• polynomial—a monomial or a sum or difference of monomials
• binomial—a polynomial made up of 2 monomials
• trinomial—a polynomial made up of 3 monomials
Identify Polynomials
State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.
A. A
B. B
C. C
D. D
A. yes, monomial
B. yes, binomial
C. yes, trinomial
D. not a polynomial
A. State whether 3x2 + 2y + z is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.
A. A
B. B
C. C
D. D
A. yes, monomial
B. yes, binomial
C. yes, trinomial
D. not a polynomial
B. State whether 4a2 – b–2 is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.
A. A
B. B
C. C
D. D
A. yes, monomial
B. yes, binomial
C. yes, trinomial
D. not a polynomial
C. State whether 8r – 5s is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.
A. A
B. B
C. C
D. D
A. yes, monomial
B. yes, binomial
C. yes, trinomial
D. not a polynomial
D. State whether 3y5 is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.
• degree of monomial—the sum of the exponents of all of its variables
• degree of polynomial—the greatest degree on any monomial in the polynomial
Degree of a Polynomial
A. Find the degree of 12 + 5b + 6bc + 8bc2.
Step 1 Find the degree of each term.
12: degree = 05b: degree = 16bc: degree = 1 + 1 or 28bc2: degree = 1 + 2 or 3
Step 2 The degree of the polynomial is the greatest degree, 3.
Answer: 3
Degree of a Polynomial
B. Find the degree of 9x2 – 2x – 4.
Find the degree of each term.
9x2: degree = 2 2x: degree = 14: degree = 0
Answer: The degree of the polynomial is 2.
A. A
B. B
C. C
D. D
A. 3
B. 2
C. 0
D. 1
A. Find the degree of 11ab + 6b +2ac2 – 7.
A. A
B. B
C. C
D. D
A. 0
B. 2
C. 4
D. 3
B. Find the degree of 3r2 + 5r2s2 – s3.
• standard form of a polynomial—when a polynomial is written with the monomials arranged in decreasing degrees
• leading coefficient—the coefficient of the term in a polynomial with the highest degree
Standard Form of a Polynomial
A. Write 9x2 + 3x6 – 4x in standard form. Identify the leading coefficient.
Answer: 3x6 + 9x2 – 4xthe leading coefficient is 3.
Step 2 Write the terms in descending order.
Step 1 Find the degree of each term.Degree: 2 6 1
Polynomial: 9x2 + 3x6 – 4x
Standard Form of a Polynomial
B. Write 12 + 5y + 6xy + 8xy2 in standard form. Identify the leading coefficient.
Answer: 8xy2 + 6xy + 5y + 12the leading coefficient is 8.
Step 2 Write the terms in descending order.
Step 1 Find the degree of each term.Degree: 1 1 2 3
Polynomial: 12 + 5y + 6xy + 8xy2
A. A
B. B
C. C
D. D
A. 3x7 + 9x4 – 4x2 –34x
B. 9x4 + 3x7 – 4x2 –34x
C. –4x2+ 9x4 + 3x7 –34x
D. 3x7 – 4x2 + 9x4–34x
A. Write –34x + 9x4 + 3x7 – 4x2 in standard form.
A. A
B. B
C. C
D. D
A. –72
B. 8
C. –6
D. 72
B. Identify the leading coefficient of 5m + 21 –6mn + 8mn3 – 72n3 when it is written in standard form.
Use a Polynomial
MEDICINE From 2000 to 2006, the number N (in thousands) of patients seen by a medical facility can be modeled by the equation N = t2 + 2.1t + 0.8 where t is the number of years since 2000. How many patients were seen in 2005?
Find the value of t, and substitute the value of t to find the number of patients.
Since t is the number of years since 2000, t equals 2005 – 2000 or 5.
Use a Polynomial
N = t 2 + 2.1t + 0.8 Original equation
N = 52 + 2.1(5) + 0.8 t = 5
N = 25 + 2.1(5) + 0.8 Simplify.
N = 25 + 10.5 + 0.8 Multiply.
N = 36.3 Simplify.
Answer: The number of patients in 2005 was 36.3 thousand or 36,300.
A. A
B. B
C. C
D. D
A. 63 pianos
B. 43 pianos
C. 87 pianos
D. 29 pianos
INSTRUMENTS From 1997 to 2005 the number P of grand pianos sold at a metropolitan store can be modeled by the equation P = 2t2 – 2t + 3, where t is the number of years since 1997. How many grand pianos were sold in 2004?