Operations on Polynomials

Definition A monomial is the product of real numbers and one or more variables with whole number exponents Some examples 2 𝑥 2 1 3 𝑎 𝑏 3 3 𝑤 𝑥 2 𝑦 3 𝑧 4 Although a monomial can have more than one variable, as two of the above examples show, we will mostly be concerned with monomials including a single variable (usually x)

Definition A polynomial is the sum of one or more monomials Note that a monomial by itself is also a polynomial (one with a special name); monomials are the same as the terms of a polynomial Two other polynomials with special names are binomial (2 terms) and trinomial (3 terms) Other examples 3 𝑎 2 𝑏−4𝑐 𝑑 2 (a binomial) 4 𝑥 3 +2 𝑥 2 +𝑥−1 6 (a monomial) 𝑥+𝑦+𝑧 (a trinomial)

Polynomials We will mostly be concerned with polynomials of one variable (usually x) So for our purposes, a polynomial expression is one of the form 𝑎 𝑛 𝑥 𝑛 + 𝑎 𝑛−1 𝑥 𝑛−1 +⋯+ 𝑎 1 𝑥+ 𝑎 0 Here, each 𝑎 𝑖 (where i is a number between 0 and n inclusive) is a real number and called the coefficient, and n is a whole number The values 𝑎 𝑛 is called the lead coefficient and must not equal zero The value 𝑎 0 is the constant; note that it is the only monomial without a variable x

Polynomials Let’s look at an example to see how this definition works 𝑎 𝑛 𝑥 𝑛 + 𝑎 𝑛−1 𝑥 𝑛−1 +⋯+ 𝑎 1 𝑥+ 𝑎 0 Suppose that we have a polynomial where 𝑛=4 is the largest exponent on the variable x Our polynomial is then 𝑎 4 𝑥 4 + 𝑎 3 𝑥 3 + 𝑎 2 𝑥 2 + 𝑎 1 𝑥+ 𝑎 0 Here, 𝑎 4 is the lead coefficient and 𝑎 0 is the constant In this example, no particular values have been given to the coefficients 𝑎 4 , 𝑎 3 , 𝑎 2 , 𝑎 1 , nor to 𝑎 0

Polynomials You will be working with polynomials where the coefficients and the constant have values, usually integer values In the previous example, let’s assign the following values 𝑎 4 =2, 𝑎 3 =1, 𝑎 2 =0, 𝑎 1 =−3, 𝑎 0 =−5 Then the polynomial becomes 2 𝑥 4 + 𝑥 3 −3𝑥−5 Note that the term with 𝑥 2 is missing since its coefficient is zero The lead coefficient is 2 while the constant is −5

Polynomials Because of the commutative property, 2 𝑥 4 + 𝑥 3 −3𝑥−5=−5+2 𝑥 4 −3𝑥+ 𝑥 3 However, most of the time we will want to write polynomials as shown on the left side of the equal sign That is, the term with the largest exponent is written first, then the term with the next largest exponent is written second, and so on, the last term being the constant This is know as the standard form of a polynomial

Guided Practice Use the commutative property of addition to rewrite the following polynomials in standard form −4 𝑥 2 +2 𝑥 4 −3 𝑥 5 −1 −7+2 𝑥 2 3 𝑥 2 −𝑥+4 𝑥 3 3+2 𝑥 4 −4𝑥

Guided Practice Use the commutative property of addition to rewrite the following polynomials in standard form −4 𝑥 2 +2 𝑥 4 −3 𝑥 5 −1=−3 𝑥 5 +2 𝑥 4 −4 𝑥 2 −1 −7+2 𝑥 2 =2 𝑥 2 −7 3 𝑥 2 −𝑥+4 𝑥 3 =4 𝑥 3 +3 𝑥 2 −𝑥 3+2 𝑥 4 −4𝑥=2 𝑥 4 −4𝑥+3

Degree of a Polynomial A number that will be useful in classifying polynomials later is the degree of a polynomial The degree of a polynomial is the value of its greatest exponent We define the degree of a constant to be zero, while zero itself has no degree

Degree of a Polynomial The following polynomials are of degree 4 𝑥 4 − 𝑥 4 −3 𝑥 3 +4 𝑥 2 +7𝑥+1 4−6 𝑥 4 2 𝑥 4 +𝑥 We could also say that the polynomials listed above are 4th degree polynomials

Guided Practice Give the degree of the polynomial. 3𝑥−5 𝑥 3 +7+ 𝑥 6 5 𝑥−1 4𝑥−1+2 𝑥 2

Guided Practice Give the degree of the polynomial. (Note that it is easier to tell at a glance the degree of a polynomial if it is written in standard form)

Operations on Polynomials Note the similarity between a polynomial, say 3 𝑥 4 +2 𝑥 2 +5𝑥+1, and the numeral 30251 They are not the same (necessarily), but if we expand the numeral we get 3⋅ 10 4 +2⋅ 10 2 +5⋅10+1 This suggest that we should think of a polynomial as, in a sense, a single number, and we can perform operations on pairs of polynomials Specifically, we can add, subtract, multiply, and divide them

Adding/Subtracting Polynomials When adding numbers in columns, as shown below, why is it necessary to line the digits up in a particular way? 4235 +41 The result will be incorrect if, say, the 4 in 41 is lined up with the 4 in 4235, and the 1 lined up with the 2, but why? Expanding both numbers gives us a clue 4⋅ 10 3 +2⋅ 10 2 +3⋅10+5 + 4⋅10+1

Adding/Subtracting Polynomials Following this example, how would you expect to add the following 4 𝑥 3 +2 𝑥 2 +3𝑥+5 +(4𝑥+1)

Adding/Subtracting Polynomials Following this example, how would you expect to add the following 4 𝑥 3 +2 𝑥 2 +3𝑥+5 +(4𝑥+1) We can take a lesson from how we numerals and add these in columns 4 𝑥 3 +2 𝑥 2 +3𝑥+5 + 4𝑥+1 The result is 4 𝑥 3 +2 𝑥 2 +7𝑥+6

Adding/Subtracting Polynomials Although our use of analogy to the addition algorithm you learned in elementary school suggests how to proceed with addition and subtraction of polynomials, it is not enough In fact, we must turn to the distributive property to come up with a rule for adding/subtracting polynomials Is the following true: 5+3 𝑥=5𝑥+3𝑥 Is this true: 5+3 𝑥=8𝑥

Adding/Subtracting Polynomials If 5+3 𝑥=5𝑥+3𝑥 and 5+3 𝑥=8𝑥, then is this true? 5𝑥+3𝑥=8𝑥 We can formulate a rule: we will say that all terms with the same variable and the same exponent on the variable are like terms Now we can say simply that like terms can be simplified by adding/subtracting their coefficients We are now ready to apply this to addition/subtraction of polynomials

Adding/Subtracting Polynomials Example: simplify the following 3 𝑥 3 −2 𝑥 2 +10𝑥−5 + 6 𝑥 2 −7𝑥+1 One way to do this is to add in columns, lining up the like terms 3 𝑥 3 −2 𝑥 2 +10𝑥−5 + 6 𝑥 2 −7𝑥+1 Now combine the coefficients; this is like adding numerals in columns, but remember that there is no carrying The result is 3 𝑥 3 +4 𝑥 2 +3𝑥−4

Adding/Subtracting Polynomials If a polynomial is “missing” a term, then use zero in its place Example: add 4 𝑥 4 +2 𝑥 2 −1 + − 𝑥 4 +3 𝑥 3 +2𝑥+1 4 𝑥 4 +0 𝑥 3 +2 𝑥 2 +0𝑥−1 − 𝑥 4 +3 𝑥 3 +0 𝑥 2 +2𝑥+1 3 𝑥 4 +3 𝑥 3 +2 𝑥 2 +2𝑥+0 We can leave zero out, so the final sum is 3 𝑥 4 +3 𝑥 3 +2 𝑥 2 +2𝑥

Adding/Subtracting Polynomials Subtracting polynomials requires special attention Remember that −𝑛=−1⋅𝑛 and also that 𝑎−𝑏=𝑎+(−𝑏) Now, consider this difference of polynomials 5 𝑥 2 +3𝑥−4 −(−2 𝑥 3 +4𝑥−1) Using our definition of subtraction this becomes 5 𝑥 2 +3𝑥−4 + − −2 𝑥 3 +4𝑥−1 But since the opposite of a number is the same as −1 times the number 5 𝑥 2 +3𝑥−4 + −1 −2 𝑥 3 +4𝑥−1

Adding/Subtracting Polynomials 5 𝑥 2 +3𝑥−4 + −1 −2 𝑥 3 +4𝑥−1 Now using the distributive property 5 𝑥 2 +3𝑥−4 + 2 𝑥 3 −4𝑥+1 We can now use column addition 2 𝑥 3 +0 𝑥 2 −4𝑥+1 + 5 𝑥 2 +3𝑥−4 2 𝑥 3 +5 𝑥 2 −𝑥−3

Adding/Subtracting Polynomials The effect of the previous example was to change the signs inside the parentheses and change subtraction to addition 5 𝑥 2 +3𝑥−4 − −2 𝑥 3 +4𝑥−1 This is the same as 5 𝑥 2 +3𝑥−4 +(2 𝑥 3 −4𝑥+1)

Guided Practice Add or subtract as required. 3 𝑥 3 −2 𝑥 2 +4 + −5 𝑥 3 +6𝑥+3 7𝑥−1 − 2 𝑥 2 +4𝑥+7 𝑥 4 +4 𝑥 2 −5 − 𝑥 4 +3 𝑥 3 −4 𝑥 2 −5 3 𝑥 4 −1 −(5 𝑥 4 +2 𝑥 3 +3𝑥−3)

Guided Practice Add or subtract as required. 3 𝑥 3 −2 𝑥 2 +4 + −5 𝑥 3 +6𝑥+3 =−2 𝑥 3 −2 𝑥 2 +6𝑥+7 7𝑥−1 − 2 𝑥 2 +4𝑥+7 =−2 𝑥 2 +3𝑥−8 𝑥 4 +4 𝑥 2 −5 − 𝑥 4 +3 𝑥 3 −4 𝑥 2 −5 =−3 𝑥 3 +8 𝑥 2 3 𝑥 4 −1 − 5 𝑥 4 +2 𝑥 3 +3𝑥−3 =−2 𝑥 4 −2 𝑥 3 −3𝑥+2

Practice 2 Handout