This document provides an introduction to polynomial functions including definitions of key terms like monomial, polynomial, standard form, degree of terms and polynomials, classifying polynomials by number of terms and degree, examples of graphs of low-degree polynomials, and how to combine like terms. It defines a monomial as an expression with variables and numbers, a polynomial as a sum of terms with whole number exponents. Standard form writes polynomials in descending order of exponents. Degree is determined by highest exponent of terms or polynomial. Polynomials are classified by number of terms (monomial, binomial, trinomial, etc.) or degree (linear, quadratic, cubic, etc.). Examples show graphs changing shape with increasing degree. Combining like terms adds coefficients of

1. Introduction to Polynomial Functions Karen Shoskey

2. Definitions Monomial An expression that is either a real number, a variable, or a product of real numbers and variables Examples 3𝑐 7𝑥2 2𝑥𝑦3

3. Definitions Polynomial An algebraic expression that is a sum of terms Each term contains only variables with whole number exponents and real number coefficients Examples 3𝑐+7 7𝑥2−5𝑥+3 2𝑥4𝑦3+ 3𝑥𝑦2

4. Standard Form A polynomial is in standard form when its terms are written in descending order of exponents from left to right Examples 2𝑥+7 14𝑐3−5𝑐+8 4𝑎𝑏2−3𝑎𝑏+2

5. Standard Form Parts of a polynomial 2𝑥3− 5𝑥2−2𝑥+5 Constant Leading Coefficient Cubic Term Linear Term QuadraticTerm

6. Degree of the Term The exponent of the variable in the term determines the degree of the term Example The degree of 12𝑑5 is 5 or fifth degree What is the degree of 4𝑐3?

7. Degree of the Term The exponent of the variable in the term determines the degree of the term Example The degree of 12𝑑5 is 5 or fifth degree What is the degree of 4𝑐3? Answer: Since the exponent is 3, the term is of degree three or cubic.

8. Degree of the Polynomial The degree of the polynomial is equal to the largest degree of any term of the polynomial Example What is the degree of 6𝑝2−7𝑝+3? This is second degree, or quadratic, polynomial since the highest exponent is 2. What is the degree of 7𝑥4 −2?

9. Degree of the Polynomial The degree of the polynomial is equal to the largest degree of any term of the polynomial Example What is the degree of 6𝑝2−7𝑝+3? This is second degree, or quadratic, polynomial since the highest exponent is 2. What is the degree of 7𝑥4 −2? Answer: This polynomial is of degree 4, or quartic, since the largest exponent is 4.

10. Multiple Variable Terms Polynomials and terms can have more than one variable. Here is another example of a polynomial. 𝑡4−6𝑠3𝑡2 −12𝑠𝑡+4𝑠4−5 The positive integer exponents confirm this example is a polynomial. The polynomial has five terms.

11. Multiple Variable Terms 𝑡4−6𝑠3𝑡2 −12𝑠𝑡+4𝑠4−5 When a term has multiple variables, the degree of the term is the sum of the exponentswithin the term. t4 has a degree of 4, so it's a 4th order term,-6s3t2 has a degree of 5 (3+2), so it's a 5th order term, -12st has a degree of 2 (1+1), so it's a 2nd order term,4s4 has a degree of 4, so it's a 4th order term,-5 is a constant, so its degree is 0. Since the largest degree of a term in this polynomial is 5, then this is a polynomial of degree 5 or a 5th order polynomial.

12. Classifying Polynomialsby Number of Terms Number Name Example Of Terms 1 Monomial 4𝑥 2 Binomial 2𝑥−7 3 Trinomial 14𝑥2+8𝑥 −5 4 + Polynomial 5𝑥3+2𝑥2−𝑥+1

13. Classifying Polynomials by Degree Degree Name Example 0 Constant 3 1 Linear 2𝑥−7 2 Quadratic 7𝑥2−18𝑥+15 3 Cubic 9𝑥3+16 4 Quartic 23𝑐4+7𝑐−2 5 Quintic−12h5−3h3

14. Classify the Polynomial Write each polynomial in standard form and classify it by degree and number of terms. −7𝑥+5𝑥4 𝑥2−4𝑥+3𝑥3+2

15. Classify the Polynomial Write each polynomial in standard form and classify it by degree and number of terms. −7𝑥+5𝑥4 Answer: 5𝑥4−7𝑥 This is a fourth degree (quartic) binomial 𝑥2−4𝑥+3𝑥3+2 Answer: 3𝑥3+𝑥2−4𝑥+2 This is a third degree (cubic) trinomial

19. Notice that the graphs of polynomials with even degrees have a similar shape to 𝑓𝑥= 𝑥2 and those with odd degrees have a similar shape to 𝑓𝑥= 𝑥3.

20. Combining Like Terms A polynomial is in simplest form if all like terms have been combined (added). Like terms have the same variable(s) wit the same exponents, but can have different coefficients. 2𝑥𝑦2 𝑎𝑛𝑑 15𝑥𝑦2 are like terms 6𝑥2𝑦 𝑎𝑛𝑑 6𝑥𝑦2 are NOT like terms

21. Combining Like Terms If a polynomial has like terms, we simplify it by combining (adding) them. 𝑥2+6𝑥𝑦 −4𝑥𝑦+𝑦2 This polynomial is simplified by combining the like terms of 𝟔𝒙𝒚 𝑎𝑛𝑑 −𝟒𝒙𝒚, giving us 𝟐𝒙𝒚. 𝑥2+𝟐𝒙𝒚+𝑦2

22. Combining Like Terms Simplify the polynomials 2𝑐2+9 − 3𝑐2+7 7𝑥2+8𝑥−5+9𝑥2−9𝑥

23. Combining Like Terms Simplify the polynomials 2𝑐2+9 − 3𝑐2+7 2𝑐2−3𝑐2+ (9+7) −𝑐2+16 7𝑥2+8𝑥−5+9𝑥2−9𝑥 7𝑥2+9𝑥2+8𝑥−9𝑥−5 16𝑥2−𝑥−5

24. References Bellman, A., Bragg, S., Charles, R., Hall, B., Handlin, D., Kennedy, D. (2009). Algebra 2. Boston, MA: Pearson Holt. (n.d.). Online graphing calculator. Retrieved from http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html