Vinh Tuong Education Departure Vinh Tuong Secondary School 1 2 3 4 5 6 Unit 8: Period 20 th PRACTICE PROBLEMS ON POLYNOMIALS We have: n 3 - 8n 2 + 20n - 13 = (n - 1)(n 2 - 7n +13) Because n 3 - 8n 2 + 20n - 13 are prime numbers 2 1 1 7 13 is n SO n n a prime − =    − +   2 1 is or 7 13 1 n a prime n n −    − + =   Problem 1: How many positive integers n are A= prime numbers 3 2 8 20 13n n n − + + SOLUTION 2 1 1 If 7 13 is 2 n n n a prime n − =    − + ∴ =   2 1 is If 3 or 4 7 13 1 n a prime n n n n  −  ∴ = =  − + =   PRACTICE PROBLEMS ON POLYNOMIALS Therefore n = 2; n = 3; n = 4 then n 3 – 8n 2 + 20n - 13 are prime numbers Problem 2: Solve the following exercises: If a, b, c are real numbers so that a 2 + 4b = 7; b 2 +8c = -10 and c 2 + 6a = -26. Find T = a 2 + b 3 + c 4 . Solution 2 2 2 2 2 2 4 7 8 10 a + 4b+b +8c+c +6a = 7+(-10)+(-26) 6 26 a b We have b c c a  + =  + = − ∴   + = −  ∴ a 2 + 4b + b 2 + 8c + c 2 + 6a + 29 = 0 PRACTICE PROBLEMS ON POLYNOMIALS ∴ (a + 3) 2+ (b + 2) 2+ (c + 4) 2= 0 ( ) 2 3 4 2 3 4 9 3 0 3 2 0 2 8 4 0 4 256 a b c 9 8 256 257 a a a b b b c c c  = + = = −      ∴ + = ∴ = − ∴ = −       + = = − =    ∴ + + = + − + = (∴ a 2 + 6a + 9) + ( b 2 + 4b + 4) + (c 2 + 8c + 16) = 0 PRACTICE PROBLEMS ON POLYNOMIALS Therefore T = a 2 + b 3 + c 4 = 257 PRACTICE PROBLEMS ON POLYNOMIALS Problem 3: Find the balance polynomial divided by polynomial P(x) =5 + x + x 3 + x 9 + x 27 + x 81 for polynomial Q(x) = x 2 - 1 Solution We have: P(x) = 5 +5 x + (x 3 - x)+(x 9 - x)+(x 27 - x) +(x 81 - x) PRACTICE PROBLEMS ON POLYNOMIALS = x(x 2 - 1) + x(x 8 - 1) + x(x 26 - 1)+x(x 80 - 1) + 5x + 5 Note that a 2n – b 2n (a - b) from n∈N.So (x 2n -1)(x 2 - 1) ∴ P(x) : Q(x) balance polynomial 5x + 5. Therefore balance polynomial divided by polynomial P(x) for polynomial Q(x) is 5x +5. Second way: PRACTICE PROBLEMS ON POLYNOMIALS Let balance polynomial divided by polynomial P(x) for polynomial Q(x) is R(x) = ax + b (a; b ∈ R) We have: P(x) = (x 2 - 1). A(x) + ax + b (A(x) is quotient polynomial). ( ) ( ) or . 1 10 5 1 10 er ( ) or ( ) is 5 5 Apply the Bezout the em We have P a b a b P a b Th efore balance polynomial divided by polynomial P x f polynomial Q x x = + =  ∴ = =  − =− + =   + [...]... Similar exercises: Find the balance polynomial divided by polynomial P(x) = x81 + x49 + x25 + x9 +x + 1 for polynomial Q(x) = x3 - 1 PRACTICE PROBLEMS ON POLYNOMIALS V) Homework: - Review all the exercises that we do today - Solve the following exercises: Question 1: Find the numbers of different positive integer triples (x; y; z) that satisfy equations x2 + y - z = 100 and x + y2 - z = 124 Question... x; y; z that satisfy the following conditions: x3 + y3 = 2z3 x + y + z is a prime number Question 3: What is the smallest possible value of: P = x2 + y2 - x - y - xy Q = 6x2 + 6y2 - 3x + 3y + 6xy +2 Thank you so much for attending our class . 2 3 4 5 6 Unit 8: Period 20 th PRACTICE PROBLEMS ON POLYNOMIALS We have: n 3 - 8n 2 + 20n - 13 = (n - 1)(n 2 - 7n +13) Because n 3 - 8n 2 + 20n - 13. n 3 – 8n 2 + 20n - 13 are prime numbers Problem 2: Solve the following exercises: If a, b, c are real numbers so that a 2 + 4b = 7; b 2 +8c = -10 and c