• 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?