polynomial function

1. polynomial function n lead degree a coefficient undefined f(x) = 0 zero function 0 f(x) = 5 constant function 1 f(x) = 2x + 5 linear function f(x) = x2 + 2x + 5 2 quadratic function slope linear constant non-zero

2. roots or solutions x = -1 or 3.5 vertex: (h, k) complete the square vertex: (–4, –1) axis of symmetry: x = –4

3. vertex: (1, 5) vertex: x – intercepts:

4. constants constant of variation or proportion power is proportional to varies as power function power: –4 constant of variation: 2 not a power function: power isn’t a constant power function independent variable: r power: 2 constant of variation:  power is 1, constant of variation is 2 power is 2, constant of variation is 1 direct variation

5. d = k F d = k t 2 non-negative integer monomial degree: 0 lead coefficient: 4 not monomial power is ½ (not an integer) monomial degree: 3 lead coefficient: 13 not monomial power is a variable

6. vertical stretch / shrink vertical stretch / shrink reflection across the x-axis domain range continuity increasing decreasing symmetry boundedness extrema asymptotes end behavior

7. divisor dividend quotient remainder

8. (3)2 – 4(3) – 5 = 9 – 12 – 5 = –8 k = 3 (–2)2 – 4(–2) – 5 = 4 + 8 – 5 = 7 k = –2 (5)2 – 4(5) – 5 = 25 – 20 – 5 = 0 k = 5 divides evenly zero x - intercept solution root

9. so factors are: x + 4, x – 3, x + 1 3(x + 4)(x – 3)(x + 1) = 3x3 + 6x2 – 33x – 36 so factors are: x + 3, x + 2, x – 5 2(x + 3)(x + 2)(x – 5) = 2x3 – 38x – 60

10. f(x) = x2 – 16 (x + 4)(x – 4) = 0 x = 4, x = –4 rational zeros

11. Use the rational zeros theorem to find the rational zeros of f(x) = 2x3 + 3x2 – 8x + 3 p = integer factors of the constant q = integer factors of the lead coefficient potential:

13. complex (real and non-real) zeros * non-real zeros are not x – intercepts zeros: 3i, – 3i, – 5 x-intercepts: – 5 complex conjugate (a + bi and a – bi)

14. x4 – 14x3 + 78x2 – 206x + 221

15. denominator the x – axis ( y = 0 ) the line y = an / bm there is no quotient output input

16. vertical asymptote: x – intercept none none y – intercept horizontal asymptote: y = 0 (0, 4) vertical asymptote: x – intercept x = –1 (0, 0) (1, 0) y – intercept horizontal asymptote: none (0, 0) slant asymptote: y = x – 2

17. (–3, 4) U (4, ) [ –3, ) (– , –3) (– , –3) because the graph crosses the x-axis because the graph does not cross the x-axis

18. –3 1 2 1, –3, 2 + – + + (– , –3) U (1, 2) U (2, ) (–3, 1)

19. Write a standard form polynomial function of degree 4 whose zeros include 1 + 2i and 3 – i. quiz

20. Solve the following inequality using a sign chart: x3 + 2x2 – 11x – 12 < 0 quiz

21. zeros: x – intercepts: Write the following polynomial function in standard form. Then identify the zeros and the x – intercepts. f(x) = (x – 3i) (x + 3i) (x + 4) quiz

22. a.) 2, –1 , –6 b.) (–6, –1) U (2, ) c.) (–, –6) U (-1, 2) • Without graphing, using a sign chart, find the values of x that cause f(x) = (x – 2) (x + 6) (x + 1) to be: • a.) zero ( f(x) = 0 ) • b.) positive ( f(x) > 0 ) • c.) negative ( f(x) < 0 ) quiz

23. Use the quadratic equation to find the zeros of f(x) = 5x2 – 2x + 5. Your answer must be in exact simplified form. quiz

24. Find all zeros of f(x) = x4 + 3x3 – 5x2 – 21x + 22 and write f(x) in its linear factorization form

25. 2i is a zero of f(x) = 2x4 – x3 + 7x2 – 4x – 4. Find all remaining zeros and write f(x) in its linear factorization form. quiz