Tại x = 16 => x +1 = 17

Thay vào A ta được:

A = x⁴ - (x+1)x³ + (x+1)x² - (x+1)x + 20

A= x⁴ -(x⁴ + x³)  + (x³ + x²)  -(x² + x) +20

A= x⁴ - x⁴ - x³ + x³ + x² - x² -x + 20

A= - x+20

Mà  x = 16

=> A= -16 + 20 = 4

Vậy A= 4 khi x =16

( x + 1 ) ^2 - ( x -1) ^2 - 3 ( x - 1 ) ( x + 1 ) 

= x^2 + 2x + 1 - ( x^2 - 2x + 1 ) - 3 ( x^2 -1 ) 

= x^2 + 2x + 1 - x^2 + 2x -1 - 3x^2 + 3 

= -3x^2 + 4x + 3 

( x + y ) ^3 - ( x - y ) ^3 - 6x^2y 

= x^3 + 3x^2y + 3xy^2 + y^3 - ( x^3 - 3x^2y + 3xy^2 - y^3 ) - 6x^2y 

= x^3 + 3x^2y + 3xy^2 + y^3 - x^3 + 3x^2y - 3xy^2 + y^3 - 6x^2y 

= 2y^3 

Ta có x⁵ + x⁴ - 2x³ - 3x² - x + 1 

= x⁵ + x⁴ - 3x³ - 3x² + (x³ - x) + 1 

= x⁴(x + 1) - 3x²(x + 1) + 2 + 1 

= (x + 1)(x⁴ - 3x²) + 3 

= x(x + 1)(x³ - 3x) + 3 

= x(x + 1)(x³ - x - 2x) + 3 

= x(x + 1)(2 - 2x) + 3 

 = -2x(x + 1)(x - 1) + 3 

= -2(x³ - x) + 3 = -2.2 + 3 = -1

Khi đó P(x) = (x⁵ + x⁴ - 2x³ - 3x² - x + 1)²⁰¹⁸ = (-1)²⁰¹⁸ = 1

12,ĐK : x >= 0

 \(\left(\sqrt{2x}-y\right)^2=2x-2\sqrt{2x}y+y^2\)

13, \(\left(\frac{3}{2}x+3y\right)^2=\frac{9}{4}x+\frac{2.3x}{2}.3y+9y^2=\frac{9}{4}x+9xy+9y^2\)

14, ĐK : x ; y >= 0 

 \(\left(\sqrt{2x}+\sqrt{8y}\right)^2=2x+2\sqrt{16xy}+8y=2x+8\sqrt{xy}+8y\)

15, \(\left(x+\frac{1}{6}y+3\right)^2=x^2+\frac{1}{36}y^2+9-\frac{1}{3}xy-y-6x\)

sửa 13 = \(\frac{9}{4}x^2+9xy+9y^2\)

16, \(\left(\frac{1}{2}x-4y\right)^2=\frac{1}{4}x^2-4xy+16y^2\)

17, \(\left(\frac{x}{2}+2y^2\right)\left(\frac{x}{2}-2y^2\right)=\frac{x^2}{4}-4y^4\)

18, \(\left(x^2-4\right)\left(x^2+4\right)=x^4-16\)

19, \(\left(x+y\right)^2+\left(x-y\right)^2=\left(x+y+x-y\right)^2-2\left(x+y\right)\left(x-y\right)\)

\(=4x^2-2\left(x^2-y^2\right)=2x^2+2y^2\)

20, \(\left(2x+3\right)^2-\left(x+1\right)^2=\left(2x+3-x-1\right)\left(2x+3+x+1\right)=\left(x+2\right)\left(3x+4\right)\)

Ta có: (x+y)³ = 1³

=> x³ + 3x²y + 3xy² + y³ = 1

=> x³+ y³ + 3xy ( x+y) =1

=> x³ + y³ + 3.(-6).1 =1

=> x³ + y³ =19

Từ x+y=1

=> (x+y)⁵ = 1⁵

=> x⁵ + 5x⁴y + 10x³y² + 10x²y³ + 5xy⁴ + y⁵ = 1

=> x⁵ + y⁵ + 5xy(x³ + y³) +  10x²y²( x + y) = 1

=> x⁵ + y⁵ + 5.(-6) . 19 + 10 . (-6)² . 1 =1

=> x⁵ + y⁵ - 210 = 1

=> x⁵ + y⁵= 1 +210 = 211

Vậy x⁵ + y⁵ = 211

(a+b)(a² - ab + b²) [ a⁶ - (ab)³ + b⁶]

= (a³ + b³) (a⁶ - a³b³ + b⁶)

= (a³ + b³) [(a³)² - a³ b³ + (b³)²]

= a⁹ + b⁹

\(B=2x^2-6x+7\)

\(=2\left(x^2-3x+\frac{9}{4}\right)-\frac{9}{2}+7\)

\(=2\left(x-\frac{3}{2}\right)^2+\frac{5}{2}\ge\frac{5}{2}\)

Vậy \(MinB=\frac{5}{2}\Leftrightarrow x=\frac{3}{2}\)

\(C=\left(2x-5\right)^2-4\left(2x-5\right)\)

\(=\left(2x-5\right)\left(2x-5-4\right)=2x-5\)

\(=[\left(2x-5\right)^2-4\left(2x-5\right)+4]-4\)

\(=\left(2x-5-2\right)^2-4\)

\(=\left(2x-7\right)^2-4\ge-4\)

Vậy \(MinC=-4\Leftrightarrow x=\frac{7}{2}\)

(2x-5)^2 -4(2x-5)=(2x-5)^2 -4(2x-5)+4-4=(2x-7)^2 -4>=-4 suy ra C đạt gtnn là -4

2xⁿ⁺¹ (x^n-1 - y^n-1) - y^n-1 (2xⁿ⁺¹ - yⁿ⁺¹)

= 2x²ⁿ -2xⁿ⁺¹ y^n-1 - y^n-1 2xⁿ⁺¹ +y²ⁿ

= 2x²ⁿ + y²ⁿ - 4xⁿ⁺¹ y^n-1