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Mathematical Investigations III Name: Mathematical Investigations III - A View of the World Sum/Product of Roots 1. Let f(x) = x2 – 3x – 10 and let r1 and r2 be the zeroes (roots) of f(x). (a) Solve the equation f ( x) 0 to find r1 and r2. (b) Find the sum of the roots, r1 r2 . (c) Find the product of the roots, r1 r2 . (d) What is the relationship between your answers to (b) and (c) and the coefficients of f (x ) ? Let's look at the relationships between the roots of a polynomial and its coefficients. Let P2 x x r1 x r2 . Then the roots of P ( x) are r and r . Let P1 x x r1 . P1 has a single root r1. 2 1 2 In expanded form, we have P2 ( x) x (r1 r2 ) x r1r2 , or P2 ( x) x 2 (sum of roots) x product of roots 2 2. Let P3 x x r1 x r2 x r3 . Expand P3 ( x) and combine like terms as above. 3. Repeat this process with P4 x x r1 x r2 x r3 x r4 . (Note: Wolfram|Alpha will do this for you if you just input P4 ( x) in factored form and ask it to “expand”.) Poly 10.1 Rev.S11 Mathematical Investigations III Name: 4. Let f ( x) ( x 1)5 . (a) Expand f (x) . (b) What are the sum and product of the roots of f (x) ? 5. Let g ( x) ( x 1)6 . (a) Expand g (x ) . (b) What are the sum and product of the roots of g (x ) ? 6. Reflecting on your answers to the last four questions, describe how you can use the coefficients of a polynomial to determine the sum and product of its roots? Be precise. Apply what you have observed above: 7. Given each of the following polynomials, find both the sum and the product of the roots. 2 3 2 (a) f (x) x 12x 25 (b) g(x) x 4x 2x 20 8. Find the sum and products of the solutions to the following polynomial equations. 4 3 2 (a) x 5x x 6x 3 = 0 3 (b) x 12x 28 0 Poly 10.2 (Caution) Rev.S11 Mathematical Investigations III Name: 2 9. The sum of the solutions of the polynomial equation 2x 10x 28 0 is NOT 10 and the product is NOT –28. (a) Solve the polynomial equation above to find its actual roots. What is their sum? What is their product? (b) Why is the sum NOT 10 and the product NOT -28? (c) Explain what must be done to find the sum and product of the roots when the leading coefficient is not 1. 10. Find the sum and product of the solutions of: 2 (a) 6x 20x 16 0 3 2 (b) 2x 3x 6x 9 0 11. The polynomial p( x) x3 4 x 2 7 x 10 has roots at x 1 and x 2 . Where is the third root? How do you know? 12. The polynomial w( x) 2 x3 17 x 2 80 x 125 has a root at x 3 4i . From previous experience you should be able to write down a second root. Then using the ideas in this packet, you should be able to find the third root. What are these two roots? Explain how you found them. Poly 10.3 Rev.S11 Mathematical Investigations III Name: Extra practice. 13. Using the pattern you observed in Question 6, find a quadratic polynomial function with integer coefficients whose zeros are 3 5 and 3 5 . Write your answer in the form f ( x) ax 2 bx c . 14. Stu Dent solved a quadratic equation with integer coefficients by factoring and setting each factor equal to zero. His answers are: x 2 and x 4 . Use the pattern you found in 3 3 Question 6 to find a possible equation that Stu could have solved. 15. Without actually solving for x, find the sum of the squares of the solutions of the 3 2 polynomial equation x 9x 12x 14 0 . (Hint: Look back to Question 2.) 16. The polynomial q( x) x3 5 x 2 9 x 45 has only one real root, namely at x 5 . Without doing any sort of computation, explain why the other two roots of q(x) are purely imaginary (that is, bi for some real number b). Poly 10.4 Rev.S11 Mathematical Investigations III Name: Poly 10.5 Rev.S11