# The whys of subtracting polynomials

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Okay, so we're working our way through polynomials, with Jacobs, btw. No problems there. The concept of reversing the sign of each term being subtracted is understood. BUT...we have no idea why we are doing it. Any help? I've googled until my eyes are swimming, but am just not finding any explanations. We'd like a bit more info than "just do it, nevermind why".

Thank you so very much!

~Killian (a math weary mom)

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Do you mean something like "Why does (2x + 4) - (5x - 3) become 2x + 4 - 5x + 3? If not, disregard everything else I'm about to say.

Subtraction is essentially adding the negative; thus, 5 - 3 is the same as 5 + (-3). To make something negative, we multiply by -1. So (2x + 4) - (5x - 3) really means (2x + 4) + (-1)(5x - 3). When we multiply 5x - 3 by -1, we need to distribute the -1, so we get 2x + 4 + (-1)(5x) + (-1)(-3). Now (-1)(5x) is -5x, and (-1)(-3) is +3 (recall that a negative times a negative will be positive), so we get 2x + 4 - 5x + 3.

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So (2x + 4) - (5x - 3) really means (2x + 4) + (-1)(5x - 3).

This is my understanding too and how I would describe it.

Put that (-1) in paratheses, distribute, and then carry on.

:seeya:

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Okay, so we're working our way through polynomials, with Jacobs, btw. No problems there. The concept of reversing the sign of each term being subtracted is understood. BUT...we have no idea why we are doing it. Any help? I've googled until my eyes are swimming, but am just not finding any explanations. We'd like a bit more info than "just do it, nevermind why".

Thank you so very much!

~Killian (a math weary mom)

The why has to do with fundamentals of the real number system.

Be careful of the distinction:

Looking at the set of real numbers we have: negative numbers, zero, and the positive numbers.

The negative of a number a is defined to be the number -a such that

a + -a = 0.

A negative number is defined to be the negative of a positive number.

Now:

If a and b are arbitrary real numbers, then their difference, a-b is a real number. So either a-b is an element of 0 (that is a=b), a-b is an element of the positive numbers (that is a<b), or -(a-b)= (b-a) is an element of the positive numbers (that is b>a).

This is a reformulation of the Axiom which states: If a is a real number, then one and only one of the following statements is true: a is the unique member 0 of the set O; a is a positive number of the set P of positive numbers; -a is a member of the set P.

Here is a proof:

To show: (-a) + (-b) = -(a + b).

(1) (-a) + a = 0

(2) (-b) + b = 0

(3) 0 + (-b) + b = 0

(4) (-a) + a + (-b) + b = 0

(5) ((-a) + (-b)) + (a + b) = 0

(6) (-a) + (-b) = 0 - (a + b)

(7) (-a) + (-b) = -(a + b)

We can always think of subtraction as an addition problem (adding the opposite).

This distributive property of multiplication over addition to real numbers can be extended to polynomials as well.

In other words, it has to do with the order relationships of the real number scale and how we define them.

Edited by fractalgal
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I guess I read the question differently. I think the OP is asking for an application of subtracting polynomials. One answer would be in calculus when you are trying to find the area bounded by two polynomial curves.

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Killian,

This isn't what you asked for...but (Don't i sound like a typical homeschooler,always full of advice;)

Book 4 of The "Key to Algebra" series by Key Curriculum Press teaches this well. The key to.. series are booklets that teach different algebra concepts. They are inexpensive (~\$3.75 each), and your dc can work through them quickly and independently.

My dc had to take a few 'detours' in algebra to reinforce concepts.

You can vie it here.

http://www.christianbook.com/Christian/Books/product?item_no=53124&event=CF#curr

Susan

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Thank you all for taking the time to answer. Susan, we have already worked through the Keys to Algebra...but thank you for the recommendation. :smiles: And thank you, Caroline, Moni, and Kiana for working through various examples.

Fractagal, I certainly appreciate your detailed explanation of the whys. I have a very, very faint glimpse of understanding. :chuckles: Yes, this is what I was asking ( the "I know what to do, but why do we do it"). I do understand the application of the formula, but feel like, for lack of a better analogy, that we can speak a language fluently, but haven't the slightest idea what we are saying! It seems to me that the answer should be the same when subtracting polynomials whether one reverses the signs or not...but that is not always the case. Why can't we subtract without reversing them? What makes it necessary to reverse them? I know that "when one subtracts polynomials, one must reverse the signs of the polynomial being subtracted and then add"..that's the formula. But it's like a foreign language to me. If I understand you correctly, it has to do with the distributive property of multiplication over addition. (I'm still a bit murky on what that has to do with subtracting in this case.) But it still makes no sense that one cannot simply subtract.

At any rate, I certainly appreciate all your responses. And I will continue digging around, rooting deeper, learning more..about this mysterious world called math. :grins:

Thanks again,

~Killian

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Thanks...yes, that is what I am asking. I do understand that subtracting is adding the negative. I just don't understand why one HAS to change it to addition. Why can't one simply subtract? It doesn't seem as though the formula is being given to make the problem easier to solve. It seems to be being taught as a concrete concept. And there are times when if one works the problem both ways (as a straight subtraction problem and as a reversed sign addition problem) that the answers do not concur with one another. Why? :grins:

Oh, for another math class in lieu of all the foreign lang/eng classes I took!

Thanks again,

~Killian

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While not absolutely correct (in mathematical terms) I have interpreted subtraction in this way (for some students who just can't seem to grasp the whole sign-change thing and can't handle big words and straight Math theory).

To put it simply--Subtraction does not exist. It never did and it never will. What we have always called 'subtraction' has ALWAYS been addition--addition in the opposite direction.

5-3= 2 right? Well that 3 in the problem is a positive 3. I have 5 apples and I remove 3 apples. Those 3 apples were real--they could be counted. The problem really should be written 5 - +3 = 2

Now read/map the problem using a number line. The "-" sign means to turn around (go backwards/change direction). You (always) start at the '0' move right 5, turn around and move 3. You will land on the 2.

Now try the problem 7 - (-5) = 12

Start at the '0' move right to the 7, turn around (left) then turn around again (right) and move 5. You will land on the 12.

In the first problem if we think 'the subtraction sign means to go backward' then you can ignore the positive sign in front of the 3.

In the second problem if you think 'the subtraction sign means to go backward' then you have to go backwards and then immediately go backwards again--they cancel each other out.

When you are subtracting polynomials and you have a pesky parenthesis you have to remember that you will be subtracting EVERYTHING inside those parenthesis... To keep from forgetting one it is usually easier to quickly change the signs (go backwards) on everything inside the parenthesis (getting rid of that pesky subtraction sign) and then combine the results (add).

5x - (2x-3) would be 5x + (-2x) + 3 and cleaned up that is 3x +3

Map it out... Start at 5x go back 2x and go back, turn around, 3.

I usually adjust the script of the above comments according to how much my student comprehends.

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What makes it necessary to reverse them? If I understand you correctly, it has to do with the distributive property of multiplication over addition. (I'm still a bit murky on what that has to do with subtracting in this case.) But it still makes no sense that one cannot simply subtract.

The reason that we want to redefine subtraction in terms of addition is that we want to use the algebraic field properties of the real number system: such as commutative, associative, and distributive laws. All order relationships in the real number system - all algebraic inequalities rest on two simple axioms regarding the set of positive numbers:

Axiom I: If a is a real number, then one and only one of the following statements is true: a is 0, a is a positive number, or a is a negative number (ETA: which means -a is positive when a is a negative number)

Axiom II: If a and b are members of the positive numbers, then the sum a+b and the product a*b are members of the positive numbers.

When subtraction is redefined in terms of addition then we can work with it much easier since addition is commutative and associative, but subtraction is neither commutative nor associative.

In other words, addition is much easier to work with than subtraction, so we don't want to mess with subtraction.

Edited by fractalgal
clarification

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