1. Graphing Polynomial Functions

2. Graphs of Polynomial Functions 1. Polynomials have smooth, continuous curves 2. Continuous means it can be drawn without picking up a pencil 3. Smooth means it have no sharp points

3. 1. Even powers are tangent to the x-axis 2. Odd powers go across the x-axis at the origin 3. As the power increases, the curve gets wider at the point of tangency or intersection

4. Steps for graphing polynomial functions: 1. find zeroes 2. Make a number line 3. determine positive and negative values 4. sketch graph

5. Examples

6. Double Roots

9. Polynomial Division

10. Long Division Check for placeholders, leave variables Change signs

11. Synthetic Division Find zeros Check for placeholders Divide zero into coefficients of polynomial function (multiply, add, repeat)

12. Remainder Theorem Let f be a polynomial function. If f(x) is divdied by x-c, then remainder is f(c) PROOF When f(x) is divided by x-c, the remainder must be a constant, r, (because the remainder must have a smaller degree than x-c) so by the division algorithm: f(x)=q(x)(x-c)+r In order to find f(c), substitute c in for x f(c)=q(c)(c-c)+r = q(c)0+r = r So f(c)=r

13. Factor Theorem Let f be a polynomial functions. Then x-c is a factor of f(x) if and only if (iff) f(c)=0 PROOF (must prove both ways) 1. If x-c is a factor, then when f(x) is divided by x-c the remainder is 0. By the remainder theorem, f(c)=the remainder, so f(c)=0 2. If f(c)=0, the remainder theorem, the remainder when f(x) is divided by x-c is 0. If the remainder is 0, then that means x-c goes into f(x), so x-c is a factor of f(x)

15. Things you know given f(3)=0 1. When x=3, y=0 2. (3, 0) is an ordered pair on the graph 3. x-3 is a factor of f(x) 4. when f(x) is divided by x-3, the remainder is 0 5. 3 is a root 6. 3 is an x-intercept 7. 3 is a solution if f(x)=0 8. f(x) touches the x-axis at 3 9. f(x) has a zero at 3 10. if the inverse of f(x) exists, then (0, 3) is on it

16. Things you know if f(-2)=5 1. When f(x) is divided by x+2, the remainder is 5 2. x+2 is not a factor of f(x) 3. The point (-2, 5) is on f(x) 4. when x= -2, y=5 5. If the inverse of f(x) exists, then the point (5, - 2) is on it 6. -2 is not an x-intercept of f(x) 7. -2 is not a solution to f(x)=0 8. -2 is not a root of f(x) 9. -2 is not a zero of f(x)

17. Rational Zeros Theorem (P/Q) =

18. After you identify possible zeros, find actual Substitute possible zeros into f(x) OR do synthetic division 1 1 -4 -4 16 1 -3 -7 1 -3 -7 -9 Remainder is not 0, so 1 isn’t a root -1 1 -4 -4 16 -1 5 -1 1 -5 1 15 Remainder is not 0, so -1 isn’t a root 2 1 -4 -4 16 2 -4 -16 1 -2 -8 0 Remainder is 0, so 2 is a root. Take coefficients and continue solving for other roots.

19. Fundamental Theorem of Algebra If f(x) is a polynomial function of degree n, where n>0, f(x) has n zeros in the complex number system Number of answers= degree

20. Linear Factorization Theorem

21. Irrational Conjugate Theorem

22. Complex Conjugate Theorem

23. Odd Degree Theorem Any polynomial with real coefficients and with odd degree must have at least one real zero (x+2i)(x-2i)(x-5)  5 is real

24. Factors of a Polynomial