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Bell-ringer

Bell-ringer. Holt Algebra II text page 431 #72-75, 77-80. 7.1 Introduction to Polynomials. Definitions. Monomial - is an expression that is a number, a variable, or a product of a number and variables. i.e. 2, y, 3x, 45x 2 … Constant - is a monomial containing no variables.

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Bell-ringer

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  1. Bell-ringer • Holt Algebra II text page 431 #72-75, 77-80

  2. 7.1 Introduction to Polynomials

  3. Definitions • Monomial - is an expression that is a number, a variable, or a product of a number and variables. • i.e. 2, y, 3x, 45x2… • Constant - is a monomial containing no variables. • i.e. 3, ½, 9 … • Coefficient - is a numerical factor of a monomial. • i.e. 3x, 12y, 2/3x3, 7x4 … • Degree - is the sum of the exponents of a monomial’s variables. • i.e. x3y2z is of degree 6 because x3y2z1 = 3 + 2+ 1 = 6

  4. Definitions • Polynomial- is a monomial or a sum of terms that are monomials. • These monomials have variables which are raised to whole-number exponents. • The degree of a polynomial is the same as that of its term with the greatest degree.

  5. Examples v. Non-examples • Examples 5x + 4 x4 + 3x3 – 2x2 + 5x -1 √7x2 – 3x + 5 • Non – examples x3/2 + 2x – 1 3/x2 – 4x3 + 3x – 13 3√x +x4 +3x3 +9x +7

  6. Classification • We classify polynomials by… …the number of terms or monomials it contains AND … by its degree.

  7. Classification of Polynomials • Classifying polynomials by the number of terms… monomial: one term binomial: two terms trinomial: three terms Poylnomial: anything with four or more terms

  8. Classification of a Polynomial n = 0 constant 3 linear n = 1 5x + 4 quadratic n = 2 2x2 + 3x - 2 cubic n = 3 5x3 + 3x2 – x + 9 quartic 3x4 – 2x3 + 8x2 – 6x + 5 n = 4 n = 5 -2x5 + 3x4 – x3 + 3x2 – 2x + 6 quintic

  9. Compare the Two Expressions • How do these expressions compare to one another? 3(x2 -1) - x2 + 5x and 5x – 3 + 2x2 • How would it be easier to compare? • Standard form - put the terms in descending order by degree.

  10. Examples • Write each polynomial in standard form, classifying by degree and number of terms. 1). 3x2 – 4 + 8x4 = 8x4+ 3x2 – 4 quartic trinomial 2). 3x2 +2x6 - + x3 - 4x4 – 1 –x3 = 2x6- 4x4 + 3x2 – 1 6th degree polynomial with four terms.

  11. Adding & Subtracting Polynomials • To add/subtract polynomials, combine like terms, and then write in standard form. • Recall: In order to have like terms, the variable and exponent must be the same for each term you are trying to add or subtract.

  12. Examples • Add the polynomial and write answer in standard form. 1). (3x2 + 7 + x) - (14x3 + 2 + x2 - x) = =- 14x3 + (3x2 - x2) +(x -x) + (7- 2) = - 14x3 + 2x2 + 5

  13. Example Add (-3x4y3 + 6x3y3 – 6x2 + 5xy5 + 1) + (5x5 – 3x3y3 – 5xy5) -3x4y3 + 6x3y3 – 6x2 + 5xy5 + 1 5x5 - 3x3y3 - 5xy5 + 1 5x5 – 3x4y3 + 3x3y3 – 6x2

  14. Example Subtract. (2x2y2 + 3xy3 – 4y4) - (x2y2 – 5xy3 + 3y – 2y4) = 2x2y2 + 3xy3 – 4y4 - x2y2 + 5xy3 – 3y + 2y4 = x2y2 + 8xy3 – 2y4 – 3y

  15. Evaluating Polynomials • Evaluating polynomials is just like evaluating any function. *Substitute the given value for each variable and then do the arithmetic.

  16. Application • The cost of manufacturing a certain product can be approximated by f(x) = 3x3 – 18x + 45, where x is the number of units of the product in hundreds. Evaluate f(0) and f(200) and describe what they represent. • f(0) = 45 represents the initial cost before manufacturing any products f(200) = 23,996,445 represents the cost of manufacturing 20,000 units of the product.

  17. Exploring Graphs of Polynomial Functions Activity • Copy the table on page 427 • Answer/complete each question/step.

  18. Graphs of Polynomial Functions Graph each function below. 2 1 y = x2 + x - 2 3 2 y = 3x3 – 12x + 4 3 2 y = -2x3 + 4x2 + x - 2 4 3 y = x4 + 5x3 + 5x2 – x - 6 4 3 y = x4 + 2x3 – 5x2 – 6x Make a conjecture about the degree of a function and the # of “U-turns” in the graph.

  19. Graphs of Polynomial Functions Graph each function below. 3 0 y = x3 3 0 y = x3 – 3x2 + 3x - 1 4 1 y = x4 Now make another conjecture about the degree of a function and the # of “U-turns” in the graph. The number of “U-turns” in a graph is less than or equal to one less than the degree of a polynomial.

  20. Now You • Graph each function. Describe its general shape. • P(x) = -3x3 – 2x2 +2x – 1 • An S-shaped graph that always rises on the left and falls on the right. • Q(x) = 2x4 – 3x2 – x + 2 • W-shape that always rises on the right and the left.

  21. Check Your Understanding • Create a polynomial. • Trade polynomials with the second person to your left. • Put your new polynomial in standard form then… …identify by degree and number of terms …identify the number of U - turns. • Turn the papers in with both names.

  22. Homework • Page 429-430 #12-48 by 3’s.

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