1. 3.2 Introduction to Polynomials

2. Identify Monomials Monomial or Term: An expression that is a constant or a product of a constant and variables that are raised to whole number powers.

3. Determine whether the expression is a monomial 5x -4xy 3 X 6 8

4. Determine whether the expression is a monomial 5x -4xy 3 X 6 8 Yes Monomial or Term: An expression that is a constant or a product of a constant and variables that are raised to whole number powers.

5. The following are NOT monomials 5/xy 3x 2 + 5x – 4 9a + 5

6. Identify the coefficient and degree of a monomial Two important parts of a monomial are its coefficient and its degree Coefficient: The numerical factor in a monomial Degree of a Monomial: The sum of the exponents of all the variables in the monomial

7. Example Problems -7x 2 Coefficient: -7 Degree: 2 -xy 3 Coefficient: -1 Degree: 4 17 Coefficient: 17 Degree: 0

8. Questions ???

9. Try Some M 6 -7x 2 y 3 14ab -10

10. Answers M 6 Coefficient: 1 Degree: 6 -7x 2 y 3 Coefficient: -7 Degree: 5 14ab Coefficient: 14 Degree: 2 -10 Coefficient: -10 Degree: 0

11. Identify like terms Like Terms: Monomials that have the same variables raised to the same exponents. Note: the definition does not say anything about the coefficient… just the variable and exponent…

12. Examples 5x 2 and 9x 2 9xyz and 5xy 7x 2 y and 7xy 2 5a and 9A

13. Examples YES NO 5x 2 and 9x 2 9xyz and 5xy 7x 2 y and 7xy 2 5a and 9A

14. You Try… -8x 3 and 5x 3 12mn and 12m -5y 3 z and y 3 z 9t and 12T

16. Identify polynomials and their terms Polynomial: A monomial or an expression that can be written as a sum of monomials Simplest form: An expression written with the fewest symbols possible Polynomial in one variable: A polynomial in which every variable term has the same variable Multivariable Polynomial: A polynomial with more than one variable

17. Some Examples Please Polynomials: 2x 3 + 5x + 8 5x + 7 9x 3 – 4x 2 + 8x – 6 Simplest form: 9x 3 + (– 4x 2 ) + 8x + (– 6) 9x 3 – 4x 2 + 8x – 6  Simplest Form Both acceptable but we prefer simplest form

18. More ?.....okay Polynomial in one variable: p 3 – 5p +2 12x 3 – 8x + 9 Multivariable Polynomial: q 3 + 8q 3 t 9px 2 – 4Px 2

19. Monomials are referred to as terms but polynomials are not because the often have more than one term. In other words polynomials are made of multiple monomials. Identify the terms in the polynomial and their coefficients: 9x 3 – 4x 2 + 8x – 6 First Term: 9x 3 Coefficient: 9 Second Term: - 4x 2 Coefficient: - 4 Third Term: 8xCoefficient: 8 Fourth Term: - 6Coefficient: - 6

20. Monomial: A single-term polynomial Binomial: A polynomial with exactly two terms Trinomial: A polynomial that has exactly three terms Polynomials that have more than three terms have no special name

21. Identify whether the polynomial is a monomial, binomial, trinomial, or has no special name 5x 3 - 3x 2 + 5x 4m 2 + 6m – 9 x 3 + 4x 2 – 9x + 2

22. Identify the degree of a polynomial The greatest degree of any of the terms in the polynomial. 5x 3 + 9x 6 – 10x 2 + 8x – 2 Degree = 6 9x 6 has the largest exponent in its term, the degree of the exponent is 6, so 6 is the degree of the polynomial.

23. Write polynomials in descending order of degree To write a polynomial in descending order of degree, write the term with the greatest degree first, then the term with the next greatest degree, and so on. 5x 3 + 4x 2 – 3x 5 – 7 + 9x - 3x 5 + 5x 3 + 4x 2 + 9x - 7

24. 9m 4 +7m 9 – 3m 2 + 15m – 6m 7 + 2 Your turn…

25. Practice Worksheet