Reasonable Basic Algebra by A. Schremmer - HTML preview

14.3 Terms

We now take a major step in the development of the "power language" by allowing unspecified numerators when writing monomials.

1. We begin by going back to the distinction between a formula and a sentence. Recall that by itself a formula, for instance an inequation or an equation, is neither TRUE nor FALSE and that only a sentence can represent a relationship among collections in the real-world. EXAMPLE 11. The inequation in Apples

x < 5

is neither true nor false because it does not represent a relationship among collections term in the real world. (2 Apples represent a collection in the real world but x Apples does not represent a collection in the real world.)

Given a formula, it is only when we replace the unspecified numerator by a specific numerator that we get a sentence which is then either true or FALSE depending on whether it fits the real world or not.

EXAMPLE 12. Given the formula in Apples

x < 5 when we replace the unspecified numerator x by the specific numerator 8 we get the sentence in Apples

X<5 \ X :=8

that is the sentence

8 Apples < 5 Apples which is false but if, instead, we replace the unspecified numerator x by the specific numerator 3 we get the sentence in Apples

X < 5 \x:=3

that is the sentence

3 Apples < 5 Apples

which is TRUE

2. Similarly, just as a formula can be viewed as an "incomplete" sentence, a term will be an "incomplete" specifying-phrase.

EXAMPLE 13. Given the term in Apples

x + 5 when we replace the unspecified numerator x by the specific numerator 8 we get the specifying-phrase in Apples

X + 5 \x:=S

that is the specifying-phrase

8 Apples + 5 Apples which we may or may not chose to identify.

Of course, an unspecified numerator is the simplest possible kind of term. EXAMPLE 14. Given the term in Apples

x when we replace the unspecified numerator x by the specific numerator 8

X + 5 \x:=8

we get

8 Apples

3. When replacing in a monomial specifying-phrase a specific numerator by an unspecified numerator to get a term, we will use

• The letters a,b,c,d... for unspecified signed coefficients,

• The letters x,y,x ... for unspecified signed bases,

• The letters m,n,p... for unspecified plain exponents. EXAMPLE 15.

monomial term monomial

c x y m

The reason we will use the letters m,n,p... to stand only for plain exponents (rather than for signed exponents) is that the sign of a exponent is most important since it distinguishes between multiplication and division and we will almost always have to specify it as in the above example.

In the rare cases when the sign of the exponent will not matter, we will write the symbol ±, read "plus or minus" in front of the letter as in the following example.

Example 16.

c x x ±n is intended to cover both the case

c x x +n and the case

c x x~ n It is also customary to let the separator x go without saying. However, this tends to cause mistakes unless we make sure we read the monomial specifying-phrase according to whether the signed exponent is positive or negative, as

• "Coefficient multiplied by number of copies of the base" when the exponent is positive,

• "Coefficient divided by number of copies of the base" when the exponent is negative.

Example 17.

■ We read cx +n as "c multiplied by n copies of x" because the exponent is positive,

■ We read ay~ p as "a divided by p copies of y" because the exponent is negative.

14.4 Monomials

In the rest of this text, coefficients and exponents will always be specified and only the base will remain unspecified. Out of habit, we shall mostly use the letter x for the base.

1. Monomial specifying-phrases in which the base is unspecified are called monomial terms or monomials for short.

Example 18. The following

-3x+ 5

+5.23X" 3

-1600a;4

+Ax+ 2

are monomials but

+4a ,+2.5

coefficient power

14.4. MONOMIALS 175

is not a monomial because 2.5 copies doesn't make sense. Laurent monomial

T ,. ,. ,. . , , T . . , .„ . plain monomial

a. Just as, earlier on, we distinguished Laurent monomial specilymg-

phrases (those whose exponent can have any sign) from plain monomial specifying phrases (those whose exponent can be only positive or 0), we could distinguish in the same manner Laurent monomials from plain monomials. However, since we will be using mostly Laurent monomials, we will just use monomial to mean Laurent monomial.

b. In a monomial we will distinguish:

• the coefficient, which is the number to be multiplied or divided by the copies of the base

• the power, which is the base together with the exponent. In other words, the separator x, whether it is actually written or goes without saying, separates the coefficient from the power. EXAMPLE 19. In the monomial —3s +4 , —3 is the coefficient and x +A is the power.

c. Thus, monomials, as well as monomial specifying-phrases, look very much like ordinary number-phrases (as opposed to specifying number-phrases):

• The coefficient in a monomial—or monomial specifying-phrase—is like the numerator in an ordinary number-phrase,

• The power in a monomial—or monomial specifying-phrase—is like the denominator in an ordinary number-phrase.

EXAMPLE 20. Monomial specifying-phrases like

17.52 x 2+ 3 (with x as separator) and monomials like

17.52 x +3 (without separator) look, and to a large extent will behave, very much like:

■ Ordinary number-phrases like

17.52 Meters in which there is no need for a separator between the numerator and the denominator,

■ Metric number-phrases like

17.52 Kilo Meters in which there is no need for a separator between the numerator and the denominator,

■ Base TEN number-phrases like

17.52 x Ten +3 Meters where x is a separator between the numerator and the denominator,

■ Exponential number-phrases like

17.52 x 10 +3 Meters where x is a separator between the numerator and the denominator.

We will investigate how far the similarity goes in the following chapters.

2. When we multiply or divide a first monomial by a second monomial, we proceed just as we did with monomial specifying-phrases, that is we can proceed either:

• The long way which is to go back to in-line templates and then proceed according to whether we are dealing with multiplication or division

• The short way which is to use the following

THEOREM 8 (EXPONENT THEOREM). In order to:

i. Multiply two monomials ax ■ and bx , we multiply the coefficients

and oplus the exponents. -

ax x

bx ±n = abx ±m ® ±n

ii. Divide two monomials ax ±m and bx ±n , we divide the coefficients and ominus the exponents:

ax ± m _i_ bx^ n = -r^ m Si n b

We now look at a few examples. Example 21. Given

[-17.89 x x +547 ~\ x ["-11.06 x x+ 312 ]

instead of replacing each monomial by the corresponding in-line template, change the order of the multiplications and write the resulting monomial:

-17.89 x x +M7 \ x -11.06 xx +312

-17.89 x x x x x • ■ ■ x x

547 copies of x

X

-11.06 x x x x x • • ■ x x

312 copies of x

-17.89 x —11.06 x x x x x • • • x x

547+312 copies of x

-17.89 X -11.06] X . T + (547+312)

17.89 x 11.06] x x +859

we can use the EXPONENT THEOREM:

-I7.M) <x+ M7 } x [-11.06 x X+ 312 ] = -I7.S!) > -11.00 < x +547 ( ' ;! -

859

+ 17.89 x 11.06] x x +

Example 22. Given

[+17.89 x x +547 } x [-11.06 x x312 } instead of replacing each monomial by the corresponding in-line template, change the

14.4. MONOMIALS

177

order of the multiplications and write the resulting monomial:

+ 17.89 x x +547 l x ["-11.06 x re" 312 !

+ 17.89 x x x x x • • • x x

547 copies of x

-11.06

x x x x • • • x x

-17.89 x -11.06 x

312 copies of x X X X X ■ ■ • XI"

17.89 x 11.06 x

547 copies of x X X X X • ■ • X x

312 copies of x X X X X --ou- X X X X X ■ ■ ■ X X '

12 copies of x 547—312 copies of x

X X X X

12 copies of x

17.89 x 11.06 x x

,+(547-312)

17.89 x 11.06 x x

,+235

+547 0 -312

we can use the EXPONENT THEOREM:

["+17.89 x .x +547 j x [-11.06 x a:" 312 ] = ["+17.89 x —11.06] x x +5

= -[l7.89x 11.06] x a;+( 547 312 ) = -I"l7.89 x 11.06] x x +235

Example 23. Given

["-17.89 x a;" 547 ] x ["+11.06 x x +312 ]

instead of replacing each monomial by the corresponding in-line template, change the order of the multiplications and write the resulting monomial:

-17.89 x x

-547

] x 1+11.06 x a; +312 ]

-17.

X X X X • ■ ■ X X

547 copies of x

-17.89 x +11.06 x

+11.06 x x x x x ■ ■ ■ x x

312 copies of x X X X X • • • X X~

312 copies of x X X X X • ■ • X X

CHAPTER 14. LAURENT MONOMIALS

17.89 x 11.06 x

X X £ X

■_^3

X X X X

2 copies of x

17.89 x 11.06] x £-(547-312) 17.89 x 11.061 x a; -235

x x x x x

547—312 copies of x.

we can use the EXPONENT THEOREM:

-17.

x x

-547

x 11.06 x x

+312

547 © +312

17.89 x +11.06 x x

17.89 x 11.06] X ^-(547-312)

17.89 x 11.06 x x

-235

Example 24. Given

[+17.89 x X+ 547 ] +- 1+11.06 x x+ 312 ]

instead of replacing each monomial by the corresponding in-line template, change the order of the multiplications, rewrite as fraction, multiply by the reciprocal instead of divide, and write the resulting monomial:

+17.

x x

+547

+ 11.06 x x

+312

"+11.06 x x x x x

+ 17.89 x x x x x • ■ • x x

" v '

547 copies of x

+ 17.89 X X X X X • • ■ X X

s v- '

547 copies of x

X X X X ■ • • X X

312 copies of x

+ 17.89 + 11.06

17.89 11.06

17.89 11.06

547 copies of x

X X X X

X X

312 copies of x X X X X • -OCX- X X X X X

X X

547-312 copies of x

X X X X

■jji~&

X X

+ (547-312)

14.4. MONOMIALS

179

17.

11.06

-2:55

it is easier to use the EXPONENT THEOREM:

17.89 x x +547 \ -=- +11.06 x a;+ 312

+ 17.89" + 11.06. " 17.89 11.06 17.89 11.06 17.89 11.06

x x

+547 9 +312

X X

+547 0 -312

X X

+ (547-312)

X X

+235

Example 25. Given

[l7.89 x x~ 547 j -r- [ll.06 x a;" 312 !

instead of replacing each monomial by the corresponding in-line template, change the order of the multiplications, rewrite as fraction, multiply by the reciprocal instead of divide, and write the resulting monomial:

17.89 x a;" 547

11.06 x a: -312 ]

17.6

547 copies of x

11.06

a; x a; x

x x

17.89

x x

x

547 copies of x

~X X X X

11.06

17.89 11.06

17.89 11.06

17.89 11.06

x

x x

312 copies of x

X X X X

X X

547 copies of x

X X X X ■ ■■X^Xr

X X

XX XX

-(547-312)

-X XX XX

X X

547—312 copies of a;-

CHAPTER 14. LAURENT MONOMIALS

17.89 11.06

x x

-235

it is easier to use the EXPONENT THEOREM:

17.89 x a:' 547 ! -r- 111.06 x a; -312

17.

X X

11.06

X X

-547 Q -312

-547 0 +312

-(547-312)

-235

Chapter 15

Polynomials 1: Addition,

Subtraction

While, as we saw in the preceding chapter, monomials behave very well with respect to multiplication and division in the sense that we can always multiply or divide a first monomial by a second monomial and get a monomial as a result, we will see that monomials behave very badly with respect to addition and subtraction. This, though, gives raise to a new type of term which will in fact play a fundamental role—to be described in the Epilogue at the end of this text—in the investigation of FUNCTIONS.

In the rest of this text, we will introduce and discuss the way this new type of terms behaves with respect to the four operations. These are the basics of what is called Polynomial Algebra.

15.1 Monomials and Addition

We begin by looking at the way monomials behave with regard to addition. The short of it is that, most of the time, monomials cannot be added.

1. One way to look at why monomials usually cannot be added is to observe that powers are to monomials much the same as denominators are to number-phrases. • Just like ordinary number-phrases need to involve the same denominator

in order to be added, monomials need to involve the same power to be

added.

Example 1. Just like

17.52 Meters + 4.84 Meters = 22.36 Meters we have that

17.52.T+ 6 + 4.84x+ 6 = 22.36a;+ 6

• Just like ordinary number-phrases that involve different denominators cannot be added and just make up a combination, monomials that involve different powers cannot be added and just make up a combination. Example 2. Just like

17.52 Feet + 4.84 Inches is a combination we have that

17.52x+ 6 + 4.84a; +4 is a combination 2. A more technical way to look at why monomials cannot be added when the powers are different is to try various ways of "adding" monomials and then to see what the results would be when we replace the unspecified numerator x by specific numerators.

EXAMPLE 3. Suppose we think that the rule for adding the monomials should be "add the coefficients and add the exponents". Then, given for instance the monomials

+7x~ 2 and - 3x+ 3 the rule "add the coefficients and add the exponents" would give us the following monomial as a result:

(+7©-3)x2 ®+ 3 that is

+4x +1 Now while, on the one hand, there is no obvious reason why this should not be an acceptable result, on the other hand, monomials are waiting for x to be replaced by some specific numerator. So, say we replace x by +4. The given monomials would then give:

^7 -21 + 7

+7x '

and

*:=+4 ( + 4) • (+4)

_ +7_ ~ +16 = 0.4375

3 * +3 L=+4 = 3 '(+ 4 )'(+ 4 )'(+ 4 )

= -192

which, when we add them up, gives us

-191.5625 But, when we replace x by +4 in the supposed result, we get

+ 4a;+1 U + 4 = + 4 '(+ 4 )

_IQ Laurent polynomial

reduced

So, in the end, the rule "add the coefficients and add the exponents" would not produce an acceptable result.

Even though, as it happens, no rule for adding monomials will survive replacement of x by a specific numerator, the reader is encouraged to try so as to convince her/him self that this is really the case.

15.2 Laurent Polynomials

A Laurent polynomial is a combination of powers involving:

• exponents that can be any signed counting numerator (including 0).

• coefficients that can be any signed decimal numerator EXAMPLE 4. All of the following are Laurent polynomials:

+22.71x+ 3 + 0.3a: 0 - 47.03a;+ 2 + 57.89a;3

+21.09a;4 - 33.99a;+ 2 + 45.02a;1 + 52.74a:+ 1 - 34.82a;+ 7

-30.18a: +6 - 41.02a: +5 + 5.6a;+ 4

+20.13a; +3 + 0.03a;+ 5 + 50.01a; 0 - 0.04x +1

-0.02a;7 + 18.03a; +6

1. While there is nothing difficult about what Laurent polynomials are, we need to agree on a few rules to make them easier to work with since, otherwise, it is not always easy even just to see if two Laurent polynomials are the same or not.

EXAMPLE 5. The following two Laurent polynomials are the same

+0.3a; 0 - 47.03a;+ 2 + 22.71a;+ 3 + 57.89a;3 +57.89a;3 + 22.71a;+ 3 + 0.3a; 0 - 47.03x+ 2 but the following two Laurent polynomials are not the same

+0.3a; 0 - 47.03a; +2 - 22.71a;+ 3 + 57.89a;3 +57.89a;3 + 22.71a; +3 + 0.3a; 0 - 47.03x+ 2

EXAMPLE 6. The following two Laurent polynomials are in fact the same

+2a:+ 3 + 6a;4 -6a; +3 + 4x~ 4 + 8a;+ 3 + 2a;4 a. The first thing we have to agree on is that Laurent polynomials must always be reduced, that is that monomials in a given Laurent polynomial that can be added (because they involve the same power) must in fact be added. EXAMPLE 7. Given the following Laurent polynomial

-6a: +3 + 4a;4 + 8a;+ 3 + 2x~ 4 it must be reduced to

CHAPTER 15. POLYNOMIALS 1: ADDITION, SUBTRACTION

ascending order of

exponents descending order of

exponents

+2x +3 +6x~ 4 before we do anything else.

b. The second thing we have to do is to agree on some order in which to write the monomials in a Laurent polynomial. i. We will agree that:

The monomials in a Laurent polynomial will and can only be written in either one of two orders:

• ascending order of exponents, that is, as we read or write a Laurent polynomial from left to right, the exponents must get larger and larger regardless of the coefficients.

• descending order of exponents, that is, as we read or write a Laurent polynomial from left to right, the exponents must get smaller and smaller regardless of the coefficients.

EXAMPLE 8. The following Laurent polynomial

-47.03a;+ 2 + 57.89a;

22.71a; +4 + 0.3a; 0

can only be written either in ascending order of exponents

+57.89 " 3 +0.3° - 47.03a;+ 2 + 22.71a;+ 4 or in descending order of exponents

+22.71a;+ 4 - 47.03a;+ 2 + 0.3x° + 57.89a;3 regardless of the coefficients.

ii. Which of the two orders is to be used depends on the size of the numerators with which x can be replaced:

• The ascending order must be used when x can be replaced only by small numerators,

• The descending order must be used when when x can be replaced only by for large numerators.

We will see the reason in a short while.

NOTE. When the size of what x stands for is unknown, it is customary,

even if for no special reason, to use the descending order of exponents.

c. The third thing we have to do is to introduce customary practices even though these practices will not be followed here. i. It is usual to write just plain exponents instead of positive exponents. EXAMPLE 9. Instead of writing

+57.89a;3 + 0.3a; 0 - 47.03a; + 2 + 22.71a; + 4 it is usual to write

+57.89aT 3 + 0.3a; 0 - 47.03a; 2 + 22.71a; 4 ii. It is usual not to write the exponent +1 at all. EXAMPLE 10. Instead of writing

+57.89x+ 3 + 0.3a; +2 - 47.03a; + 1 + 29.77x+ 4

15.2. LAURENT POLYNOMIALS

185

f- 29.77a; +4

-22.71a; +4 22.71x+ 4

it is usual to write

+57.89a; +3 + 0.3a; +2 - 47.03a; iii. It is usual not to write the power x° at all. Example 11. instead of

+57.89a;3 + 0.3a; ° - 47.03a; +2 -it is usual to write

+57.89a;3 + 0.3 - 47.03a;+ 2 -iv. Most of the time, the exponents of the powers will be consecutive but occasionally there can be missing powers.

EXAMPLE 12. The following Laurent polynomials in which the powers are consecutive are fairly typical of those that we will usually encounter.

-47.03a; +3 + 57.89a; +2 + 22.71a; +1 + 0.3a; 0 -47.03a; +1 + 57.89a; 0 + 22.71a;1

-47.03a;1 + 57.89a; 0 + 22.71a;+ 1 + 0.3.T+ 2

EXAMPLE 13. The following Laurent polynomials in which at least one power is missing are fairly typical of those that we will occasionally encounter.

-47.03a;+ 3 + 0.3a; 0

-47.03a; +2 + 57.89a; 0 + 22.71a;1

-47.03a;1 + 57.89a; 0 + 22.71a;+ 1 + 0.3.T+ 3

When working with a Laurent polynomial in which powers are missing, it is

much safer to insert in their place powers with coefficient 0.

EXAMPLE 14. Instead of working with

-47.03a; +3 + 13.3a; 0 it is much safer to work with

-47.03a;+ 3 +0a;+ 2 + 0x +1 + 13.3a; 0 2. Laurent polynomials are specifying-phrases and we evaluate Laurent polynomials in the usual manner, that is we replace x by the required numerator and we then compute the result.

a. EXAMPLE 15. Given the Laurent polynomial

-47.03a; +2 ©+13.3a;3 when x := —5

consecutive missing power evaluate

-47.03a; H

-13.3a;

-31

-47.03(-5)+ 2 © +13.3(-5)3 -47.03 © (-5)(-5)] '

-47.03 ©+25]©

L(-5)(-5)(-5) +13.3

-125

-1175.75©+0.1064 -1175.6436

diminishing }-,_ \yh e ri the coefficients are all single-digit counting numerators and

p am po ynomia we re pi ace x by ten, the result shows an interesting connection between

Laurent polynomials and decimal numbers.

EXAMPLE 16. Given the Laurent polynomial

Ax+ 3 + 7 x+ 2 + 9 x +1 + 8 x a + 2X1 + 5 a;" 2 + 6 ir~ 3 when x := 10 we get: 4x+ 3 + 7x +2 + 9x +1 +4x° + 2x~ x + 7x~ 2 + 7x~ 3 1 ln =

\x— 10

= 4 x 10+ 3 + 7 x 10+ 2 + 9 x lQx +1 + 8 x 10° + 2 x 10 _1 + 5 x 1(T 2 + 6 x 10~ 3 = 4 x 1000. + 7 x 100. + 9 x 10. + 8 x 1. + 2 x 0.1 + 5 x 0.01 + 6 x 0.001 = 4000. + 700. + 90. + 8. + 0.2 + 0.05 + 0.006

= 479 8. 256 which is the decimal number whose digits are the coefficients of the Laurent polynomial.

3. We are now in a position at least to state the reason for allowing only the ascending order of exponents and the descending order of exponents:

When we replace a; by a specific numerator and go about evaluating the Laurent polynomial, we evaluate, one by one, each one of the monomials in the Laurent polynomial. But what happens is that

• When x is replaced by a numerator that is large in size, the more copies there are in a monomial, the larger in size the result will be.

• When x is replaced by a numerator that is small in size, the more copies there are in a monomial, the smaller in size the result will be.

But what we want, no matter what, is that the size of the successive results go diminishing. So,

• When x is to be replaced by a numerator that is going to be large in size, we will want the Laurent polynomial to be written in descending order of exponents.

• When x is to be replaced by a numerator that is going to be small in size, we will want the Laurent polynomial to be written in ascending order of exponents.

For lack of time, we cannot go here into any more detail but the interested reader will find this discussed at some length in the Epilogue.

15.3 Plain Polynomials

A plain polynomial is a combination of powers involving:

• exponents that can be any positive counting numerator as well as 0.

• coefficients that can be any signed decimal numerator

In other words, a plain polynomial is a combination of powers that do polynomial not involve any negative exponent—but can involve the exponent 0. EXAMPLE 17. The following are plain polynomials:

-47.03a; +3 + 57.89a;+ 2 + 22.71a; +1 + 0.3a; 0 0.3a; 0 - 47.03a; +1 + 57.89a; +2 + 22.71a; +3

The following are not plain polynomials:

-47.03x+ 3 + 57.89x +2 + 22.71a; +1 + 0.3a; 0 - 22.43a;" 1 -22.43a;" 1 + 0.3a; 0 - 47.03a;+ 1 + 57.89a; +2 + 22.71a: +3

1. When we replace x by TEN in a plain polynomial whose coefficients are all single-digit counting numerators, the result is a counting number. EXAMPLE 18. Given the plain polynomial

4a;+ 3 + 7a;+ 2 + 9a; +1 + 8.t° when x := 10 we get:

4a;+ 3 + 7a; +2 + 9a; +1 + 4a;°| , =4x 10+ 3 + 7 x 10+ 2 + 9 x 10x +1 + 8 x 10°

= 4 x 1000 + 7 x 100 + 9 x 10 + 8 x 1

= 4000 + 700 + 90 + 8

= 4798 which is the counting number whose digits are the coefficients of the plain polynomial.

2. Just like decimal numerators are not really more difficult to use than just counting numerators—they just require understanding that the decimal point indicates which of the digits in the decimal numerator corresponds to the denominator 1 , Laurent polynomials are just as easy to use as just plain polynomials. This is particularly the case since, in the case of polynomials, we do not have to worry about the "place" of a monomial in a polynomial since the place is always given by the exponent

3. Just like decimal numbers are vastly more useful than just counting numbers, Laurent polynomials will be vastly more useful than plain polynomials for our purposes as the discussion in the EPILOGUE will show.

4. Since, from the point of view of handling them, there is not going to be any difference between Laurent polynomials and plain polynomials, we will just the word polynomial.

J But then of course, since Educologists have a deep aversion to denominators, they are sure to disagree.

CHAPTER 15. POLYNOMIALS 1: ADDITION, SUBTRACTION

add

like monomials

ffl

addition of polynomials

15.4 Addition

Just like combinations can always be added to give another combination, polynomials can always be added to give another polynomial. EXAMPLE 19. Just like the combinations

17 Apples & 4 Bananas and can be added to give another combination:

7 Bananas & 8 Carrots

17 Apples & 4 Bananas

7 Bananas & 8 Carrots

17 Apples & 11 Bananas & 8 Carrots

the polynomials

-17x+ 6 + 4x"

and

+7x~ 3 +8x+ 2

can be added to give another polynomial:

-17x+ 6 +4x3

+7x~ 3 4

8x+ 2

-17x+ 6 + lla;" 3 + 8a;+ 2 1. To add two polynomials with signed coefficients, we oplus the coefficients of like monomials that is of monomials with the same exponent. We will use the symbol EH to write the specifying-phrase that corresponds to the addition of polynomials.

EXAMPLE 20. Given the polynomials

-17a; +6 + 4a;^ 3 and +7x~ 3 + 8x +2 the specifying-phrase for addition will be

-17a;+ 6 + Ax3 ffl +7x3 + 8x+ 2 and to identify it, we will write

-17a;

+6

4x

+7x3 + 8x+ 2 = -17X+ 6 + [+4 © +7],

„-3

8x^

-\7x

+6

liar

8a;

+ 2

2. The only difficulties when adding polynomials occur when one is not careful to write them:

• in order—whether ascending or descending

• with missing monomials written-in with 0 coefficient

EXAMPLE 21. Given the polynomials

-17a;+ 3 -14a;+ 2 -8x° + 4a;1 and +7x+ 4 + 8.T+ 3 - lla;+ 1 - 4a;~ 2 consider the difference between the following two ways to write the addition of two polynomials:

When we do not write the polynomials in order and do not write-in missing monomials with a 0 coefficient, we get:

-17x+ 3 - Ux+ 2 -8x° +4X' 1 +7x +i + 8x+ 3 - llx +1 - Ax2

and it is not easy to do the addition and get the result:

+7x+ 4 - 9x+ 3 - Ux+ 2 - lla; +1 - 8x° + Ax" 1 - 4 X 2

■ When we do write the polynomials in order and we do write-in the missing monomials with a 0 coefficient, we get:

0x+ 4 - 17x+ 3 - Ux+ 2 + 0x +1 - 8x° + 4a;" 1 + Ox" 2 +7x+ 4 + 8x+ 3 + 0x+ 2 - llx +1 + 0x° + Ox" 1 - 4x" 2

+7x+ 4 - 9x+ 3 - Ux+ 2 - ll:r +1 - 8x° + 4a;" 1 - 4x" 2

where the result is much easier to get.

3. One way in which polynomials are easier than numerators to deal with is that when we add them there is no so-called "carry-over". The reason we have "carry-over" in arithmetic is that when dealing with combinations of powers of ten, the coefficients can only be digits. So, when we add, say, the hundreds, if the result is still a single digit, we can can write it down but if the result is more than nine, we have no single digit to write the result down and we must change TEN o