 112 from 112 . Notice that I “brought down the 2” and placed it next to my“11” to make it 112. I show this with the arrow. Finally, notice that my final remainder is 0. That means that 16 went into 272 perfectly, which means that 16 × 17 is 272. If you’ve seen this before, I apologize for possibly wasting your time, bu

FAQs

How does the chapter connect long division to the inverse of multiplication?

Division is presented as the inverse operation of multiplication. By starting with number division (e.g., dividing 272 by 16), the chapter shows how repeated “multiply and subtract” recovers the quotient and remainder in a way that mirrors multiplication’s structure.

Why does place value matter in long division according to the chapter?

The chapter highlights that powers of ten must line up during the process. This alignment is treated as essential groundwork for polynomial division, where powers of x must line up so subtraction can be carried out cleanly.

How is polynomial division demonstrated using binomials like (x + 3)(x − 5)?

The chapter expands binomials to form a polynomial (x^2 − 2x − 15), then performs polynomial long division by repeatedly asking what term multiplies the divisor to produce the leading term of the dividend, multiplying back, subtracting, and continuing until the remainder is obtained.

Oct 16, 2026

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