Operation of polynomials (Mathematics) 1. It is shown that for any positive integer n, the value of formula n (n + 5) - (n-3) (n + 2) can be divisible by 6 2. If the product of (X & # 178; + PX + 8) (X & # 178; - 3x + Q) does not contain X & # 178; and X & # 179; terms, find the value of P, Q 3. If 3x & # 179; - x = 1, find the value of 9x Λ 4 + 12x & # 179; - 3x & # 178; - 7x 4. It is known that: X-Y = 4, X & # 178; + Y & # 178; = 26, please find the values of: X Λ 4 + y Λ 4, X Λ 8 + y Λ 8, X Λ 4-y Λ 4 respectively

Operation of polynomials (Mathematics) 1. It is shown that for any positive integer n, the value of formula n (n + 5) - (n-3) (n + 2) can be divisible by 6 2. If the product of (X & # 178; + PX + 8) (X & # 178; - 3x + Q) does not contain X & # 178; and X & # 179; terms, find the value of P, Q 3. If 3x & # 179; - x = 1, find the value of 9x Λ 4 + 12x & # 179; - 3x & # 178; - 7x 4. It is known that: X-Y = 4, X & # 178; + Y & # 178; = 26, please find the values of: X Λ 4 + y Λ 4, X Λ 8 + y Λ 8, X Λ 4-y Λ 4 respectively

1. N (n + 5) - (n-3) (n + 2) = n ^ 2 + 5N - (n ^ 2-n-6) = 6N + 6; the original formula divided by 6 equals n + 1
2、（x²+px+8)(x²-3x+q)
=(x^4-3x^3+qx^2)+p(x^3-3x^2+qx)+(8x²-24x+8q)
=x^4+(-3+p)x^3+(q-3p+8)x^2+(pq-24)x+8q
There are (- 3 + P) = 0, (q-3p + 8) = 0
p=3,q=1；
3. From 3x & # 179; - x = 1,3x & # 179; = x + 1,9x Λ 4 = 3x (3x & # 179;) = 3x (x + 1)
Original formula = 3x (x + 1) + 12x & # 179; - 3x & # 178; - 7x
=3x²+3x+12x³-3x²-4x
=12x³-4x
=4（3 x³-x）
=4
（x²+y²）²= x^4+y^4+2 x²y²=26²
x^4+y^4=26²-2 x²y²
4² =(x-y) ²= (x^2+y^2)-2 xy=26-2xy
Xy=(26-4²)/2=5
x^4+y^4=26²-2 x²y²=26²-2*(5^2)=144=12²
Similarly:
x∧8+y∧8=144²-2*(5^4)=20736-2*625=19486
x∧4-y∧4
= （x²+y²）（x²-y²）
=（x²+y²）（x-y）（x+y）
=26*4（x+y）
(x+y)^2
=(x^2+y^2)+2*xy
=26+2*5
=36
X + y = plus or minus 6
X Λ 4-y Λ 4 = 26 * 4 (x + y) = plus or minus 26 * 24 = 25 ^ 2-1 = plus or minus 624