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Comprehensive Guide to Polynomial Functions

Learn all about polynomial functions, from lead degree and coefficients to zeros and asymtotes. Master concepts like quadratic and linear functions with detailed explanations and examples.

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Comprehensive Guide to Polynomial Functions

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  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:

  12. 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)

  13. x4 – 14x3 + 78x2 – 206x + 221

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

  15. 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

  16. (–3, 4) U (4, ) [ –3, ) (– , –3) (– , –3) because the graph crosses the x-axis because the graph does not cross the x-axis

  17. –3 1 2 1, –3, 2 + – + + (– , –3) U (1, 2) U (2, ) (–3, 1)

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

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

  20. 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

  21. 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

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

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

  24. 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

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