1. Vocabulary 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

2. Example 1 Identify Polynomials State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

3. A.A B.B C.C D.D Example 1 A.yes, monomial B.yes, binomial C.yes, trinomial D.not a polynomial A. State whether 3x 2 + 2y + z is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

4. A.A B.B C.C D.D Example 1 A.yes, monomial B.yes, binomial C.yes, trinomial D.not a polynomial B. State whether 4a 2 – b –2 is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

5. A.A B.B C.C D.D Example 1 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.

6. A.A B.B C.C D.D Example 1 A.yes, monomial B.yes, binomial C.yes, trinomial D.not a polynomial D. State whether 3y 5 is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

7. 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

8. Example 2 Degree of a Polynomial A. Find the degree of 12 + 5b + 6bc + 8bc 2. Step 1Find the degree of each term. 12: degree = 0 5b: degree = 1 6bc: degree = 1 + 1 or 2 8bc 2 : degree = 1 + 2 or 3 Step 2The degree of the polynomial is the greatest degree, 3. Answer: 3

9. Example 2 Degree of a Polynomial B. Find the degree of 9x 2 – 2x – 4. Find the degree of each term. 9x 2 : degree = 22x: degree = 1 4: degree = 0 Answer: The degree of the polynomial is 2.

10. A.A B.B C.C D.D Example 2 A.3 B.2 C.0 D.1 A. Find the degree of 11ab + 6b +2ac 2 – 7.

11. A.A B.B C.C D.D Example 2 A.0 B.2 C.4 D.3 B. Find the degree of 3r 2 + 5r 2 s 2 – s 3.

12. 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

13. Example 3 Standard Form of a Polynomial A. Write 9x 2 + 3x 6 – 4x in standard form. Identify the leading coefficient. Answer: 3x 6 + 9x 2 – 4x the leading coefficient is 3. Step 2Write the terms in descending order. Step 1Find the degree of each term. Degree: 261 Polynomial:9x 2 + 3x 6 – 4x

14. Example 3 Standard Form of a Polynomial B. Write 12 + 5y + 6xy + 8xy 2 in standard form. Identify the leading coefficient. Answer: 8xy 2 + 6xy + 5y + 12 the leading coefficient is 8. Step 2Write the terms in descending order. Step 1Find the degree of each term. Degree: 1123 Polynomial: 12 + 5y + 6xy + 8xy 2

15. A.A B.B C.C D.D Example 3 A.3x 7 + 9x 4 – 4x 2 –34x B. 9x 4 + 3x 7 – 4x 2 –34x C. –4x 2 + 9x 4 + 3x 7 –34x D.3x 7 – 4x 2 + 9x 4 –34x A. Write –34x + 9x 4 + 3x 7 – 4x 2 in standard form.

16. A.A B.B C.C D.D Example 3 A.–72 B. 8 C. –6 D.72 B. Identify the leading coefficient of 5m + 21 –6mn + 8mn 3 – 72n 3 when it is written in standard form.

17. Example 4 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 = t 2 + 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.

18. Example 4 Use a Polynomial N= t 2 + 2.1t + 0.8Original equation N= 5 2 + 2.1(5) + 0.8t = 5 N= 25 + 2.1(5) + 0.8Simplify. N= 25 + 10.5 + 0.8Multiply. N= 36.3Simplify. Answer: The number of patients in 2005 was 36.3 thousand or 36,300.

19. A.A B.B C.C D.D Example 4 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 = 2t 2 – 2t + 3, where t is the number of years since 1997. How many grand pianos were sold in 2004?