GIAO AN DAY TOAN BANG TIENG ANH LOP 8

Vinh Tuong Education Departure Vinh Tuong Secondary School... Therefore balance polynomial divided by polynomial Px for polynomial Qx is 5x +5... Thank you so much for attending our clas

Vinh Tuong Education Departure Vinh Tuong Secondary School

1 2 3

Unit 8: Period 20th

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

 







 

1 is or

7 13 1









  



Problem 1: How many positive integers n are

A= prime numbersn3  8 n2  20 n  13

SOLUTION

1 1 If

n

 









2

1 is

7 13 1

 









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 a2 + 4b = 7;

b2 +8c = -10 and c2+ 6a = -26 Find T = a2+ b3+ c4

Solution

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

  



  





 



 a2+ 4b + b2+ 8c + c2+ 6a + 29 = 0

PRACTICE PROBLEMS ON POLYNOMIALS

 (a + 3)2+ (b + 2)2+ (c + 4)2= 0

2 3 4

9

a





( a2+ 6a + 9) + ( b2+ 4b + 4) + (c2 + 8c + 16) = 0

PRACTICE PROBLEMS ON POLYNOMIALS

Therefore T = a2+ b3+ c4= 257

PRACTICE PROBLEMS ON POLYNOMIALS

Problem 3: Find the balance polynomial divided by

polynomial P(x) =5 + x + x3 + x9 + x27 + x81 for

polynomial Q(x) = x2 - 1

Solution

We have:

P(x) = 5 +5 x + (x3 - x)+(x9 - x)+(x27 - x) +(x81 - x)

PRACTICE PROBLEMS ON POLYNOMIALS

= x(x2 - 1) + x(x8 - 1) + x(x26- 1)+x(x80 - 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) = (x2 - 1) A(x) + ax + b

(A(x) is quotient polynomial)

 

 

or

5

er

Apply the Bezout the em

We have

P a b

a b

Th efore balance polynomial divided by

  





   



PRACTICE PROBLEMS ON POLYNOMIALS

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:

integer triples (x; y; z) that satisfy equations

x2 + y - z = 100 and x + y2 - z = 124

satisfy the following conditions:

x3 + y3 = 2z3 x + y + z is a prime number

Thank you so much for attending our class