Exponents and Polynomials - XYZ Custom Plus

Adding and Subtracting PolynomialsPreviously we wrote numbers in expanded form to show the place value of eachof the digits. For example, the number 456 in expanded form looks like this:456 = 4 ⋅ 100 + 5 ⋅ 10 + 6 ⋅ 1If we replace 100 with 10 2 , we have9.2ObjectivesA Add polynomials.B Subtract polynomials.C Find the value of a polynomial.456 = 4 ⋅ 10 2 + 5 ⋅ 10 + 6 ⋅ 1If we replace the 10’s with x’s, we get what is called a polynomial. It looks likethis:Examples now playing atMathTV.com/books4x 2 + 5x + 6Polynomials are to algebra what whole numbers written in expanded form areto arithmetic. As in other expressions in algebra, we can use any variable wechoose. Here are some other examples of polynomials:5a + 3 y 2 − 2y + 4 x 3 − 2x 2 + 5x − 1A Adding PolynomialsWhen we add two whole numbers, we add in columns. That is, if we add 234and 345, we write one number under the other and add the numbers in the onescolumn, then the numbers in the tens column, and finally the numbers in the hundredscolumn. Here is how it looks:Instructor NotePolynomials are introduced hereby analogy rather than by a formaldefinition. I avoid formal definitionswhen I think it will make a simpleidea too complicated. In this case Ithink students can add and subtractpolynomials without knowing theformal definition of a polynomial.2343455792 ⋅ 10 2 + 3 ⋅ 10 + 43 ⋅ 10 2 + 4 ⋅ 10 + 55 ⋅ 10 2 + 7 ⋅ 10 + 9We add polynomials in the same manner. If we want to add 2x 2 + 3x + 4 and3x 2 + 4x + 5, we write one polynomial under the other, and then add in columns:2x 2 + 3x + 43x 2 + 4x + 55x 2 + 7x + 9The sum of the two polynomials is the polynomial 5x 2 + 7x + 9. We add only thedigits. Notice that the variable parts (the x’s) stay the same, just as the powers of10 did when we added 234 and 345. The reason we add the numbers, while thevariable parts of each term stay the same, can be explained with the commutative,associative, and distributive properties. Here is the same problem again, butthis time we show the properties in use:(2x 2 + 3x + 4) + (3x 2 + 4x + 5)= (2x 2 + 3x 2 ) + (3x + 4x) + (4 + 5) Commutative andassociative properties= (2 + 3)x 2 + (3 + 4)x + (4 + 5) Distributive property= 5x 2 + 7x + 9 Addition9.2 Adding and Subtracting Polynomials575