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An in-depth explanation of polynomials, their classification based on terms and degree, and includes examples of monomials, binomials, and trinomials. It covers the concepts of degree of a monomial, degree of a polynomial, linear and quadratic polynomials, and the use of function notation.
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Classifying Polynomials 663
Lesson 11-
Classifying Polynomials by Numbers of Terms
Recall that a term can be a single number, variable, or product of numbers and variables. In an expression, addition (or subtraction, which is “adding the opposite”) separates terms.
Polynomials are identified by their number of terms. A monomial is a single term in which the exponent for every variable is a positive integer. A polynomial is an expression that is either a monomial or sum of monomials. Polynomials with two or three terms are used so often they have special names. A binomial is a polynomial that has two terms. A trinomial is a polynomial that has three terms. Here are some examples.
Monomials Not Monomials
6 x –16 t^2 x^2 y^4
6 x + y (a binomial) –16 t –2^ (negative exponent on a variable) __^ x^2 y^4 (variables divided) Binomials Not Binomials
x + 26 √ 2 x __ 3 -^ y^ 3 0.44 - 2 –10 pq^4
26 x √ 2 (monomial)
Trinomials Not Trinomials
18 x^2 + 5 x + 9 a^2 + 2 ab - b^20 pq + qr + rp
(15 x^2 )(5 x )(9) (monomial) a –2^ + 2 ab - b –20^ (negative exponent on variables) _________^1 pq + qr + rp (variables divided)
There are no special names for polynomials with more than three terms.
Vocabulary monomial polynomial binomial trinomial degree of a monomial degree of a polynomial linear polynomial quadratic polynomial
BIG IDEA Polynomials are classified by their number of terms and by their degree.
Refer to the graph of a function.
1 ļ 2 ļ 1 ļ 3 ļ 4 ļ 5 ļ 6 ļ 7
2 3 4 5 67
ļ 5 ļ 4 ļ 3 ļ 2 ļ 1 1 2 3 4 5
y
x
a. State the domain of the function. b. State the range of the function. c. State the x-intercepts. d. State the y-intercept.
Mental Math
664 Polynomials
Classifying Polynomials by Degree
Every nonconstant term of a polynomial has one or more exponents. For example, 3 x^2 has 2 as its exponent. 10 t has an unwritten exponent of 1, since t^1 = t. 15 a^2 b^3 c^4 has 2, 3, and 4 as its exponents.
The degree of a monomial is the sum of the exponents of the variables in the expression.
3 x^2 has degree 2. 10 t has degree 1. 15 a^2 b^3 c^4 has degree 2 + 3 + 4, or 9.
The degree of a single number, such as 15, is considered to be 0 because 15 = 15 x^0. However, the number 0 is said not to have any degree, because 0 = 0 · x n , where n could be any number. The degree of a polynomial is the highest degree of any of its monomial terms after the polynomial has been simplified. For example, 6 x - 17 x^4 + 8 + x^2 has degree 4. p + q^2 + pq^2 + p^2 q^3 has degree 5 (because 2 + 3 = 5).
QY
When a polynomial has only one variable, writing it in standard form makes it easy to determine its degree. When the polynomial in x above is written in standard form, the degree is the exponent of the leftmost term.
–17 x^4 + x^2 + 6 x + 8 has degree 4.
Function notation can be used to represent a polynomial in a variable. For example, let p ( x ) = –17 x^4 + x^2 + 6 x + 8. Then values of the polynomial are easily described. For example, p (2) = –17 · 2 4 + 2 2 + 6 · 2 + 8 = –248.
QY
The polynomial p + q^2 + pq^2 + p^2 q^3 is a polynomial in p and q. There is no standard form for writing polynomials that have more than one variable, like this one. However, sometimes one variable is picked and the polynomial is written in decreasing powers of that variable. For example, written in decreasing powers of q, this polynomial is p^2 q^3 + pq^2 + q^2 + p , or, to emphasize the powers of q, p^2 q^3 + ( p + 1) q^2 + p.
A polynomial of degree 1, such as 13 t - 6, is called a linear polynomial. A polynomial of degree 2, such as 2 x^2 + 3 x + 1 or w, is called a quadratic polynomial. Linear and quadratic polynomials whose coefficients are positive integers can be represented by tiles.
Chapter 11
QY Classify each polynomial by the number of its terms and its degree. a. x^6 + x^7 + x^5 b. 8 y^3 z^2 - 40 yz^6 c. 4 __ 3 πr 3
QY If p(x) = –x + 3 + 4x^4 , what is p(–2)?
666 Polynomials
In 3–6, an expression is given.
a. Tell whether the expression is a monomial. b. If it is a monomial, state its degree.
In 11 and 12, what polynomial is represented by the tiles?
1 1 1 1 1
x^2
x
x^2
x
x^2
x
x
1
x
1
x
n
n 1
n n 1
In 14–18, an expression is given.
a. Show that the expression can be simplified into a monomial. b. Give the degree of the monomial.
Chapter 11
Classifying Polynomials 667
In 21–24, give the degree of these polynomials used to find length, area, and volume of geometric figures.
b. Write a monomial with two variables whose degree is 70.
8 x - 5
?
REVIEW
Lesson 11-