This document discusses polynomials. It defines key terms like monomial, polynomial, binomial, and trinomial. It explains how to determine the degree of a polynomial, classify polynomials by degree and number of terms, and write polynomials in standard form. It also provides examples of how to add and subtract polynomials by lining them up and combining like terms.

1. Polynomials

2. What's the Deal? Be able to determine the degree of a polynomial. Be able to classify a polynomial. Be able to write a polynomial in standard form. Be able to add and subtract polynomials

3. Vocabulary Monomial: A number, a variable or the product of a number and one or more variables. Polynomial: A monomial or a sum of monomials. Binomial: A polynomial with exactly two terms. Trinomial: A polynomial with exactly three terms. Coefficient: A numerical factor in a term of an algebraic expression.

4. Vocabulary Degree of a monomial: The sum of the exponents of all of the variables in the monomial. Degree of a polynomial in one variable: The largest exponent of that variable. Standard form: When the terms of a polynomial are arranged from the largest exponent to the smallest exponent in decreasing order.

5. Polynomials in One Variable A polynomial is a monomial or the sum of monomials Each monomial in a polynomial is a term of the polynomial. The number factor of a term is called the coefficient. The coefficient of the first term in a polynomial is the lead coefficient . A polynomial with two terms is called a binomial. A polynomial with three terms is called a trinomial.

6. Polynomials in One Variable The degree of a polynomial in one variable is the largest exponent of that variable. A constant has no variable. It is a 0 degree polynomial. This is a 1 st degree polynomial. 1 st degree polynomials are linear . This is a 2 nd degree polynomial. 2 nd degree polynomials are quadratic . This is a 3 rd degree polynomial. 3 rd degree polynomials are cubic.

7. Classify the polynomials by degree and number of terms. Examples Polynomial a. b. c. d. Degree Number of Terms Classify by number of terms Zero 1 Monomial First 2 Binomial Second 2 Binomial Third 3 Trinomial

8. Standard Form To rewrite a polynomial in standard form, rearrange the terms of the polynomial starting with the largest degree term and ending with the lowest degree term. The leading coefficient , the coefficient of the first term in a polynomial written in standard form, should be positive.

9. Examples Write the polynomials in standard form. Remember: The lead coefficient should be positive in standard form. To do this, multiply the polynomial by –1 using the distributive property.

10. Practice Write the polynomials in standard form and identify the polynomial by degree and number of terms. 1. 2.

11. Problem 1 This is a trinomial. The trinomial’s degree is 3.

12. Problem 2 This is a 2 nd degree, or quadratic, trinomial.

13. Find the Sum Add (x 2 + x + 1) to (2x 2 + 3x + 2) You might decide to add like terms as the next slide demonstrates.

14. Add Like Terms + 2x 2 + 3x + 2 x 2 + x + 1 = 3x 2 + 4x +3 Or you could add the trinomials in a column

15. Just like adding like-terms + 2x 2 + 3x + 2 x 2 + x + 1 3x 2 + 4x +3 Start with the trinomials in a column + 2x 2 + 3x + 2 Combine the trinomials going down

16. Problem #2 Try one. (3x 2 +5x) + (4 -6x -2x 2 ) Make sure you put the polynomials in standard form and line them up by degree.

17. (3x 2 +5x) + (4 -6x -2x 2 ) 3x 2 +5x -2x 2 -6x + 4 + 0 x 2 -x + 4 It might be helpful to use a zero as a placeholder.

18. Find the difference - (3x 2 - 2x + 3) x 2 + 2x - 4 -2x 2 + 4x - 7 Start with the trinomials in a column - (3x 2 - 2x + 3) The negative sign outside of the parentheses is really a negative 1 that is multiplied by all the terms inside. - 3x 2 + 2x - 3)

19. Try One. - (10x 2 + 3x + 2) 12x 2 +5x + 11 2x 2 + 2x + 9 Reminder: Start with the trinomials in a column - (10x 2 + 3x + 2) The negative sign outside of the parentheses is really a negative 1 that is multiplied by all the terms inside. - 10x 2 - 3x - 2

20. Special Thanks to Public Television Station KLVX for the basic outline of the first 12 slides of this presentation