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(1)(2)1 2 3
(3)(4)PRACTICE PROBLEMS ON POLYNOMIALS
We have:
n3 - 8n2 + 20n - 13 = (n - 1)(n2 - 7n +13)
Because n3 - 8n2 + 20n - 13 are prime numbers
2
1
7 13 is n
SO
n n a prime is or
7 13
n a prime n n
Problem 1: How many positive integers n are A= prime numbersn3 8n2 20n 13
(5)2
1 1 If
7 13 is 2
n
n n a prime n
1 is
If 3 or 4
7 13 1
n a prime
n n n n
PRACTICE PROBLEMS ON POLYNOMIALS
(6)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 2
2
4
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 =
(7) (a + 3)2+ (b + 2)2+ (c + 4)2=
2
2
9
3 0 3
2 0 2 8
4 0 4 256
a b c 256 257 a
a a
b b b
c c c
( a2+ 6a + 9) + ( b2+ 4b + 4) + (c2 + 8c + 16) =
PRACTICE PROBLEMS ON POLYNOMIALS
(8)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 -
Solution
We have:
(9)PRACTICE PROBLEMS ON POLYNOMIALS
= x(x2 - 1) + x(x8 - 1) + x(x26- 1)+x(x80 - 1) + 5x +
Note that a2n – b2n(a - b) from nN.So (x2n-1)(x2 - 1)
P(x) : Q(x) balance polynomial 5x +
Therefore balance polynomial divided by
polynomial P(x) for polynomial Q(x) is 5x +5
(10)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 10 10
Apply the Bezout the em We have
P a b
a b
P a b
(11)PRACTICE PROBLEMS ON POLYNOMIALS
Similar exercises:
Find the balance polynomial divided by polynomial P(x) = x81 + x49 + x25 + x9 +x +
(12)PRACTICE PROBLEMS ON POLYNOMIALS
V) Homework:
- Review all the exercises that we 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 2: Find the natural numbers x; y; z that satisfy the following conditions:
x3 + y3 = 2z3 x + y + z is a prime number
(13)