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Mastering Polynomials: Types, Graphs, and Evaluation Methods

Learn about different types of polynomials, standard forms, leading coefficients, and degrees. Explore graphing techniques and evaluating polynomials using synthetic substitution. Improve your skills in this comprehensive guide.

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Mastering Polynomials: Types, Graphs, and Evaluation Methods

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  1. Polynomials

  2. Polynomials • Types of Polynomials • About Polynomials • Graphing Polynomials • Evaluating Polynomials

  3. Types of Polynomials • Monomial • Binomial • Trinomial • Polynomial

  4. Types of Polynomials A polynomial is a broad term for a function that only has many terms with positive integer exponents. is not a polynomial.

  5. About Polynomials • Standard form of a polynomial is when the terms of a polynomial are written in descending order of exponents. is not in standard form. is in standard form.

  6. Rewrite the Polynomials in Standard Form

  7. About Polynomials • The leading coefficient is number of the term with the highest degree.

  8. Identify the Leading Coefficient

  9. About Polynomials • The degree of the polynomial is the the greatest exponent.

  10. Identify the Degree of the Polynomial

  11. y 10 10 8 6 4 2 x -10 -10 -8 -6 -4 -2 2 4 6 8 10 10 -2 -4 -6 -8 -10 -10 Graphs of Polynomials

  12. y y 10 10 8 8 6 6 4 4 2 2 x x -10 -10 -10 -10 -8 -8 -6 -6 -4 -4 -2 -2 2 2 4 4 6 6 8 8 10 10 -2 -2 -4 -4 -6 -6 -8 -8 -10 -10 -10 -10 Even Exponents • Positive Leading Coefficient • Negative Leading Coefficient

  13. y y 10 10 8 8 6 6 4 4 2 2 x x -10 -10 -10 -10 -8 -8 -6 -6 -4 -4 -2 -2 2 2 4 4 6 6 8 8 10 10 -2 -2 -4 -4 -6 -6 -8 -8 -10 -10 -10 -10 Odd Exponents • Positive Leading Coefficient • Negative Leading Coefficient

  14. Graph on your Calculator

  15. Evaluating Polynomials Using Synthetic Substitution

  16. Why not just “plug it in?” • Don’t really need a calculator. • We’ll use the idea to make dividing polynomials, which can be difficult, an easier task.

  17. 1. Rewrite the coefficients and put the “2” we’re evaluating to the side. 2 -3 -4 8 2|

  18. 2. Bring down the “2”. 2 -3 -4 8 2| 2

  19. 3. Multiply the “2” on the side by the “2” that was brought down and put product below “-3”. 2 -3 -4 8 2| 4 2 x =

  20. 4. Add -3 and 4 and write sum below. 2 -3 -4 8 2| +4 2 1

  21. 5. Multiply the “2” on the side by the “1” that was brought down and put product below “-4”. 2 -3 -4 8 2| +4 2 2 1 x =

  22. 6. Add -4 and 2 and write sum below. 2 -3 -4 8 2| 4 +2 2 1 -2

  23. 7. Multiply the “2” on the side by the “-2” that was brought down and put product below “8”. 2 -3 -4 8 2| 4 +2 -4 2 1 -2 x =

  24. 8. Add 8 and -4 and write sum below. 2 -3 -4 8 2| 4 +2 -4 2 1 -2 4

  25. You try one!

  26. Try a Tricky One!

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