1) Даны многочлены P(X)=5X² - 1 ; Q(X)=5X² + 1;R(X)=X⁴ + 2X² + 1 Найдите 1) P(X)+Q(X)-R(X) 2)P(X) x Q(X) - R(X) 3)P(X)

1) P (X) = 5 * X^2 - 1; Q (X) = 5 * X^2 + 1; R (X) = X^4 + 2 * X^2 + 1; 

Найдем: 

1) P (X) + Q (X) - R(X) =  5 * x^2 - 1 + 5 *  x^2 + 1 - x^4 - 2 * x^2 - 1 = 10 * x^2 - x^4 - 2 * x^2 - 1 = 8 * x^2 - x^4 - 1; 

2) P (X) x Q (X) - R (X) = (5 * x^2 - 1) * (5 * x^2 + 1) - x^4 - 2 * x^2 - 1 = 25 * x^4 - 1 - x^4 - 2 * x^2 - 1 = 24 * x^4 - 2 * x^2 - 2; 

3) P (X) x Q (X) = (5 * x^2 - 1) * (5 * x^2 + 1) = 25*  x^4 - 1; 

4) P^2 (X)  = (5 * x^2 -1) = 25 * x^4 - 10 * x^2 + 1;

5) P(X) x R(X)  = (5 * x^2 - 1) * (x^4 + 2 * x^2 + 1) = 5 * x^6 + 10 * x^4 + 5 * x^2 - x^4 - 2 * x^2 - 1 = 5 * x^6 + 9 * x^4 + 3 * x^2 - 1; 

6) P^ (X) x Q^2 (X) = (5 * x^2 - 1)^2 * (5 * x^2 + 1)^2 = (25 * x^4 - 1)^2 = 625 * x^8 - 50 * x^4 + 1; 

2) P (X)= -X^4 - 8 * X^3 + X^2 - 3 * X и Q (X) = X^3 + 5 * X^2; 

1) P(X) + Q(X) = S(X); 

S (X) = -x^4 - 8 * x^3 + x^2 - 3 * x + x^3 + 5 * x^2 = -x^4 - 7 * x^3 + 6 * x^2 - 3 * x; 

P (X) - Q (X) = D(X); 

D (X) = -X^4 - 8 * X^3 + X^2 - 3 * X - x^3 - 5 * x^2 = -x^4 - 9 * x^3 - 4 * x^2 - 3 * x; 

Q (X) - P (X) = R (X); 

R (x) = x^3 + 5 * x^2 + x^4 + 8 * x^3 - x^2 + 3 * x = 9 * x^3 + 4 * x^2 + x^4 + 3 * x; 

Вычислим и сравним: 

a) S (3) c P (3) + Q (3); 

S (3) = -81 - 9 * 27 - 4 * 9 - 3 * 3 = -81 - 243 - 36 - 9 = -369;  

P (3) = -81 - 8 * 27 + 9 - 9 = -81 - 216 = -297; 

Q (3) = 27 + 8 * 9 = 27 + 72 = 99; 

Тогда: 

-369 и -297 + 99; 

-369 и -198; 

-369 < - 198; 

Значит,  S (3) < P (3) + Q (3); 

б) D (-1) c P (-1) - Q(-1); 

D (-1) = -1 + 9 - 4 + 3 = 7; 

P (-1) = -1 + 8 + 1 + 3 = 8 + 3 = 11; 

Q (-1) = -1 + 5 = 4; 

Тогда: 

7 и 11 - 4; 

7 = 7; 

Значит, D (-1) = P (-1) - Q(-1); 

3) Укажем степени многочленов S(X), D(X) , R(X);  

S (X) и D (X) - четвертая степень; 

R (X) - третья степень; 

4) D(X)/R(X) = (-x^4 - 9 * x^3 - 4 * x^2 - 3 * x)/(9 * x^3 + 4 * x^2 + x^4 + 3 * x).