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Polynomial Functions

Learn about polynomial functions, types, evaluation methods, real zeros, and the Fundamental Theorem of Algebra. Identify polynomial functions, evaluate values, and analyze end behavior. Practice exercises and solutions provided.

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Polynomial Functions

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  1. Polynomial Functions Chapter 7 Algebra 2B

  2. an a0 n– 1 n n leading coefficient an a0 degree n an 0 descending order of exponents from left to right. A polynomial function is a function of the form f(x) = an xn+ an– 1xn– 1+· · ·+ a1x + a0 For this polynomial function, only one variable (x) an is the leading coefficient, a0 is the constant term, and nis the degree. Where an 0 and the exponents are all whole numbers. A polynomial function is in standard form if its terms are written in descending order of exponents from left to right.

  3. Degree Type Standard Form You are already familiar with some types of polynomial functions. Here is a summary of common types ofpolynomial functions. 0 Constant f (x) = a0 1 Linear f (x) = a1x + a0 2 Quadratic f (x) = a2x2+a1x + a0 3 Cubic f (x) = a3x3+ a2x2+a1x + a0 4 Quartic f (x) = a4x4 + a3x3+ a2x2+a1x + a0

  4. Value of a function: f (k) One way to evaluate polynomial functions is to usedirect substitution. Another way to evaluate a polynomialis to use synthetic substitution.

  5. Real Zeros of a polynomial function: Maximum number of real zeros is equal to the degree of the polynomial.Real zeros: where the graph crosses the x-axis. How many (total) zeros do the following functions have? f(x) = x + 2 g(x) = x2– 4 h(x) = x3– 2x2 – 10x + 20 p(x) = x4 + 8x2 - 10 one: –2 two: 2, –2 three: -3.16, 2, 3.16 four: -1, 1 & 2 complex The zeros may be real or complex... The Fundamental Theorem of Algebra: Counting complex and repeated solutions, an nth degree polynomial equation has exactly n solutions.

  6. The end behavior of a polynomial function’s graphis the behavior of the graph, which isf(x), as x approaches infinity(+ ) or negative infinity (– ). The expression x+ is read as “x approaches positive infinity.” END BEHAVIOR OFPOLYNOMIAL FUNCTIONS

  7. A B C D

  8. Identifying Polynomial Functions 1 f(x) = x2– 3x4– 7 2 1 Its standard form is f(x) = –3x4+x2 – 7. 2 Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, typeand leading coefficient. SOLUTION The function is a polynomial function. It has degree 4, so it is a quartic function. The leading coefficient is – 3.

  9. Identifying Polynomial Functions f(x) = x3+ 3x Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, typeand leading coefficient. SOLUTION The function is not a polynomial function because the term 3xdoes not have a variable base and an exponentthat is a whole number.

  10. Identifying Polynomial Functions f(x) = 6x2+ 2x–1+ x Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, typeand leading coefficient. SOLUTION The function is not a polynomial function because the term2x–1has an exponent that is not a whole number.

  11. Identifying Polynomial Functions f(x) = –0.5x+x2– 2 Its standard form is f(x) = x2– 0.5x– 2. Decide whether the function is a polynomial function. If it is, write the function in standard form and state its degree, typeand leading coefficient. SOLUTION The function is a polynomial function. It has degree 2, so it is a quadratic function. The leading coefficient is .

  12. Identifying Polynomial Functions 1 f(x) = x2– 3x4– 7 2 f(x) = –0.5x+ x2– 2 Polynomial function? f(x) = x3+ 3x f(x) = 6x2+ 2x–1+ x

  13. Your Turn 1:What are the degree and leading coefficient? a) 3x2 - 2x4 – 7 + x3 b) 100 -5x3 + 10x7 c) 4x2 – 3xy + 16y2 d) 4x6 + 6x4 + 8x8 – 10x2 + 20

  14. Value of a function Using Direct Substitution Use direct substitution to evaluate Solution: f(x) = 2x4- 8x2 + 5x- 7 when x = 3. f(3) = 2(3)4- 8(3)2 + 5(3)- 7 = 98

  15. Value of a function Using Synthetic Substitution Use synthetic substitution to evaluate f(x) = 2x4 + -8x2 + 5x- 7 when x = 3.

  16. Using Synthetic Substitution Polynomial in standard form 3 Coefficients x-value SOLUTION 2x4 + 0x3 + (–8x2) + 5x + (–7) Polynomial in standard form 2 0 –8 5 –7 3• Coefficients 6 18 30 105 35 10 98 2 6 The value of f(3) is the last number you write, In the bottom right-hand corner.

  17. Your Turn 2: Use direct substitution. • f(x)= 2x2 - 3x + 1 f(-4)

  18. Your Turn 3: Use direct substitution. • f(x)= x2 – 4x – 5 f(a2-1)

  19. Your Turn 4: Use Synthetic SubstitutionFind f(2) a. 3x2 - 2x4 – 7 + x3 Your Turn 5: Use Synthetic SubstitutionFind f(-5) b. 100 – 5x3 + 10x4

  20. f(x)= f(x)= f(x)= f(x)= f(x)= f(x)= f(x)= f(x)= END BEHAVIOR f(x)= GRAPHING POLYNOMIAL FUNCTIONS

  21. For each graph below,  describe the end behavior,  determine whether it represents an odd-degree or an even-degree function, and  state the number of real zeros. Your Turn 6: Degree: ___ End behavior: # real zeros: _____

  22. For each graph below,  describe the end behavior,  determine whether it represents an odd-degree or an even-degree function, and  state the number of real zeros. Your Turn 7: Degree: ___ End Behavior # real zeros: _____

  23. For each graph below,  describe the end behavior,  determine whether it represents an odd-degree or an even-degree function, and  state the number of real zeros. Your Turn 8: Degree: ___ End behavior: # real zeros: _____

  24. For each graph below,  describe the end behavior,  determine whether it represents an odd-degree or an even-degree function, and  state the number of real zeros. Your Turn 9: Your own: Degree: ___ End behavior: # real zeros: _____

  25. Closure 7.1

  26. Lesson 7.1 • Check for understanding: Closure 7.1 • Homework: Practice 7.1

  27. Warm-up 7.1

  28. Warm-up 7.1

  29. 7.2 Graphs of Polynomial Functions

  30. We have learned how to graph functions with the following degrees: 0 Example: f(x) = 2 horizontal line 1 Example: f(x) = 2x – 3 line 2 Example: f(x) = x2 + 2x – 3 parabola How do you graph polynomial functions with degrees higher than 2?

  31. We’ll make a table of values, then graph... Graphs of Polynomial Functions: are continuous (there are no breaks) have smooth turns with degree n, have at most n – 1turns Follows end behavior according to n (even or odd) and to an (positive or negative).

  32. End behavior of a polynomial function: END BEHAVIOR FOR POLYNOMIAL FUNCTIONS annx – x + > 0 even f(x) + f(x) +  > 0 odd f(x) – f(x) +  < 0 even f(x) – f(x) –  < 0 odd f(x) + f(x) –  A B C D

  33. Graphing Polynomial Functions –3 –2 –1 0 1 2 3 x The degree is odd and the leading coefficient is positive, so f(x) –7 3 3 –1 –3 3 23 f(x) . and f(x) – – + + as x as x Graph: f(x) =x3 + x2 - 4x - 1 n: __ an: ___ # turns: at most ____ # total zeros: ____ # real zeros: at most _____ # Real Zeros: ___ between ___ and ___ x = _________ between ___ and ___ x = _________ between ___ and ___ x = _________ Relative Maxima value: ____ @ x = ____ Relative Minima value: ____ @ x = ____ left right

  34. # Real zeros: ______ Zeros: ______ (exact) & _______(exact) between ____ and ____ ( x = _____) between ____ and _____ ( x = _____) Relative Maxima: _____ @ ________ _____ @ ________ Relative Minima: _____ @ ________

  35. # Real zeros: ______ Zeros: ______ (exact) & _______(exact) between ____ and ____ ( x = _____) between ____ and _____ ( x = _____) Relative Maxima: _____ @ ________ _____ @ ________ Relative Minima: _____ @ ________

  36. # Real zeros: ______ Zeros: ______ (exact) & _______(exact) between ____ and ____ ( x = _____) between ____ and _____ ( x = _____) Relative Maxima: _____ @ ________ _____ @ ________ Relative Minima: _____ @ ________

  37. Lesson 7.2 • Check for understanding: Closure 7.2 • Homework: Practice 7.2

  38. Closure 7.2

  39. Warm-up/Review Lesson 7.2 a) Real Zeros: check table ___________________________________ b) x-coord. ______ (max/min?) Value: ______ x-coord. ______ (max/min?) Value: ______ a) Real Zeros: use 2nd calc___________________________________ b) x-coord. ______ (max/min?) Value: ______ x-coord. ______ (max/min?) Value: ______ x-coord. ______ (max/min?) Value: ______

  40. Warm-up/Review Lesson 7.2 a) Real Zeros: ___________________________________ b) x-coord. ______ (max/min?) Value: ______ x-coord. ______ (max/min?) Value: ______ a) Real Zeros: ___________________________________ b) x-coord. ______ (max/min?) Value: ______ x-coord. ______ (max/min?) Value: ______ x-coord. ______ (max/min?) Value: ______

  41. Zeros: -.879; 1.347; 2.532 Zeros: -.1.911; 0.160; 3.2509 Min @ x = 0value:-3 Max @ x = 2value:1 Max @ x = -1.0value:4.5 Min @ x = 2value:-9

  42. Min @ x = -1value:-4 Min @ x = -2value:-4 Zeros: -2.574; -1.121 Min @ x = 1value:2.75 Max @ x = 0value:4 Zeros: -2.414; .414

  43. 7.3: Solving Polynomial Equations by using quadratic techniques

  44. Vocabulary Quadratic form: Quadratic formula: u

  45. Example 1: Write the given expression in quadratic form, if possible Answer:

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