Question

determine o polinómio p(x) de grau 2 tal que p(x+1)-p(x)=x

Answer 1

$\sf P(x) = ax^2+bx+c \\\\ Fa{\c c}amos : \\\\ P(x+1)-P(x) = x \\\\ a(x+1)^2+b(x+1)+c -(ax^2+bx+c)=x \\\\ a(x^2+2x+1)+bx+b+c-ax^2-bx-c=x \\\\ ax^2+2ax+a+bx+b+c-ax^2-bx-c=x \\\\ ax^2-ax^2 +bx-bx+c-c + 2ax +a+b=x \\\\ 2ax+a+b =x$

para que dois polinômios sejam iguais seus coeficientes devem ser iguais, ou seja :

$\displaystyle \sf \displaystyle \sf 2ax = x\to a = \frac{1}{2}\\\\ \displaystyle \sf a+b = 0 \to b = -a \to b = \frac{-1}{2}$

Daí :

$\displaystyle \sf P(x) = ax^2+bx+c \\\\ a=\frac{1}{2}\ \ ; \ \ b=\frac{-1}{2} \ \ ;\ \ c = 0 \\\\\\ \boxed{\sf \displaystyle \sf P(x) = \frac{x^2}{2}-\frac{x}{2} \ }\checkmark$