7 5 Polynomials Objectives Classify polynomials and write

7 -5 Polynomials Objectives Classify polynomials and write polynomials in standard form. Evaluate polynomial expressions. Holt Algebra 1

7 -5 Polynomials A monomial is numbers or letters all multiplying (there are no + or – signs) The degree of a monomial is all the exponents added up. *If there is just a number, the degree is 0 Holt Algebra 1

7 -5 Polynomials 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. Holt Algebra 1 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.

7 -5 Polynomials Check It Out! Example 1 Find the degree of each monomial. a. 1. 5 k 2 m The degree is 3. b. 4 x The degree is 1. c. 2 c 3 The degree is 3. Holt Algebra 1 Add the exponents of the variables: 2 + 1 = 3. Add the exponents of the variables: 1 = 1. Add the exponents of the variables: 3 = 3.

7 -5 Polynomials A polynomial is a bunch of monomials with + or – signs. The degree of a polynomial is the degree of the monomial with the biggest degree. Holt Algebra 1

7 -5 Polynomials 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 : degree 4 Find the degree of each term. – 5: degree 0 The degree of the polynomial is the greatest degree, 4. Holt Algebra 1

7 -5 Polynomials 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. Holt Algebra 1 Find the degree of each term.

7 -5 Polynomials Standard Form Putting all the monomials in order of their degrees from biggest to smallest Leading Coefficient Th number in front of the first letter when written in standard form Holt Algebra 1

7 -5 Polynomials Example 3 A: Writing Polynomials in Standard Form Write the polynomial in standard form. Then give the leading coefficient. 6 x – 7 x 5 + 4 x 2 + 9 Find the degree of each term. Then arrange them in descending order: 6 x – 7 x 5 + 4 x 2 + 9 Degree 1 5 2 0 – 7 x 5 + 4 x 2 + 6 x + 9 5 2 1 0 The standard form is – 7 x 5 + 4 x 2 + 6 x + 9. The leading coefficient is – 7. Holt Algebra 1

7 -5 Polynomials Example 3 B: Writing Polynomials in Standard Form Write the polynomial in standard form. Then give the leading coefficient. y 2 + y 6 − 3 y Find the degree of each term. Then arrange them in descending order: y 2 + y 6 – 3 y Degree 2 6 1 y 6 + y 2 – 3 y 6 2 1 The standard form is y 6 + y 2 – 3 y. The leading coefficient is 1. Holt Algebra 1

7 -5 Polynomials Check It Out! Example 3 a Write the polynomial in standard form. Then give the leading coefficient. 16 – 4 x 2 + x 5 + 9 x 3 Find the degree of each term. Then arrange them in descending order: 16 – 4 x 2 + x 5 + 9 x 3 Degree 0 2 5 3 x 5 + 9 x 3 – 4 x 2 + 16 5 3 2 0 The standard form is x 5 + 9 x 3 – 4 x 2 + 16. The leading coefficient is 1. Holt Algebra 1

7 -5 Polynomials Some polynomials have special names based on their degree and the number of terms they have. Degree Name Terms Name 0 Constant 1 Monomial 1 Linear 2 Binomial 2 Quadratic Trinomial 3 4 Cubic Quartic 3 4 or more 5 Quintic 6 or more Holt Algebra 1 6 th, 7 th, degree and so on Polynomial

7 -5 Polynomials 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. Holt Algebra 1 6 th-degree polynomial.

7 -5 Polynomials Check It Out! Example 4 Classify each polynomial according to its degree and number of terms. a. x 3 + x 2 – x + 2 Degree 3 Terms 4 x 3 + x 2 – x + 2 is a cubic polynomial. b. 6 Degree 0 Terms 1 6 is a constant monomial. c. – 3 y 4 + 18 y 3 + 14 y Degree 4 Terms 3 – 3 y 4 + 18 y 3 + 14 y is an Quartic trinomial. Holt Algebra 1