Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.

Introduction to Polynomial Functions

8,408 views

Published on

Published in: Education, Technology

Introduction to Polynomial Functions

  1. 1. Introduction to Polynomial Functions<br />Karen Shoskey<br />
  2. 2. Definitions<br />Monomial<br />An expression that is either a real number, a variable, or a product of real numbers and variables<br />Examples<br />3𝑐<br />7π‘₯2<br />2π‘₯𝑦3<br />Β <br />
  3. 3. Definitions<br />Polynomial<br />An algebraic expression that is a sum of terms<br />Each term contains only variables with whole number exponents and real number coefficients<br />Examples<br />3𝑐+7<br />7π‘₯2βˆ’5π‘₯+3<br />2π‘₯4𝑦3+Β 3π‘₯𝑦2<br />Β <br />
  4. 4. Standard Form<br />A polynomial is in standard form when its terms are written in descending order of exponents from left to right<br />Examples<br />2π‘₯+7<br />14𝑐3βˆ’5𝑐+8<br />4π‘Žπ‘2βˆ’3π‘Žπ‘+2<br />Β <br />
  5. 5. Standard Form<br />Parts of a polynomial<br />2π‘₯3βˆ’Β 5π‘₯2βˆ’2π‘₯+5<br />Β <br />Constant<br />Leading Coefficient<br />Cubic Term<br />Linear <br />Term<br />QuadraticTerm<br />
  6. 6. Degree of the Term<br />The exponent of the variable in the term determines the degree of the term<br />Example<br />The degree of 12𝑑5Β is 5 or fifth degree<br />What is the degree of 4𝑐3?<br />Β <br />
  7. 7. Degree of the Term<br />The exponent of the variable in the term determines the degree of the term<br />Example<br />The degree of 12𝑑5Β is 5 or fifth degree<br />What is the degree of 4𝑐3?<br />Answer: Since the exponent is 3, the term is of degree three or cubic.<br />Β <br />
  8. 8. Degree of the Polynomial<br />The degree of the polynomial is equal to the largest degree of any term of the polynomial<br />Example<br />What is the degree of 6𝑝2βˆ’7𝑝+3?<br />This is second degree, or quadratic, polynomial since the highest exponent is 2.<br />What is the degree of 7π‘₯4Β βˆ’2?<br />Β <br />
  9. 9. Degree of the Polynomial<br />The degree of the polynomial is equal to the largest degree of any term of the polynomial<br />Example<br />What is the degree of 6𝑝2βˆ’7𝑝+3?<br />This is second degree, or quadratic, polynomial since the highest exponent is 2.<br />What is the degree of 7π‘₯4Β βˆ’2?<br />Answer: This polynomial is of degree 4, or quartic, since the largest exponent is 4.<br />Β <br />
  10. 10. Multiple Variable Terms<br />Polynomials and terms can have more than one variable. Here is another example of a polynomial.<br />𝑑4βˆ’6𝑠3𝑑2Β βˆ’12𝑠𝑑+4𝑠4βˆ’5<br />The positive integer exponents confirm this example is a polynomial. The polynomial has five terms.<br />Β <br />
  11. 11. Multiple Variable Terms<br />𝑑4βˆ’6𝑠3𝑑2Β βˆ’12𝑠𝑑+4𝑠4βˆ’5<br />When a term has multiple variables, the degree of the term is the sum of the exponentswithin the term.<br />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.<br />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.<br />Β <br />
  12. 12. Classifying Polynomialsby Number of Terms <br />Number Name Example<br />Of Terms<br /> 1 Monomial 4π‘₯<br /> 2 Binomial 2π‘₯βˆ’7<br /> 3 TrinomialΒ Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β 14π‘₯2+8π‘₯Β βˆ’5<br /> 4 + PolynomialΒ Β Β Β Β Β Β Β Β Β Β Β Β 5π‘₯3+2π‘₯2βˆ’π‘₯+1<br />Β <br />
  13. 13. Classifying Polynomials by Degree<br />Degree Name Example<br /> 0 Constant 3<br /> 1 Linear 2π‘₯βˆ’7<br /> 2 Quadratic 7π‘₯2βˆ’18π‘₯+15<br /> 3 Cubic 9π‘₯3+16<br /> 4 Quartic 23𝑐4+7π‘βˆ’2<br /> 5 Quinticβˆ’12h5βˆ’3h3<br />Β <br />
  14. 14. Classify the Polynomial<br />Write each polynomial in standard form and classify it by degree and number of terms.<br />βˆ’7π‘₯+5π‘₯4<br />π‘₯2βˆ’4π‘₯+3π‘₯3+2<br />Β <br />
  15. 15. Classify the Polynomial<br />Write each polynomial in standard form and classify it by degree and number of terms.<br />βˆ’7π‘₯+5π‘₯4<br />Answer: 5π‘₯4βˆ’7π‘₯<br />This is a fourth degree (quartic) binomial<br />π‘₯2βˆ’4π‘₯+3π‘₯3+2<br />Answer: 3π‘₯3+π‘₯2βˆ’4π‘₯+2<br />This is a third degree (cubic) trinomial<br />Β <br />
  16. 16. Sample Graphs<br /><ul><li>Linear
  17. 17. Quadratic</li></li></ul><li>Sample Graphs<br /><ul><li>Cubic
  18. 18. Quartic</li></li></ul><li>Sample Graphs<br /><ul><li>Quintic
  19. 19. Notice that the graphs of polynomials with even degrees have a similar shape to 𝑓π‘₯=Β π‘₯2 and those with odd degrees have a similar shape to 𝑓π‘₯=Β π‘₯3.</li></ul>Β <br />
  20. 20. Combining Like Terms<br />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.<br />2π‘₯𝑦2Β π‘Žπ‘›π‘‘Β 15π‘₯𝑦2 are like terms<br />6π‘₯2π‘¦Β π‘Žπ‘›π‘‘Β 6π‘₯𝑦2 are NOT like terms<br />Β <br />
  21. 21. Combining Like Terms<br />If a polynomial has like terms, we simplify it by combining (adding) them.<br />π‘₯2+6π‘₯π‘¦Β βˆ’4π‘₯𝑦+𝑦2<br />This polynomial is simplified by combining the like terms of πŸ”π’™π’šΒ π‘Žπ‘›π‘‘Β βˆ’πŸ’π’™π’š, giving us πŸπ’™π’š.<br />π‘₯2+πŸπ’™π’š+𝑦2<br />Β <br />
  22. 22. Combining Like Terms<br />Simplify the polynomials<br />2𝑐2+9Β βˆ’Β 3𝑐2+7<br />7π‘₯2+8π‘₯βˆ’5+9π‘₯2βˆ’9π‘₯<br />Β <br />
  23. 23. Combining Like Terms<br />Simplify the polynomials<br />2𝑐2+9Β βˆ’Β 3𝑐2+7<br />2𝑐2βˆ’3𝑐2+Β (9+7)<br />βˆ’π‘2+16<br />7π‘₯2+8π‘₯βˆ’5+9π‘₯2βˆ’9π‘₯<br />7π‘₯2+9π‘₯2+8π‘₯βˆ’9π‘₯βˆ’5<br />16π‘₯2βˆ’π‘₯βˆ’5<br />Β <br />
  24. 24. References<br />Bellman, A., Bragg, S., Charles, R., Hall, B., Handlin, D., Kennedy, D. (2009). Algebra 2. Boston, MA: Pearson<br />Holt. (n.d.). Online graphing calculator. Retrieved from http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html<br />

Γ—