If f(x) is divided by x-1 and the quotient is2x2+x2and the reminder is 3 what is f(x)

When a polynomial $f(x)$ is divided by a divisor $d(x)$, yielding a quotient $q(x)$ and a remainder $r(x)$, the relationship is given by:

$$f(x) = d(x) \cdot q(x) + r(x)$$

Given:
Divisor $d(x) = x-1$
Quotient $q(x) = 2x^2 + x - 2$
Remainder $r(x) = 3$

Substitute these into the formula:

$$f(x) = (x-1)(2x^2 + x - 2) + 3$$

First, multiply $(x-1)$ by $(2x^2 + x - 2)$:

$$f(x) = x(2x^2 + x - 2) - 1(2x^2 + x - 2) + 3$$ $$f(x) = (2x^3 + x^2 - 2x) - (2x^2 + x - 2) + 3$$ $$f(x) = 2x^3 + x^2 - 2x - 2x^2 - x + 2 + 3$$

Combine like terms:

$$f(x) = 2x^3 + (x^2 - 2x^2) + (-2x - x) + (2 + 3)$$ $$f(x) = 2x^3 - x^2 - 3x + 5$$

The function $f(x)$ is: