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Unformatted text preview: Contents IX Gold Problems 1 IX.A Roots of Polynomials and Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 1 IX.B Increasing and Decreasing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 IX.C Constant and Linear Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 IX.D Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 IX.E Rational Functions as x → ±∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 IX.F Complex Numbers and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 IX Gold Problems IX.1 IX Gold Problems 1. i) State the Remainder Theorem. ii) Prove that x n c n is divisible by x c and compute the quotient where n is a positive integer. iii) Under what conditions is x n + c n divisible by x + c . 2. Use the remainder theorem to show that a) p ( x ) = x 4 8 x 2 + 4 x + 3 is divisible by x + 3. b) p ( x ) = 2 x 4 7 x 3 2 x 2 + 13 x + 6 is divisible by x 2 5 x + 6. 3. For the following functions f ( x ) = P ( x ) D ( x ) express f ( x ) in the form f ( x ) = Q ( x ) + R ( x ) D ( x ) and compute f ( x ) in a form free of negative exponents. i) f ( x ) = x 2 x 2 1 . ii) f ( x ) = x 2 +1 x 2 . iii) f ( x ) = x 3 x 2 . iv) f ( x ) = x 3 b x 3 + a . v) f ( x ) = x 3 3 x 2 + x +1 . vi) f ( x ) = x +1 x 2 + x +1 . IX.A Roots of Polynomials and Rational Functions 4. Show that x 3 + ax + b = 0 has exactly one real root if a ≥ 0 and at most one real root between 1 3 p 3  a  and 1 3 p 3  a  if a < 0. 5. Show that ax 3 + bx 2 + cx + d = 0 has exactly one real root if b 2 3 ac < 0. 6. Let P ( x ) = x 3 + cx + d . Prove that if c > 0 there is exactly one real zero of P . 7. Let f be a polynomial. Suppose that f has a double root at a and at b . Show that f ( x ) has at least three roots. 8. Let P ( x ) be a polynomial. Show that ( x a ) 2 divides P ( x ) if and only if P ( a ) = 0 and P ( a ) = 0. 9. i) By evaluating P ( x ) = x 3 + x + 1 at 1 and at 0 show that P ( x ) has at least one root. ii) Show that if a < b then P ( b ) P ( a ) b a ≥ 1. iii) How many roots does P ( x ) have? 10. Does there exist a function f such that f (0) = 1, f (2) = 4 and f ( x ) ≤ 2 for all x . 11. Show that x 5 + 10 x + 3 = 0 has exactly one real root. 12. Show that x 7 + 5 x 3 + x 6 = 0 has exactly one real root. 13. Show that x 5 6 x + c = 0 has at most one distinct root in the interval [ 1 , 1]. 14. Show that x 4 + 4 x + c = 0 has at most two distinct real roots. 15. Give three examples, not all the same type, of polynomials that have no real roots. 16. Prove f α ( x ) = x 3 3 x + α never has two roots in [0 , 1]....
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This note was uploaded on 10/04/2011 for the course MATH 16121 taught by Professor Rachelbelinsky during the Spring '11 term at Georgia State.
 Spring '11
 RACHELBELINSKY

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