Title: Lesson 1–1
Author: Teri Willard
Last saved by: Glencoe Mathematics

Lesson 6–2

Example 1  Degree of a polynomial
Determine whether each expression is a polynomial. If it is a polynomial, state the degree of the polynomial.
a. 
This expression is not a polynomial because the term  is not a monomial.

b. an^3 – an + n
This expression is a polynomial because each term is a monomial. The degree of the first term is 1 + 3 or 4, the degree of the second term is 1 + 1 or 2, and the degree of the third term is 1. The degree of the polynomial is 4.

Example 2 Simplify Polynomials
Simplify each expression.
a.  (–4a^3 + a^2 –1) – (–3a^3 – a^2 + 2a + 5)

(–4a^3 + a^2 –1) – (–3a^3 – a^2 + 2a + 5) | = –4a^3 + a^2 –1 + 3a^3 + a^2 – 2a – 5 | Distribute the –1.
| = (–4a^3 + 3a^3) + (a^2 + a^2) – 2a + (–1 – 5) | Group like terms.
| = –a^3 + 2a^2 – 2a – 6 | Combine like terms.

b. (3a^2 – a –6) + (–a^2 – a + 3)

(3a^2 – a –6) + (–a^2 – a + 3) | = 3a^2 – a –6 + –a^2 – a + 3 | Remove parenthesis.
| = (3a^2 + -a^2) +(– a– a) + (–6 + 3) | Group like terms.
| = 2a^2 – 2a – 3 | Combine like terms.

Example 3 Simplify Using the Distributive Property
Find –2b(3b^3 – b^2 + 5b – 6).

–2b(3b^3 – b^2 + 5b – 6) | = –2b(3b^3) + (–2b)(– b^2) + (–2b)(5b) + (–2b)(–6) | Distributive Property
| = –6b^4 + 2b^3 – 10b^2 + 12b | Multiply the monomials.

Example 4  Multiply Polynomials
Find (m + n)(2m^2 – 5mn + 8n).
 (m + n)(2m^2 – 5mn + 8n)
= m(2m^2 – 5mn + 8n) + n(2m^2 – 5mn + 8n) | Distributive Property
= m(2m^2) + m(– 5mn) + m(8n) + n(2m^2) + n(– 5mn) + n(8n) | Distributive Property
= 2m^3 – 5m^2n + 8mn + 2m^2n – 5mn^2 + 8n^2 | Multiply monomials.
=  2m^3 – 3m^2n + 8mn - 5mn^2 + 8n^2 | Combine like terms.