Advanced Algebra Name ________________________________________ HW FOR LESSON 5.7 POLYNOMIALS AND POLYNOMIAL FUNCTIONS Seat # ______ Date _____________________________ 5.7 Fundamental Theorem of Algebra DO ALL YOUR WORK ON A SEPARATE SHEET State how many solutions or zeros each equation or function has, including “multiple” ones. 1. f x   x 4  2 x 3  x  10 2. 4x 3  9  x 5  6x 2  x 3. x7  x  6 4. hx   x  x 2  x 6  44 5. 6. Given the polynomial function g x   x 4  3x 3  7 x 2  15x  18 , a) How many zeros does the function have? b) The given polynomial can be factored as g x   x  1  x  32 x  2 . Identify all distinct zeros of the polynomial. Does your answer contradict what you answered to part (a)? Explain. c) How many x-intercepts will the graph of g x  have? Does your answer contradict what you answered to parts (a) and/or (b)? Explain. Given the polynomial function px   x 3  5x 2  2 x  10 , a) How many zeros does the function have? b) The given polynomial can be factored as px   x  5  x 2  2 . Identify all distinct zeros of the polynomial. (Note: include complex numbers.) c) How many x-intercepts will the graph of px  have? Does your answer contradict what you answered to parts (a) and/or (b)? Explain.  7.  Given the polynomial function r x   2 x 4  x 3  14 x 2  7 x , a) How many zeros does the function have? b) The given polynomial can be factored as r x   x  2 x  1  x 2  7 . Identify all distinct zeros of the polynomial. c) How many x-intercepts will the graph of r x  have? Does your answer contradict what you answered to parts (a) and/or (b)? Explain.   8. What would be the least degree possible for a polynomial with known roots equal to 5,  7 , 6  3i , and 4  7 ? Explain your answer. 9. What would be the least degree possible for a polynomial with known roots equal to  7 , 2i , and  4 ? Explain your answer. 10. While discussing the homework, Andy told Joan that he was only able to find two real solutions to a cubic equation. Andy concluded that the third solution must be a complex solution. Joan told Andy that he is wrong. Write a paragraph to help Joan explain to Andy the reason(s) he must be wrong. Write a polynomial function of least degree with rational coefficients, a leading coefficient of 1, and the given zeros. Write your answers in standard form. 11. 1, −4, 3 12. −1, −5, 2, 0 Advanced Algebra ANSWER SHEET HW FOR LESSON 5.7 POLYNOMIALS AND POLYNOMIAL FUNCTIONS 5.7 Fundamental Theorem of Algebra 1. 4 zeros 2. 5 solutions 3. 7 solutions 4. 6 zeros 5. a) 4 zeros b) Zeros are x  1 , x  3 , and x  2 . There are 3 distinct zeros. The answer does not contradict (a) because x  3 is a multiple zero (it counts as two zeros.) c) The graph will have three different x-intercepts. Since x  3 is a multiple zero this does not contradict the answer to (a). 6. a) 3 zeros b) Zeros are x  5 and x  i 2 . There are 3 distinct zeros, two of them are complex numbers. c) The graph will have one x-intercept. Since two of the zeros are complex this does not contradict the answer to (a). 7. a) 4 zeros b) Zeros are x  0 , x   12 and x   7 . There are 4 distinct zeros. c) The graph will have four different x-intercepts. In this case, since all zeros are real numbers and there are no multiple zeros, the number of zeros of the polynomial is equal to the number of xintercepts. 8. The degree should be six. Since complex and irrational roots come in pairs (due to the conjugate roots theorem), x  6  3i and x  4  7 must also be roots of the polynomial. 9. The degree should be five. Since complex and irrational roots come in pairs (due to the conjugate roots theorem), x  7 and x  2i must also be roots of the polynomial. 10. We know Andy is wrong because a polynomial equation cannot have a single complex solution: complex answers come in pairs as stated by the conjugate roots theorem. There are two possible explanations as to what happened. One of the roots Andy already found might be a multiple root, bringing up the total of roots to three. Andy might also have made a mistake and there might be only one real solution to the cubic equation, leaving two conjugate complex answers. 11. g x   x 3  13x  12 12. f x   x 4  4 x 3  7 x 2  10 x