Ta có:

f(x+3) = a(x+3)²+ b(x+3) +c=ax²+ (6a+b) x+ 9a+ 3b+c

f(x+2) = a(x+2)²+ b(x+2) +c=ax²+ (4a+b) x+ 4a+ 2b+c

f (x+1) = a(x+1)²+ b(x+1) +c=ax²+ (2a+b) x+ 2a+ 2b+c

Suy ra: (x+ 3) -3f( x+ 2) +3f( x+ 1)= ax²+ bx+ c

Chọn D.

\(f\left(x+3\right)-3f\left(x+2\right)+3f\left(x+1\right)\)

\(=a\left(x+3\right)^2+b\left(x+3\right)+c-3\left[a\left(x+2\right)^2+b\left(x+2\right)+c\right]+3\left[a\left(x+1\right)^2+b\left(x+1\right)+c\right]\)

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

\(=ax^2+bx+c\)

-13/32

Lời giải:

Ta có: \(f(x)=ax^2+bx+c\)

\(\Rightarrow \left\{\begin{matrix} f(x+3)=a(x+3)^2+b(x+3)+c\\ f(x+2)=a(x+2)^2+b(x+2)+c\\ f(x+1)=a(x+1)^2+b(x+1)+c\\ f(x)=ax^2+bx+c\end{matrix}\right.\)

\(\Rightarrow f(x+3)-3f(x+2)+3f(x+1)-f(x)\)

\(=[f(x+3)-f(x)]-3[f(x+2)-f(x+1)]\)

Có:

\(f(x+3)-f(x)=a(x+3)^2+b(x+3)+c-[ax^2+bx+c]\)

\(=a[(x+3)^2-x^2]+b(x+3-x)\)

\(=3a(2x+3)+3b(1)\)

Và: \(f(x+2)-f(x+1)=a[(x+2)^2-(x+1)^2]+b[(x+2)-(x+1)]\)

\(=a(2x+3)+b\)

\(\Rightarrow 3[f(x+2)-f(x+1)]=3a(2x+3)+3b(2)\)

Từ (1)(2) suy ra:

\(f(x+3)-3f(x+2)+3f(x+1)-f(x)=3a(2x+3)+3b-[3a(2x+3)+3b]=0\)