Bài 3:
x=19 nên x+1=20
\(M=x^6-20x^5+20x^4-20x^3+20x^2-20x+20\)
\(=x^6-x^5\left(x+1\right)+x^4\left(x+1\right)-x^3\left(x+1\right)+x^2\left(x+1\right)-x\left(x+1\right)+x+1\)
\(=x^6-x^6-x^5+x^5+\cdots+x^3+x^2-x^2-x+x+1\)
=1
Bài 4:
Q(-1)=6 nên \(a\cdot\left(-1\right)^2+b\cdot\left(-1\right)+c=6\)
=>a-b+c=6
Tổng các hệ số là 0 nên a+b+c=0
mà a-b+c=6
nên a-b+c-(a+b+c)=6-0=6
=>-2b=6
=>b=-3
=>a+c=-b=-(-3)=3
b=-3 nên \(Q\left(x\right)=a\cdot x^2-3\cdot x+c\)
Q(2)=3 nên \(a\cdot2^2-3\cdot2+c=3\)
=>4a+c=3+6=9
mà a+c=3
nên 4a+c-a-c=9-3
=>3a=6
=>a=2
a+c=3
=>c=3-a=3-2=1
Bài 5:
P(x)=Q(x)
=>\(x^3+3x^2+3x+1=x^3+2x^2+8x-5\)
=>\(x^3+3x^2+3x+1-x^3-2x^2-8x+5=0\)
=>\(x^2-5x+6=0\)
=>(x-2)(x-3)=0
=>\(\left[\begin{array}{l}x-2=0\\ x-3=0\end{array}\right.\Rightarrow\left[\begin{array}{l}x=2\\ x=3\end{array}\right.\)
