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Ta có : x=100=>101=x+1
Thay vào f(x), ta được : x10 -(x+1)x9 +(x+1)x8 - (x+1)x7 +....-(x+1)x +100
<=> x10 - x10 -x9 +x9 + x8 -x8 -x7 +.... -x2 -x +100
<=> -x+100
=> f(100) = -x+100=-100+100=0
ta có :
\(f\left(x\right)=x^{10}-101x^9+101x^8-...-101x+101\)
\(=x^{10}-x^9-100x^9+x^8+100x^8-...-x-100x+100+1\)
ta có :
\(f\left(100\right)=100^{10}-100^9-100\times100^9+100^8+100\times100^8-...-100-100\times100+100+1\)
\(=100^{10}-100^{10}-100^9+100^9+100^8-...-100^2-100+100+1\)
\(=1\)
vậy f(100)=1
Ta có:
\(x^{10}-\left(100+1\right)x^9+\left(100+1\right)x^8-\left(100+1\right)x^7+.....-\left(100+1\right)x+100+1\)
\(=x^{10}-100x^9-x^9+100x^8+x^8-100x^7-x^7+......-100x-x+100+1\)
\(f\left(x\right)=x^{10}-\left(100+1\right)x^9+\left(100+1\right)x^8-\left(100+1\right)x^7+...-\left(100+1\right)x+100+1\)
\(=x^{10}-100x^9-x^9+100x^8+x^8-100x^7-x^7+...-100x-x+100+1\)
\(=x^9\left(x-100\right)-x^8\left(x-100\right)+x^7\left(x-100\right)-...+x\left(x-100\right)-\left(x-100\right)+1\)
\(=\left(x-100\right)\left(x^9-x^8+x^7-...+x-1\right)+1\)
Ta có: \(f\left(100\right)=\left(100-100\right)\left(100^9-100^8+100^7-...+100-1\right)+1\)
\(=0+1=1\)
Vậy f(100) = 1.
Ta hiểu tính f(100) thì tính f(x) tại x=100
Ta có:x+1=100+1=101
Thay vào biểu thức thì ta có:
\(f\left(x\right)=x^{10}-101x^9+101x^8-101x^7+...-101x+30\)
\(f\left(x\right)=x^{10}-\left(x+1\right)x^9+\left(x+1\right)x^8-\left(x+1\right)x^7+...-\left(x+1\right)x+30\)
\(f\left(x\right)=x^{10}-x^{10}-x^9+x^9+x^8-x^8-x^7+...-x^2-x+30\)
\(f\left(x\right)=-x+30\)
\(f\left(100\right)=-100+30=-70\)
Vậy f(100)=-70
f(x) = x8 - 101x7 + 101x6 - 101x5 +...+ 101x2 - 101x + 25
f(x) = x8 - 100x7 - x7 + 100x6 + x6 - 100x5 - x5 +...+ 100x2 + x2 - 100x - x + 25
f(x) = x7(x - 100) - x6(x - 100) + x5(x - 100) -...+ x(x - 100) - (x - 25)
f(100) = 1007(100 - 100) - 1006(100 - 100) + 1005(100 - 100) -...+ 100(100 - 100) - (100 - 25)
f(100) = 0 - 0 + 0 -...+ 0 - 75
f(100) = -75
Ta có : với f(100) thì x = 100 ( điều đương nhiên)
=> 101 = x + 1
Thay vào f(x) , ta lại có :
\(f\left(x\right)=x^8-101x^7+101x^6-101x^5+...+101x+25\)
\(f\left(x\right)=x^8-\left(x+1\right)x^7+\left(x+1\right)x^6-\left(x+1\right)x^5+...+\left(x+1\right)x+25\)
\(f\left(x\right)=x^8-x^8-x^7+x^7+x^6-x^6-x^5+....+x^2+x+25\)
\(f\left(x\right)=x+25\)
\(f\left(100\right)=100+25=125\)
\(f\left(x\right)=x^8-101x^7+101x^6-101x^5+...+101x^2-101x+25\)
\(f\left(x\right)=x^8-100x^7-x^7+100x^6+x^6-100x^5-x^5+...+100x^2+x-100x-x+25\)
\(f\left(x\right)=x^7\left(x-100\right)-x^6\left(x-100\right)+x^5\left(x-100\right)-...+x\left(x-100\right)-x+25\)
\(f\left(100\right)=x^7.0-x^6.0+x^5.0-...+x.0-100+25\)
\(f\left(100\right)=25-100=-75\)
\(f\left(100\right)=100^8-101.\left(100^7-100^6+100^5-100^4+100^3-100^2+100\right)+25\)
ta có: \(100.\left(100^7-100^6+...-100^2+100\right)+\left(100^7-100^6+...-100^2+100\right)=100^8+100\)
\(\Rightarrow f\left(x\right)=100^8-100^8-100+25=-75\)
p/s: cách này ngắn nên tớ làm, ko phải câu (có 1 bạn tl rồi)
\(f\left(x\right)=x^8-101x^7+101x^6-...-101x+25\)
\(f\left(100\right)=x^8-\left(x+1\right)x^7+\left(x+1\right)x^6-...-\left(x+1\right)x+25\)
\(f\left(100\right)=x^8-x^8-x^7+x^7+x^6-...-x^2-x+25\)
\(f\left(100\right)=-x+25=-100+25=-75\)
Ta có:
\(F\left(100\right)=100^{10}-101.100^9+101.100^8-101.100^7+...-101.100+101\)
\(=100-\left(100+1\right).100^9+\left(100+1\right).100^8-\left(100+1\right).100^7+...-\left(100+1\right).100+101\)
\(=100^{10}-100^{10}-100^9+100^9+100^8-100^8-100^7+...-100^2-100+101\)
\(=1\)
Ta có:\(101=100+1=x+1\)
\(\Rightarrow F\left(100\right)=x^{10}-\left(x+1\right)x^9+\left(x+1\right)x^8-...-\left(x+1\right)x+x+1\)
\(=x^{10}-x^{10}-x^9+x^9+x^8-...-x^2-x+x+1=1\)