Polynomial.docx

What is Polynomial? A Polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example in three variables is x3 + 2xyz2 − yz + 1. Polynomials appear in a wide variety of areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra and algebraic geometry. A polynomial can have: constants (like 3, −20, or ½) variables (like x and y) exponents (like the 2 in y2), but only 0, 1, 2, 3, ... etc are allowed that can be combined using addition, subtraction, multiplication and division ... ... except ... ... not division by a variable (so something like 2/x is right out) So: A polynomial can have constants, variables and exponents, but never division by a variable.

Polynomial or Not? These are polynomials: 

3x



x−2



−6y2 − (79)x



3xyz + 3xy2z − 0.1xz − 200y + 0.5



512v5 + 99w5



5 (Yes, "5" is a polynomial, one term is allowed, and it can even be just a constant!) And these are not polynomials



3xy-2 is not, because the exponent is "-2" (exponents can only be 0,1,2,...)



2/(x+2) is not, because dividing by a variable is not allowed



1/x is not either



√x is not, because the exponent is "½" (see fractional exponents) But these are allowed:



x/2 is allowed, because you can divide by a constant



also 3x/8 for the same reason



√2 is allowed, because it is a constant (= 1.4142...etc)

Monomial, Binomial, Trinomial There are special names for polynomials with 1, 2 or 3 terms: How do you remember the names? Think cycles! There is also quadrinomial (4 terms) and quintinomial (5 terms), but those names are not often used.

Can Have Lots and Lots of Terms Polynomials can have as many terms as needed, but not an infinite number of terms.

Variables Polynomials can have no variable at all Example: 21 is a polynomial. It has just one term, which is a constant. Or one variable Example: x4 − 2x2 + x has three terms, but only one variable (x) Or two or more variables Example: xy4 − 5x2z has two terms, and three variables (x, y and z)

What is Special About Polynomials? Because of the strict definition, polynomials are easy to work with. For example we know that: 

If you add polynomials you get a polynomial



If you multiply polynomials you get a polynomial

So you can do lots of additions and multiplications, and still have a polynomial as the result. Also, polynomials of one variable are easy to graph, as they have smooth and continuous lines. Example: x4−2x2+x

See how nice and smooth the curve is?

You can also divide polynomials (but the result may not be a polynomial).

Degree The degree of a polynomial with only one variable is the largest exponent of that variable. Example: The Degree is 3 (the largest exponent of x) For more complicated cases, read Degree (of an Expression) .

Standard Form The Standard Form for writing a polynomial is to put the terms with the highest degree first. Example: Put this in Standard Form: 3x2 − 7 + 4x3 + x6 The highest degree is 6, so that goes first, then 3, 2 and then the constant last: x6 + 4x3 + 3x2 − 7 You don't have to use Standard Form, but it helps.

Adding Polynomial Adding polynomials is just a matter of combining like terms, with some order of operations considerations thrown in. As long as you're careful with the "minus" signs, and don't confuse addition and multiplication, you should do fine. There are a couple formats for adding and subtracting polynomials, and they hearken back to the two methods you learned for addition and subtract of plain numbers, back when you were in grade school. First, you learned addition "horizontally", like this: 6+3=9 That is, you were given relatively small values, and you learned to do the addition — largely in your head, and by working horizontally. We can add polynomials in the same way, grouping any "like" terms and then simplifying the results. 

Simplify (2x + 5y) + (3x – 2y)

I'll clear the parentheses first. This is easy to do when adding, because there are no "minus" signs to take through any parentheticals. Then I'll group the like terms in accordance to their variables (keeping them in alphabetical order), and finally I'll simplify: (2x + 5y) + (3x – 2y) 2x + 5y + 3x – 2y 2x + 3x + 5y – 2y 5x + 3y These two terms are un-like (because they have different variables), so I cannot combine them. This means that I've gone as far as I can, so my hand-in answer is: 5x + 3y

Horizontal addition works fine for simple polynomials. But when you were adding plain old numbers, you didn't generally try to apply horizontal addition to adding numbers like 432 and 246; instead, you would stack the numbers vertically, one on top of the other, and then add down the columns (doing "carries", as necessary): 432+246672

You can do the same thing with polynomials. Here's how the above simplification exercise looks, when it is done "vertically" 

Simplify (2x + 5y) + (3x – 2y)

I'll put each variable in its own column; in this case, the first column will be the xcolumn, and the second column will be the y-column: 2x3x5x+5y−2y+3y I get the same solution vertically as I got horizontally. 5x + 3y The format you use, horizontal or vertical, is a matter of taste (unless the instructions explicitly tell you otherwise). Given a choice, you should use whichever format that you're more comfortable and successful with. Note that, for simple additions, horizontal addition (so you don't have to rewrite the problem) is probably simplest, but, once the polynomials get complicated, vertical is probably safest bet (so you don't "drop", or lose, terms and minus signs). One advantage of vertical polynomial addition over vertical numerical addition: there is never anything to "carry" from one column to the next. 

Simplify (3x3 + 3x2 – 4x + 5) + (x3 – 2x2 + x – 4)

I can add horizontally: (3x3 + 3x2 – 4x + 5) + (x3 – 2x2 + x – 4) 3x3 + 3x2 – 4x + 5 + x3 – 2x2 + x – 4 3x3 + x3 + 3x2 – 2x2 – 4x + x + 5 – 4 4x3 + 1x2 – 3x + 1 ...or vertically: 3x31x34x3+3x2−2x2+1x2−4x+1x−3x+5−4+1 Either way, I get the same answer. For my final hand-in answer, I'll remove the "understood" 1s. 4x3 + x2 – 3x + 1 Note that each column in the vertical addition above contains only one degree of x: the first column above (that is, the left-most column being added down) was the x3 column, the second column was the x2 column, the third column was the x column, and the fourth column was the constants column. This is

analogous to having a thousands column, a hundreds column, a tens column, and a ones column when doing strictly-numerical addition. And, just as we need to use zeroes to fill empty slots in hundreds columns (or whichever column has no digit), we need to leave spaces in vertical addition for any gaps in the powers of variables. 

Simplify (7x2 – x – 4) + (x2 – 2x – 3) + (–2x2 + 3x + 5)

It's perfectly okay to have to add three or more polynomials at once. I'll just go slowly and do each step throroughly, and it should work out right. First, I'll do the adding horizontally: (7x2 – x – 4) + (x2 – 2x – 3) + (–2x2 + 3x + 5) 7x2 – x – 4 + x2 – 2x – 3 + –2x2 + 3x + 5 7x2 + 1x2 – 2x2 – 1x – 2x + 3x – 4 – 3 + 5 8x2 – 2x2 – 3x + 3x – 7 + 5 6x2 – 2 Note the 1's in the third line. Any time I have a variable without a coefficient, there is an "understood" 1 as the coefficient. If it's helpful to me to write that 1 in, then I'll do so. Now, I'll do the adding vertically: 7x21x2−2x26x2−1x−2x+3x+0x−4−3+5−2 Either way, I get the same answer. For my hand-in answer, I won't include the "+0x" term. 6x2 – 2

Subtracting Polynomial Subtracting polynomials is quite similar to adding polynomials, but there are those pesky "minus" signs to deal with. If the subtraction is being done horizontally, then the "minus" signs will need to be taken carefully through the parentheses. If the subtraction is done vertically, then all that's needed is flipping all of the subtracted polynomial's signs to their opposites. 

Simplify (x3 + 3x2 + 5x – 4) – (3x3 – 8x2 – 5x + 6)

The first thing I have to do is take that "minus" sign through the parentheses containing the second polynomial. Some students find it helpful to put a "1" in front of the parentheses, to help them keep track of the minus sign. Here's what the subtraction looks like, when working horizontally: (x3 + 3x2 + 5x – 4) – (3x3 – 8x2 – 5x + 6) (x3 + 3x2 + 5x – 4) – 1(3x3 – 8x2 – 5x + 6) (x3 + 3x2 + 5x – 4) – 1(3x3) – 1 (–8x2) – 1(–5x) – 1(6) x3 + 3x2 + 5x – 4 – 3x3 + 8x2 + 5x – 6 x3 – 3x3 + 3x2 + 8x2 + 5x + 5x – 4 – 6 –2x3 + 11x2 + 10x –10 And here's what the subtraction looks like, when going vertically: x3−(3x3+3x2−8x2+5x−5x−4+6) In the horizontal addition (above), you may have noticed that running the negative through the parentheses changed the sign on each and every term inside those parentheses. The shortcut when working vertically is to not bother writing in the subtaction sign or the parentheses; instead, write the second polynomial in the second row, and then just flip all the signs in that row, "plus" to "minus" and "minus" to "plus". I'll change all the signs in the second row (shown in red below), and add down: x3–4x3−2x3+3x2+8x2+11x2+5x+5x+10x−4–6−10 Either way, I get the answer: –2x3 + 11x2 + 10x – 10 

Simplify (6x3 – 2x2 + 8x) – (4x3 – 11x + 10)

Here's the subtraction, done horizontally: (6x3 – 2x2 + 8x) – (4x3 – 11x + 10)

(6x3 – 2x2 + 8x) – 1(4x3 – 11x + 10) (6x3 – 2x2 + 8x) – 1(4x3) – 1(–11x) – 1(10) 6x3 – 2x2 + 8x – 4x3 + 11x – 10 6x3 – 4x3 – 2x2 + 8x + 11x – 10 2x3 – 2x2 + 19x – 10 Going vertically, I'll write out the polynomials, leaving gaps as necessary: 6x34x3−2x2+8x−11x+10 Then I'll flip all of the signs in the second line, and then add down: 6x3–4x32x3−2x2−2x2+8x+11x+19x–10−10 Either way, I get the same answer: 2x3 – 2x2 + 19x – 10 Are we limited to only adding or subtracting pairs of polynomials? No, not at all. Especially once you get to calculus, it is very likely that it will be necessary to combine three or more polynomials, some of which are added and others which are subtracted. Just take care to write things out neatly, and don't try to do too much in any one step. 

Simplify: (3x2 – 5x – 1) – (x3 + 2x2 + 4) + (9x3 + 5x2 – 3x – 2)

Okay; to make this easier on myself, I'm first going to flip all of the signs for the second parenthetical, because there's currently a "minus" sign in front of that polynomial. So that middle polynomial becomes: –x3 – 2x2 – 4 Then I'll set up my simplification (which now involves only addition) in the vertical format: x39x38x33x22x2+5x2+10x2−5x−3x−8x−1+4−2+1 Then my hand-in answer is: 8x3 + 10x2 – 8x + 1

Multiplying Polynomials Just as we can multiply numbers, so also we can multiply polynomials. And just as some numerical multiplication is easier than others, so it is with polynomials. The simplest multiplication involving polynomials is where we're taking a number through a set of parentheses. 

Simplify the following: –5  (2x2)

All I have to do here is multiply the –5 by the 2, while carrying the x2 along for the ride: –5  (2x2) (–5)(2)(x2) –10x2 

Simplify the following: 2  (3x + 1)

I have a number (being the 2) that I need to take through (or, using the technical terms, "distribute over") the parenthetical expression (being the 3x + 1). I'll show every step: 2  (3x + 1) 2  (3x) + 2  (+1) (2)(3)(x) + (2)(1) (6)(x) + 2 6x + 2 At this point, I'm left with two un-like terms, so I cannot combine or simplify any further. My answer is: 6x + 2 You may already have seen this sort of computation when you learned about simplifying with parentheses. (You likely won't need to use so many steps as I did above, at least not once you're comfortable with the process, and your instructor almost certainly won't be expecting this much. I'm being overly complete in this lesson in hopes that, by the time you're done, you're sure of what's going on and are comfortable with the process.) Moving up in complexity, we can multiply two single-term polynomials (called "monomials").



Simplify (5x2)(–2x3)

I've already done this type of multiplication when I was first learning about exponents, negative numbers, and variables. I'll apply the rules that I already know: (5x2)(–2x3) (5)(x2)(–2)(x3) (5)(–2)(x2)(x3) (–10)(x2+3) –10x5 I can't simplify any further, so I'm done. My answer is: –10x5 Usually (and in contrast to the exercise just completed), a monomial that's going to be taken through a parenthetical doesn't have parentheses around it. Instead, the multiplication is indicated simply by the "juxtaposition" of the monomial with (that is, by putting the monomial right next to) the parenthetical expression. This is called "multiplication by juxtaposition", and looks like this: 

Simplify –3x  (1 – x)

I'll need to be careful with my "minus" signs. –3x  (1 – x) –3x(1) + (–3x)(–x) –3x + (+3)(x)(+x) –3x + 3x2 The next step up in complexity is multiplying a monomial (rather than a plain number) through a multi-term polynomial. 

Simplify –3x(4x2 – x + 10)

To do this multiplication, I have to distribute the –3x through the parentheses: –3x(4x2 – x + 10) –3x(4x2) + (–3x)(–x) + (–3x)(+10) (–12)(x1+2) + (3)(x1+1) + (–30)(x) –12x3 + 3x2 – 30x The next step up in complexity is the multiplication of one two-term polynomial by another two-term polynomial (that is, one binomial by another binomial). This

is the simplest of the "multi-term times multi-term" cases. There are actually three ways to do this. Since this is one of the most common polynomial multiplications that you likely will be doing, I'll spend a fair amount of time on this. 

Simplify (x + 3)(x + 2)

The first way I can do this multiplication is by working "horizontally". Doing so, I will have to distribute twice, taking each of the terms in the first parentheses "through" each of the terms in the second parentheses. (x + 3)(x + 2) (x + 3)(x) + (x + 3)(+2) x(x) + 3(x) + x(+2) + 3(+2) x2 + 3x + 2x + 6 x2 + 5x + 6 This horizontal multiplication, while mathematically valid, is probably the most difficult and error-prone way to do this multiplication. The "vertical" method is much simpler. Think back to when you were first learning about multiplication. When you did small numbers, it was simplest to work horizontally: 3 × 4 = 12 But when you got to larger numbers, you stacked the numbers vertically and, working from right to left, took one digit at a time from the lower number and multiplied it, right to left, across the top number. For each digit in the lower number (first the ones digit, then the tens digits, then the hundreds digits, and so forth), you formed a new row underneath, stepping the rows off to the left as you worked from right-most digit to left-most digit in the lower number. Then you added down. For instance, you would probably not want to try to multiply 121 by 32 horizontally, but it's easy when you do it vertically:

You can multiply polynomials in this same manner, so here's the same exercise as above, but done "vertically" this time: 

Simplify (x + 3)(x + 2)

I need to be sure to do my work very neatly. First, I'll set up the multiplication:

...and then I'll multiply: Multiply the bottom +2by the top +3, and carry down the +6:

Multiply the bottom +2by the top x, and carry down the +2x:

Multiply the bottom xby the top +3, and carry down the +3x:

Multiply the bottom xby the top x, and carry down the x2:

Draw a horizontal line below the two new rows:

Carry the x2 down to the bottom:

Add the +2x and the +3x, bringing down a +5x:

Carry the +6 down to the bottom:

The completed vertical multiplication:

I get the same answer as before, when I multiplied horizontally: x2 + 5x + 6

Dividing Polynomials There are two cases for dividing polynomials: either the "division" is really just a simplification and you're just reducing a fraction (albeit a fraction containing polynomials), or else you need to do long polynomial division (which is explained on the next page). We'll start with reduction of a fraction. 22x+4 This "division" is just a simplification problem, because there is only one term in the polynomial that they're having me dividing by. And, in this case, there is a common factor in the numerator (top) and denominator (bottom), so it's easy to reduce this fraction. There are two ways of proceeding. I can split the division into two fractions, each with only one term on top, and then reduce each of the two fractions separately:

...or else I can factor out the common factor from the top and bottom, and then cancel off this common factor:

Either way, my answer is the same: x+2

Note: Some students try to "cancel" before factorization. This cannot work! Fractions have "understood" parentheses around their numerators or denominators. It is necessary to include these explicitly when typing fractions out sideways, such as "(2x + 4)/2", so it's clear what, exactly, is on top and what is underneath. Otherwise, the typed version would likely be mis-understood to mean "2x + 4/2 = 2x + 2", which is not what was intended. Students may try to do this:

This is wrong! Don't do this! Even when the fractions are typeset in the math-book upright way, don't forget that there are (invisible) parentheses around the numerator and denominator, especially if the top or the bottom of the fraction has more than one term. When simplifying polynomial fractions, you can never reach inside those "understood" parentheses around the numerator and denominator, ripping arms and legs off of the polynomials within! (The poor little polynomials' big brown eyes are welling up with tears, just thinking about it!) Instead, you must factor, and then only cancel off common factors, if any. In other words, for the exercise above, you must do the following:

This is the way to go. 7x21x3−35x2 Again, I can solve this in either of two ways. One way is to simplify by splitting up the sum and then simplifying each fraction separately:

The other way is to simplify by taking the common factor of the numerator and denominator out front and then canceling it off:

Either way, my answer is the same: 3x2 – 5x Note: Most books don't talk about the domain at this point. But if your book does, you will need to note, for the above simplification, that x cannot equal zero. Why? Because, in the original (unsimplified) form, letting x equal 0 would have caused division by zero. That's not allowed. So the original form could not allow x to equal zero. However, in the simplified form, there is no way to know about this original-form restriction. For the simplified form to be mathematically equal to the original expression, the simplified form would need to be "3x2 – 5x, for all x ≠ 0". But this is a technical point and, if your book doesn't mention anything about this now, then don't worry about it for the time being. x+3x(x+3)−2(x+3) I can split the difference in the numerator to get the difference of two fractions, and then I can reduce each fraction separately. Each will have a factor of x + 3 in the numerator which will cancel with the denominator.

Or, alternatively, I can note that the terms in the numerator do indeed have a common factor; it's just that this common factor is rather large. Since both terms in the numerator contain the factor "x + 3", then this is a common factor, and it can be factored out front. Then the big factor out front will cancel with the denominator:

Either way, my simplified answer is the same: x–2