Polynomials Necessary Terms for Success Welcome back! We will now tackle the world of polynomials. Before we get started with performing operations or using polynomials for applications, we will need some basic vocabulary so we will all know what the different terms mean. Here are some terms you can expect to encounter. I have put the meanings of these words into the simplest terms possible, so that understanding them will be easier for you. Monomial letters and numbers that are multiplied. (i.e. : 5x or 3 or 6xy or Binomial - two monomials added or subtracted. (i.e. : 5x + 3 or 6xy 7 Trinomial three monomials added or subtracted. (i.e. : 6xy - 5x + 3) 3 x y ) 3 7x y ) Polynomial one or more monomials added or subtracted. (i.e. : 5x or 6xy-3 or 6xy - 5x + 3 or 5x 3 6xy 7x y 3 + + ) Descending Order (Note: this is the preferred method for ordering your answers.) Order the monomials in the polynomial so that the monomial with the highest power of the letter that comes first in the alphabet, out of the letters given, is first. Next order the monomials, so the exponents of this letter go down. If two terms have the same power for the letter that comes first in the alphabet, then look at the next letter in the alphabet. (i.e. : 3 5 7x y + 5xy 4xy y + 3 ) 5 3 Ascending Order The opposite of descending. (i.e. : 3 y 4xy + 5xy 7x y ) Like Terms monomials whose variables match perfectly, including the exponents. (i.e. : x and 3x, or 6xy and 8xy, or 5 5 7 x y and 9x y ) Coefficient the number in front of the variables in a monomial. (i.e. : in the monomial 6x, the coefficient is 6; in the monomial 7xy, the coefficient is 7)
Degree of polynomials is somewhat complicated. It is the exponent of the leading term in a polynomial. Look at the following examples: 3 x + x + x + 3 has a degree of 3. 3 x has a degree of, as x = x. 3 4 x y + x has a degree of 5 (add up the exponents of x and y in the first term). has a degree of 0 (think of it as 0 x ). Now we have a base of terms that will be used throughout this chapter, so we are ready to perform operations with polynomials. Adding and Subtracting We are now ready to add polynomials. We will begin with adding and then add one small step for subtracting. Before we start any problem, we need to refer to a rule that we did in Chapter II of Take the Fear Out of Basic Algebra. Before we started a problem that required us to simplify, we changed all of the subtraction symbols to adding the opposite. Doing this step will also help us here, so we first want to change all subtraction to adding the opposite. Once we have changed our subtraction, adding is pretty easy, because all we have to do is combine like terms by adding the coefficients. There are two ways to go about combining like terms. Both methods will make it so that terms are automatically in descending order for your answers. The first method I call the crossing out method and the second is like old-fashioned addition from grade school. Both methods work, so pick the one that you like the best and use it. We will first look at the crossing out method. For this method, you will first change all subtraction to adding the opposite, then glance through all of the terms and find all of the like terms with the highest exponent on the variable that comes first in the alphabet. Add the coefficients of these terms and write the answer down, then glance through and find all of the like terms with the next highest exponent. Add the coefficients of these terms and write down the answer as the next term in your answer to the problem. Continue this with all of the exponents until all of the terms have been combined and you will have your answer in descending order. Be sure to only combine like terms! Let s look at a problem and use this method, then we can try the other method.
Example 3 3 ( x x + ) + ( x + x ) 5 3 4 6 7 8 This is a trinomial added to another trinomial. 3 3 ( x + x + ) + ( x + + x ) First change all subtraction to adding the opposite. 5 3 4 6 7 8 Now find the highest exponent, which is 3, so we 3 5x + 6x + 5x 3 x x x 3 3 3 need to combine the - 3 and the 8 to get a 5. 3 The 5 x will be first in our answer. Now cross out the two terms we just added, so we do not look at them again. The next highest exponent is. There is only one term with a on the x, so 3 we will write the 6 after the 5. + x We also only have one term with a for the exponent, which is the 5 x, so it can be written next. (If a variable does not have a visible exponent, there is an implied one. If you need to write in the one as a reminder, for now go ahead and do it. It will look like 5 x, but you do not need the one.) Finally, we have the numbers 4 and -7, which combine to get -3. Now cross out the two terms we just added, so we do not look at them again. Notice that there aren't any terms left, so we are done. The answer is: x To do the same problem with old fashioned addition, we put the first polynomial in descending order with a space for every exponent, even if one is missing, then line up the like terms of the second one directly underneath the first one. Draw a line under them and add straight down just like old-fashioned addition as seen here for this problem: + + 3 3 x 5x 4 3 + 8x + 6 x 7 3 5x + 6x + 5x 3
Example To subtract, we add one step in at the beginning. To illustrate this step, let s look at a problem: ( 5 x 6 x 7 ) ( 8 x 7 x 4 ) + +. If we change all of the subtraction to adding the opposite, we need to remember that when there is not a number in front of a parenthesis, there is an invisible, so changing the signs gives us: ( 5x + 6x 7) + ( 8x + 7x + 4) +. We would now distribute the into the second parenthesis to get: ( 5x 6x + 7) + ( 8x + 7x + 4) + Notice the result is that the signs in the second parenthesis are changed to the opposite of what they were, and the sign between the two polynomials is now addition, so as a shortcut, we can just change all of the signs in the parenthesis that we are subtracting and change the problem to addition. Our problem is now an addition problem, so we can either use the crossing out method or line them up like old-fashioned addition. Let s line them up to get 5x + 6x + 7 + + + 8x 7x 4 + 3x 3x Example 3 Let s try one more subtraction problem. ( x + x ) ( x x + ) 6 4 5 9 First let's change all of the subtraction ( x + x + ) + ( + x + x + ) in the first parenthesis to adding the opposite, then change the subtraction in the middle to addition and change all signs in the second parenthesis, to get: 6 4 5 9 Now we can do the crossing out 6x method or line them up. Let's line them up. + 4x + + + + 5x 9x x + 3x 3 answer Now you are ready to tackle problems in your homework that deal with adding and subtracting polynomials. Keep in mind that there is nothing wrong with referring back to this book and/or your class notes as you work through your homework. Eventually wean yourself off of looking
back, but as you are learning, looking back at notes is an important part of the learning process. When you are done, come back and we will multiply polynomials.