\(x=9\Leftrightarrow x+1=10\\ \Leftrightarrow M=x^{2012}-\left(x+1\right)x^{2012}+...-\left(x+1\right)x^2+\left(x+1\right)x-\left(x+1\right)\\ M=x^{2012}-x^{2013}-x^{2012}+...-x^3-x^2+x^2+x-x-1\\ \Leftrightarrow M=-x^{2013}-1=-9^{2013}-1\)

a) 

\(P=\left(x^{14}-9x^{13}\right)-\left(x^{13}-9x^{12}\right)+\left(x^{12}-9x^{11}\right)-...+\left(x^2-9x\right)-\left(x-9\right)+1\)

\(=x^{13}\left(x-9\right)-x^{12}\left(x-9\right)+x^{11}\left(x-9\right)+...+x\left(x-9\right)-\left(x-9\right)+1\)

\(P\left(9\right)=1\)

b)

\(Q=\left(x^{15}-7x^{14}\right)-\left(x^{14}-7x^{13}\right)+\left(x^{13}-7x^{12}\right)-...-\left(x^2-7x\right)+\left(x-7\right)+2\)

\(=x^{14}\left(x-7\right)-x^{13}\left(x-7\right)+x^{12}\left(x-7\right)-...-x\left(x-7\right)+\left(x-7\right)+2\)

\(Q\left(7\right)=2\)

a, \(A=x^3-30x^2-31x+1\)

\(=x^3-31x^2+x^2-31x+1\)

\(=x^2\left(x-31\right)+x\left(x-31\right)+1\)

\(=\left(x^2+x\right)\left(x-31\right)+1\)

Thay x = 31 \(\Rightarrow A=1\)

Vậy A = 1 khi x = 31

b, tách ra làm tương tự phần a

x=9 nên x+1=10

f(9)=x^50-x^49(x+1)+x^8(x+1)-...+x^2(x+1)-x(x+1)+100

=x^50-x^50-x^49+x^49+x^48-x^48+...+x^3+x^2-x^2-x+100

=-x+100

=-9+100=91

thay 10 = x+1 vào rồi triệt tiêu

bạn nói rõ ra được không

với x=9 thì 10=9+1=x+1

thay vào biểu thức ta có

A=x²⁰¹³ -(x+1)x²⁰¹²+(x+1)x²⁰¹¹+ .....-(x+1)x² +(x+1)x-10

   =x²⁰¹³-x²⁰¹³-x²⁰¹²+x²⁰¹²+x²⁰¹¹-......-x³-x²+x²+x-10

   =x-10=9-10=-1

\(G=x^4+10x^3+10x^2+10x+10\)
\(=x^4+10\left(x^3+x^2+x+1\right)\)
\(=\left(-9^4\right)+10\left[\left(-9\right)^3+\left(-9\right)^2+-9+1\right]\)
\(=6561+10\cdot-656\)
\(=6561-6560\)
\(=1\)

Thay `x=-9` vào biểu thức G:

`G=(-9)^4+10.(-9)^3+10.(-9)^2+10.(-9)+10`

`=6561-7290+810-90+10`

`=1`

Do \(x=-9\Rightarrow x+9=0\)

Ta có:

\(G=\left(x^4+9x^3\right)+\left(x^3+9x^2\right)+\left(x^2+9x\right)+\left(x+9\right)+1\)

\(=x^3\left(x+9\right)+x^2\left(x+9\right)+x\left(x+9\right)+\left(x+9\right)+1\)

\(=x^3.0+x^2.0+x.0+0+1=1\)

x=9

=>x+1=10

\(A=x^{10}-10x^9+10x^8-...+10x^2-10x+1\)

\(=x^{10}-x^9\left(x+1\right)+x^8\left(x+1\right)-...+x^2\left(x+1\right)-x\left(x+1\right)+1\)

\(=x^{10}-x^{10}-x^9+x^8+...+x^3+x^2-x^2-x+1\)

=-x+1

=-9+1=-8

1. 10x²(x - 1) + 15xy(1 - x)

= 10x²(x - 1) - 15xy(x - 1)

= (10x² - 15xy)(x - 1)

= 5x(2x - 3y)(x - 1)

2. Hình như sai đề

2. 25x² - x² + 10x - 2

= 25x² + 10x - x² - 2

= -(25x² - 10x + x²) - 2

= -(5x - x)² - 2

= -(5x - x)² - \(\left(\sqrt{2}\right)^2\)

= -(5x + x + \(\sqrt{2}\))(5x + x - \(\sqrt{2}\))