1. Solve: 2x 3 – 4x 2 – 6x = 0. (Check with GUT) 2. Solve algebraically or graphically: x 2 – 2x – 15> 0 1

We Know: f(x) = c f(x) = mx + b f(x) = ax 2 + bx + c 2 constant linear quadratic

Pre-Cal Polynomial Functions

 Determine end behavior  Factor a polynomial function  Graph a polynomial function  Fin the zeros of a polynomial function  Write a polynomial function given its zeros  Use GUT to graph and solve polynomial function 4

f(x) = a n x n + a n-1 x n a 1 x 1 + a 0 where a n ≠ 0 Example: f(x) = 3x 4 – 2x 3 + 5x – 4 5

Standard Form means that the polynomial is written in _____________ order of _____________ A function of degree “n” has at most “n” zeros. If the degree of a function is “n”, then the number of total zeros (real or nonreal) is n. (FTA) Descending Exponents 6

F(x) = a(x – b)(x – c)(x – d)… Once a polynomial is factored is easy to find the zeros. Factor: (x – b) Solution/zero: x = b X-Intercept: (b, 0) 7

 exponents are all ______________ therefore all __________________  all coefficients are___________________  a n is called the _____________________  a 0 is called the _____________________  n is equal to the ____________________ (always the _______________ exponent) Whole numbers Positive Real numbers Leading coefficient Constant term degree highest 8

9

End behavior is what the y values are doing as the x values approach positive and negative infinity. It is written: f(x) _____ as x -∞, and f(x) _____ as x ∞ 10

 If the degree is __________ the ends of the graph go in the _________ direction.  If the degree is __________ the ends of the graph go in the _________ directions.  Look at the ________________ to see what direction the graph is going in. odd same opposite Leading coefficient even 11

 Even exponent  Odd exponent 12

1. f(x) = 3x 4 – 2x 2 + 5x – 8 D: LC: End Behavior: f(x) --->____ as x ---> f(x) --->____ as x ----> 2. f(x) = -x D: LC: End Behavior: f(x) --->____as x ---> f(x) --->____ as x ----> -∞-∞ ∞ ∞ -∞-∞ ∞ ∞ -∞-∞ -∞-∞ 3, positive 2, even -1, negative 4, even 13

3. f(x) = x 7 – 3x 3 + 2x D: LC: End Behavior: f(x) --->____ as x ---> f(x) --->____ as x ----> 4. f(x) = -2x 6 + 3x – 7 D: LC: End Behavior: f(x) --->____as x ----> -∞-∞ ∞ ∞ -∞-∞ ∞ -∞-∞-∞ -∞-∞ 1, positive 6, even -2, negative 7, odd 14

5. f(x) = -4x 3 + 3x 8 D: LC: End Behavior: f(x) --->____ as x ---> f(x) --->____ as x ----> 6. f(x) = 4x 3 + 5x 7 – 2 D: LC: End Behavior: f(x) --->____as x ----> -∞-∞ ∞ ∞ -∞-∞ ∞ ∞-∞-∞ ∞ 3, positive 7, odd 5, positive 8, even 15

16 Single Root: passes through Double Root: touches and turns Triple Root: flattens out then passes through

17 Double Root: Multiplicity of two Triple Root: Multiplicity of three Y = x 3 has a multiplicity of 3 at x=0

18

19

20

21 1. y = -x 5 2. g(x) = x f(x) = (x + 1) 3

22 1. f(x) = x 3 – x 2 – 2x x(x 2 – x – 2) x(x – 2)(x + 1) x = 0 x = 2 x = -1 x y ½ 5/

23 2. f(x) = -2x 4 + 2x 2 -2x 2 (x 2 – 1) -2x 2 (x – 1)(x + 1) x = 0 x = 1 x = -1 x y ½3/8 ½ 2-24

24 3. f(x) = 3x 4 – 4x 3 x 3 (3x – 4) x = 0 x = 4/3

25 4. f(x) = -2x 3 + 6x 2 – 9/2x 0 = -2x 3 + 6x 2 – 9/2x 2(0 = -2x 3 + 6x 2 – 9/2x ) 0 = -4x x 2 – 9x 0 = -x(4x x + 9) 0 = -x(2x – 3)(2x – 3) x = 0 x = 3/2

1. 4, -4, and 1 x = 4 x = -4 x = 1 (x – 4)(x + 4)(x – 1) (x 2 – 16)(x – 1) f(x) = x 3 –x 2 –16x , -4, 5 x = 1 x = -4 x = 5 (x – 1)(x + 4)(x – 5) (x 2 + 3x – 4)(x – 5) f(x)=x 3 –5x 2 +3x 2 –15x–4x+20 f(x) = x 3 – 2x 2 – 19x

3. 2, √11, -√11 x = 2 x = √11 x = - √11 (x – 2)(x - √11)(x + √11) (x – 2)(x 2 – 11) f(x) = x 3 – 2x 2 – 11x – 22 27

4. -3, 4i x = -3, x = 4i, x = -4i **imaginary zeros always come in conjugate pairs!! (x + 3)(x – 4i)(x + 4i) *do the imaginary first! (x + 3)(x 2 – 16i 2 ) *remember i 2 is -1! (x + 3)(x ) f(x) = x 3 + 3x x , -i x = 8, x = -i, x = i (x – 8)(x + i)(x – i) (x – 8)(x 2 – i 2 ) (x – 8)(x 2 + 1) f(x) = x 3 – 8x 2 + 1x – 8 28

29 The zero is the x value that would give you zero for y. X = 2.3

30 The zero is the x value that would give you zero for y. X = 3.3

 f(x) = x 3 + 2x 2 – 8x – 16 31