Question

Answer 1

a) A = 153² - 53²

=> A = (153+53)(153-53)

=> A = 206.100

=> A = 20600

b) B = 2020² - 2019² + 2018² - 2017² +...+2² - 1²

=>B = (2020+2019)(2020-2019)+(2018 + 2017).(2018-2017) + ... +(2-1)(2+1)

=> B = 4039 + 4035 +...+3

=> B = (3+4039). 1010 : 2 

=> B = 2041210

c) C = x³ - 3x² + 3x - 1

=> C = (x³ -1) - (3x² - 3x)

=> C = (x-1)(x² +x+1)- 3x(x-1)

=>C = (x-1)(x² +x+1 - 3x)

Thay x= 101 vào C

=> C = (101-1)(101² +101+1-3.101)

=> C = 1000000

d) D = x³ + 6x² +12x + 8

=> D = (x³ + 2³) + (6x² +12x)

=> D = (x +2)(x² - 2x + 4)+ 6x(x+2)

=> D = (x+2)(x² +4x+4)

=> D = (x+2)(x+2)²

=> D = (x+2)³

=> D = (8+2)³

=> D = 1000

e) Ta có x = 9 => 10 = (x+1)

=> E = x²⁰¹³ - (x+1).x²⁰¹² + (x+1).x²⁰¹¹ +...-(x+1).x² + (x+1).x-(x+1)

=> E = x²⁰¹³ - x²⁰¹³ - x²⁰¹² + x²⁰¹² + x²⁰¹¹ +..- x³ - x² + x² + x - x - 1

=> E = -1 

Answer 2

a) \(A=153^2-53^2=\left(153-53\right)\left(153+53\right)=100.206=20600\)

b) \(B=2020^2-2019^2+2018^2-2017^2+...+2^2-1^2\)

\(=\left(2020-2019\right)\left(2020+2019\right)+\left(2018-2017\right)\left(2018+2017\right)+...+\left(2-1\right)\left(2+1\right)\)

\(=\left(2020+2019\right)+\left(2018+2017\right)+...+\left(2+1\right)\)

\(=2020+2019+2018+2017+...+2+1\) (có \(\left(2020-1\right)+1=2020\) (số hạng))

\(=\left(2020+1\right)+\left(2019+2\right)+...+\left(1011+1010\right)\) (có \(2020:2=1010\) (cặp))

\(=2021+2021+...+2021\) (có 1010 (số hạng))

\(=1010.2021\)

\(=2041210\)

c) \(C=x^3-3x^2+3x-1=\left(x-1\right)^3\)

Thay \(x=101\) vào biểu thức C, ta được:

\(C=\left(101-1\right)^3=100^3=1000000\)

d) \(D=x^3+6x^2+12x+8=\left(x+2\right)^3\)

Thay \(x=8\) vào biểu thức D, ta được:

\(D=\left(8+2\right)^3=10^3=1000\)

e) Vì \(x=9\Rightarrow x+1=10,\) thay vào biểu thức E ta được:

\(E=x^{2013}-\left(x+1\right)x^{2012}+\left(x+1\right)x^{2011}+...-\left(x+1\right)x^2+\left(x+1\right)x-\left(x+1\right)\)

\(=x^{2013}-x^{2013}-x^{2012}+x^{2012}+x^{2011}+...-x^3-x^2+x^2+x-x-1\)

\(=-1\)