Polynomials and Polynomial Functions Definitions Term a number

Polynomials and Polynomial Functions Definitions Term: a number or a product of a number and variables raised to a power. Coefficient: the numerical factor of each term. Constant: the term without a variable. Polynomial: a finite sum of terms of the form axn, where a is a real number and n is a whole number.

Polynomials and Polynomial Functions Definitions Monomial: a polynomial with exactly one term. Binomial: a polynomial with exactly two terms. Trinomial: a polynomial with exactly three terms.

A polynomial is a monomial or a sum or difference of monomials. The degree of a polynomial is the degree of the term with the greatest degree.

Example 1: Finding the Degree of a Monomial Find the degree of each monomial. A. 4 p 4 q 3 The degree is 7. B. 7 ed The degree is 2. C. 3 The degree is 0. Add the exponents of the variables: 4 + 3 = 7. Add the exponents of the variables: 1+ 1 = 2. Add the exponents of the variables: 0 = 0.

Example 2: Finding the Degree of a Polynomial Find the degree of each polynomial. A. 11 x 7 + 3 x 3 11 x 7: degree 7 3 x 3: degree 3 The degree of the polynomial is the greatest degree, 7. Find the degree of each term. B. : degree 3 – 5: degree 0 : degree 4 Find the degree of each term. The degree of the polynomial is the greatest degree, 4.

Check It Out! Example 2 Find the degree of each polynomial. a. 5 x – 6 5 x: degree 1 – 6: degree 0 The degree of the polynomial is the greatest degree, 1. Find the degree of each term. b. x 3 y 2 + x 2 y 3 – x 4 + 2 x 3 y 2: degree 5 –x 4: degree 4 x 2 y 3: degree 5 2: degree 0 The degree of the polynomial is the greatest degree, 5. Find the degree of each term.

Polynomials and Polynomial Functions Practice Problems Identify the degrees of each term and the degree of the polynomial. 8 8 10 10

Example 4: Classifying Polynomials Classify each polynomial according to its degree and number of terms. A. 5 n 3 + 4 n Degree 3 Terms 2 5 n 3 + 4 n is a cubic binomial. B. 4 y 6 – 5 y 3 + 2 y – 9 Degree 6 Terms 4 4 y 6 – 5 y 3 + 2 y – 9 is a C. – 2 x Degree 1 Terms 1 – 2 x is a linear monomial. 6 th-degree polynomial.

Polynomials and Polynomial Functions Practice Problems Evaluate each polynomial function

7 -7 Adding and Subtracting Polynomials Example 1: Adding and Subtracting Monomials Add or subtract. A. 12 p 3 + 11 p 2 + 8 p 3 12 p 3 + 8 p 3 + 11 p 2 20 p 3 + 11 p 2 Identify like terms. Rearrange terms so that like terms are together. Combine like terms. B. 5 x 2 – 6 – 3 x + 8 5 x 2 – 3 x + 8 – 6 5 x 2 – 3 x + 2 Identify like terms. Rearrange terms so that like terms are together. Combine like terms. Holt Algebra 1

7 -7 Adding and Subtracting Polynomials Example 1: Adding and Subtracting Monomials Add or subtract. C. t 2 + 2 s 2 – 4 t 2 – s 2 t 2 – 4 t 2 + 2 s 2 – 3 t 2 + s 2 Identify like terms. Rearrange terms so that like terms are together. Combine like terms. D. 10 m 2 n + 4 m 2 n – 8 m 2 n Identify like terms. 6 m 2 n Combine like terms. Holt Algebra 1

7 -7 Adding and Subtracting Polynomials Check It Out! Example 1 Add or subtract. a. 2 s 2 + 3 s 2 + s 5 s 2 + s Identify like terms. Combine like terms. b. 4 z 4 – 8 + 16 z 4 + 2 4 z 4 + 16 z 4 – 8 + 2 20 z 4 – 6 Holt Algebra 1 Identify like terms. Rearrange terms so that like terms are together. Combine like terms.

7 -7 Adding and Subtracting Polynomials Check It Out! Example 1 Add or subtract. c. 2 x 8 + 7 y 8 – x 8 – y 8 2 x 8 – x 8 + 7 y 8 – y 8 x 8 + 6 y 8 Identify like terms. Rearrange terms so that like terms are together. Combine like terms. d. 9 b 3 c 2 + 5 b 3 c 2 – 13 b 3 c 2 b 3 c 2 Holt Algebra 1 Identify like terms. Combine like terms.

7 -7 Adding and Subtracting Polynomials Example 2: Adding Polynomials Add. A. (4 m 2 + 5) + (m 2 – m + 6) Identify like terms. (4 m 2 + m 2) + (–m) +(5 + 6) Group like terms together. Combine like terms. 5 m 2 – m + 11 B. (10 xy + x) + (– 3 xy + y) Identify like terms. (10 xy – 3 xy) + x + y Group like terms together. Combine like terms. 7 xy + x + y Holt Algebra 1

7 -7 Adding and Subtracting Polynomials Example 2 C: Adding Polynomials Add. (6 x 2 – 4 y) + (3 x 2 + 3 y – 8 x 2 – 2 y) Identify like terms. (6 x 2 – 4 y) + (– 5 x 2 + y) (6 x 2 – 5 x 2) + (– 4 y + y) x 2 – 3 y Holt Algebra 1 Combine like terms in the second polynomial. Combine like terms. Simplify.

7 -7 Adding and Subtracting Polynomials Example 2 D: Adding Polynomials Add. Identify like terms. Group like terms together. Combine like terms. Holt Algebra 1

7 -7 Adding and Subtracting Polynomials Check It Out! Example 2 Add (5 a 3 + 3 a 2 – 6 a + 12 a 2) + (7 a 3 – 10 a) Identify like terms. (5 a 3 + 7 a 3) + (3 a 2 + 12 a 2) + (– 10 a – 6 a) Group like terms together. 12 a 3 + 15 a 2 – 16 a Holt Algebra 1 Combine like terms

7 -7 Adding and Subtracting Polynomials Lesson Quiz: Part I Add or subtract. 1. 7 m 2 + 3 m + 4 m 2 11 m 2 + 3 m 2. (r 2 + s 2) – (5 r 2 + 4 s 2) (– 4 r 2 – 3 s 2) 3. (10 pq + 3 p) + (2 pq – 5 p + 6 pq) 18 pq – 2 p 4. (14 d 2 – 8) + (6 d 2 – 2 d +1) 20 d 2 – 2 d – 7 5. (2. 5 ab + 14 b) – (– 1. 5 ab + 4 b) Holt Algebra 1 4 ab + 10 b

Polynomials and Polynomial Functions Multiplication Multiplying Monomials by Monomials Examples:

Polynomials and Polynomial Functions Multiplication Multiplying Monomials by Polynomials Examples:

5. 4 – Polynomials and Polynomial Functions Multiplication Multiplying Two Polynomials Examples:

5. 4 – Multiplying Polynomials Special Products Multiplying Two Binomials using FOIL First terms Outer terms Inner terms Last terms

5. 4 – Multiplying Polynomials Special Products Squaring Binomials

5. 4 – Multiplying Polynomials Special Products Multiplying the Sum and Difference of Two Binomials

5. 4 – Multiplying Polynomials Special Products Dividing by a Monomial

Extra Problems

5. 4 – Polynomials and Polynomial Functions Multiplication Multiplying Two Polynomials Examples: