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# Introduction to Polynomial Functions

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### 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 />3π<br />7π₯2<br />2π₯π¦3<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 />3π+7<br />7π₯2β5π₯+3<br />2π₯4π¦3+Β 3π₯π¦2<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 />14π3β5π+8<br />4ππ2β3ππ+2<br />Β <br />
5. 5. Standard Form<br />Parts of a polynomial<br />2π₯3βΒ 5π₯2β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 12π5Β is 5 or fifth degree<br />What is the degree of 4π3?<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 12π5Β is 5 or fifth degree<br />What is the degree of 4π3?<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 6π2β7π+3?<br />This is second degree, or quadratic, polynomial since the highest exponent is 2.<br />What is the degree of 7π₯4Β β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 6π2β7π+3?<br />This is second degree, or quadratic, polynomial since the highest exponent is 2.<br />What is the degree of 7π₯4Β β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 />π‘4β6π 3π‘2Β β12π π‘+4π 4β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 />π‘4β6π 3π‘2Β β12π π‘+4π 4β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₯β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 β5<br /> 4 + PolynomialΒ Β Β Β Β Β Β Β Β Β Β Β Β 5π₯3+2π₯2βπ₯+1<br />Β <br />
13. 13. Classifying Polynomials by Degree<br />Degree Name Example<br /> 0 Constant 3<br /> 1 Linear 2π₯β7<br /> 2 Quadratic 7π₯2β18π₯+15<br /> 3 Cubic 9π₯3+16<br /> 4 Quartic 23π4+7πβ2<br /> 5 Quinticβ12h5β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 />β7π₯+5π₯4<br />π₯2β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 />β7π₯+5π₯4<br />Answer: 5π₯4β7π₯<br />This is a fourth degree (quartic) binomial<br />π₯2β4π₯+3π₯3+2<br />Answer: 3π₯3+π₯2β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 ππ₯=Β π₯2 and those with odd degrees have a similar shape to ππ₯=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₯π¦2Β πππΒ 15π₯π¦2 are like terms<br />6π₯2π¦Β πππΒ 6π₯π¦2 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 β4π₯π¦+π¦2<br />This polynomial is simplified by combining the like terms of πππΒ πππΒ βπππ, giving us πππ.<br />π₯2+πππ+π¦2<br />Β <br />
22. 22. Combining Like Terms<br />Simplify the polynomials<br />2π2+9Β βΒ 3π2+7<br />7π₯2+8π₯β5+9π₯2β9π₯<br />Β <br />
23. 23. Combining Like Terms<br />Simplify the polynomials<br />2π2+9Β βΒ 3π2+7<br />2π2β3π2+Β (9+7)<br />βπ2+16<br />7π₯2+8π₯β5+9π₯2β9π₯<br />7π₯2+9π₯2+8π₯β9π₯β5<br />16π₯2βπ₯β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 />