1. 1 Polynomial Functions Exploring Polynomial Functions Exploring Polynomial Functions –Examples Examples Modeling Data with Polynomial Functions Modeling Data with Polynomial Functions –Examples Examples

2. 2 Degree Name of Degree Number of Terms Name using number of terms 0Constant1Monomial 1Linear2Binomial 2Quadratic3Trinomial 3Cubic4 3 rd degree Polynomial 4Quarticn 4 th degree Polynomial with n terms 5Quintic 5 th degree polynomial with n terms

3. 3 x03569111214 y4231262117151922

4. 4

5. 5 Polynomials and Linear Factors Standard Form Standard Form –Example Example Factored Form Factored Form –Examples Examples Factors and Zeros Factors and Zeros Factors and Zeros Factors and Zeros –Examples Examples

6. 6 Writing a polynomial in standard form You must multiply: (x + 1)(x+2)(x+3) X 3 + 6x 2 + 11x + 6

7. 7 2x 3 + 10x 2 + 12x 2x(x 2 + 5x +6)

8. 8 The expression x-a is a linear factor of a polynomial if and only if the value a is a zero of the related polynomial function. Factor Theorem

9. 9 Factors and Zeros -3 -3 -2 -2 -1 -1 0 1 2 3 (x – (-3)) or (x + 3) (x – (-3)) or (x + 3) (x – (-2)) or (x + 2) (x – (-2)) or (x + 2) (x – (-1)) or (x + 1) (x – (-1)) or (x + 1) (x – 0) or x (x – 0) or x (x – 1) (x – 1) (x – 2) (x – 2) (x – 3) (x – 3) ZEROSFACTORS

10. 10 Dividing Polynomials Long Division Long Division Long Division Long Division Synthetic Division Synthetic Division Synthetic Division Synthetic Division

11. 11 Long Division The purpose of this type of division is to use one factor to find another. ) 440 Just as 4 finds the 10 ) x - 1x 3 + 6x 2 -6x - 1 The (x-1) finds the (x 2 + 7x + 1)

12. 12 Synthetic Division When dividing by x – a, use synthetic division. When dividing by x – a, use synthetic division. The Remainder Theorem The Remainder Theorem The Remainder Theorem The Remainder Theorem

13. 13 The Remainder Theorem When using Synthetic Division, the remainder is the value of f(a). When using Synthetic Division, the remainder is the value of f(a). This method is as good as “PLUGGING IN”, but may be faster. This method is as good as “PLUGGING IN”, but may be faster.

14. 14 Solving Polynomial Equations Solving by Graphing Solving by Graphing Solving by Graphing Solving by Graphing Solving by Factoring Solving by Factoring Solving by Factoring Solving by Factoring

15. 15 Solving by Graphing Set equation equal to 0, then substitute y for 0. Look at the x- intercepts. (Zeros) Set equation equal to 0, then substitute y for 0. Look at the x- intercepts. (Zeros) Let the left side be y 1 and let the right side be y 2. (Very much like solving a system of equations by graphing). Look at the points of intersection. Let the left side be y 1 and let the right side be y 2. (Very much like solving a system of equations by graphing). Look at the points of intersection.

16. 16 Solving by Factoring Sum of two cubes Sum of two cubes (a 3 + b 3 ) = (a + b)(a 2 – ab + b 2 ) Difference of two cubes Difference of two cubes (a 3 – b 3 ) = (a – b)(a 2 + ab + b 2 )

17. 17 More on Factoring If a polynomial can be factored into linear or quadratic factors, then it can be solved using techniques learned from earlier chapters. If a polynomial can be factored into linear or quadratic factors, then it can be solved using techniques learned from earlier chapters. Solving a polynomial of degrees higher than 2 can be achieved by factoring. Solving a polynomial of degrees higher than 2 can be achieved by factoring.

18. 18 Theorems about Roots Rational Root Theorem Rational Root Theorem Rational Root Theorem Rational Root Theorem Irrational Root Theorem Irrational Root Theorem Imaginary Root Theorem Imaginary Root Theorem

19. 19 Rational Root Theorem What are Rational Roots? What are Rational Roots? What are Rational Roots? What are Rational Roots? P’s and Q’s ………. ;) P’s and Q’s ………. ;) Using the calculator to speed up the process. Using the calculator to speed up the process. Using the calculator to speed up the process. Using the calculator to speed up the process.

20. 20 …And the Rational Roots are….. P includes all of the factors of the constant. Q includes all of the factors of the leading coefficient. f(x) = x 3 – 13x - 12 p = 12 q = 1 The possible rational roots are:

21. 21 Test the Possible Roots… In this case all roots are real and rational, but you need only to find one rational root. This will become clear later.

22. 22 Since -1, -3, and 4 are the Roots, (x + 1), (x + 3), and (x – 4) are the factors. Multiply to show that (x+1)(x+3)(x-4) = x 3 – 13x – 12 (x+1)(x 2 – x – 12) x 3 – x 2 – 12x +x 2 – x – 12 x 3 – 13x – 12

23. 23 Irrational Root Theorem If is a root, then is too. These are called CONJUGATES.

24. 24 Imaginary Root Theorem These are called CONJUGATES. If is a root, then is too.

25. 25 The Fundamental Theorem of Algebra If P(x) is a polynomial of degree with complex coefficients, then P(x) = 0 has at least one complex root. If P(x) is a polynomial of degree with complex coefficients, then P(x) = 0 has at least one complex root. A polynomial equation with degree n will have exactly n roots; the related polynomial function will have exactly n zeros. A polynomial equation with degree n will have exactly n roots; the related polynomial function will have exactly n zeros.

26. 26 The Binomial Theorem Binomial Expansion and Pascal’s Triangle Binomial Expansion and Pascal’s Triangle Binomial Expansion and Pascal’s Triangle Binomial Expansion and Pascal’s Triangle The Binomial Theorem The Binomial Theorem

27. 27 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 PASCAL’S TRIANGLE