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Lesson
Classifying Polynomials 663
Lesson 11-
Classifying
11-2 Polynomials
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)
- xy ___^3 3 (monomial) 0.44 - 2 –10 p + q^4 (trinomial)
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.
- 17 x^11 4. 2 w –^4
- 1 __ 2 bh 6. 2 a^4 b^5
- Is xyz a trinomial? Explain your reasoning.
- Classify each polynomial by its degree and number of terms. a. x^2 + 10 b. x^2 + 10 x + 21 c. x^2 + 10 xy + y^2 d. x^3 + 10 x^2 + 21 x
- Write the polynomial 12 - 4 x - 3 x^5 + 8 x^2 in standard form.
- a. Write the polynomial a^3 - 3 ab^2 - b^3 - 3 a^2 b in standard form as a polynomial in a. b. Write the polynomial a^3 - 3 ab^2 - b^3 - 3 a^2 b in standard form as a polynomial in b.
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
- Fill in the blank with always, sometimes but not always , or never. Explain your answer. The degree of the sum of two polynomials is?^ greater than the degree of either polynomial addend.
APPLYING THE MATHEMATICS
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.
- 10 x - 14 x 15. 10 x (‒ 14 x )
- (5 n^3 )(6 n ) 2 17. xy + yx
- 12 x^4 - (3 x^4 + 2 x^4 + x^4 )
Chapter 11
Classifying Polynomials 667
- Let p ( x ) = 50 x^3 + 50 x^2 + 50 x + 50 and q ( x ) = 100 x^3 + 120 x^2 + 140 x + 160. Give the degree of each polynomial. a. p ( x ) b. q ( x ) c. p ( x ) + q ( x ) d. p ( x ) - q ( x )
- Repeat Question 19 if p ( x ) = x^200 - x^100 + 1 and q ( x ) = x^100 - x^200 + 1.
In 21–24, give the degree of these polynomials used to find length, area, and volume of geometric figures.
- perimeter of a triangle = a + b + c
- volume of a circular cone = 1 __ 3 π r^2 h
- area of a trapezoid = 1 __ 2 hb 1 + 1 __ 2 hb 2
- surface area of a cylinder = 2 π r^2 + 2 π rh
- a. Write a monomial with one variable whose degree is 70.
b. Write a monomial with two variables whose degree is 70.
- Complete the fact triangle below and write the related polynomial addition and subtraction facts. - + 3 x + 2
8 x - 5
?
- a. Give an example of two trinomials in x of degree 5 whose sum is of degree 5. b. Give an example of two trinomials in x of degree 5 whose sum is not of degree 5.
- a. Write 318 and 4,670 as polynomials with 10 substituted for the variable. b. Add your polynomials from Part a. Is your sum equal to the sum of 318 and 4,670?
REVIEW
- a. If you received $1,000 as a present on the day you were born, and the money was put into an account at an annual scale factor of x, how much would be in your account on your 18th birthday? b. Evaluate the amount in Part a if x = 1.05. (Lesson 11-1)
Lesson 11-
