Math Snap

STEP 1

1. The function $f(x)$ is a rational function.
2. We are tasked with simplifying or evaluating the function.
3. The denominator $x - 3$ suggests potential division or simplification by polynomial division.

STEP 2

1. Perform polynomial long division to simplify the expression $\frac{2x^3 + x^2 - 25x + 12}{x - 3}$.
2. Interpret the result of the division.

STEP 3

Set up the polynomial long division with $2x^3 + x^2 - 25x + 12$ as the dividend and $x - 3$ as the divisor.

STEP 4

Divide the leading term of the dividend $2x^3$ by the leading term of the divisor $x$ to get the first term of the quotient: $2x^2$.

STEP 5

Multiply the entire divisor $x - 3$ by the first term of the quotient $2x^2$ to get $2x^3 - 6x^2$.

STEP 6

Subtract $2x^3 - 6x^2$ from the original dividend $2x^3 + x^2 - 25x + 12$ to get the new dividend: $7x^2 - 25x + 12$.

STEP 7

Repeat the division process: divide $7x^2$ by $x$ to get $7x$.

STEP 8

Multiply the divisor $x - 3$ by $7x$ to get $7x^2 - 21x$.

STEP 9

Subtract $7x^2 - 21x$ from the new dividend $7x^2 - 25x + 12$ to get the next dividend: $-4x + 12$.

STEP 10

Divide $-4x$ by $x$ to get $-4$.

STEP 11

Multiply the divisor $x - 3$ by $-4$ to get $-4x + 12$.

STEP 12

Subtract $-4x + 12$ from the current dividend $-4x + 12$ to get a remainder of $0$.

The quotient from the division is $2x^2 + 7x - 4$ with a remainder of $0$, indicating that:
$$f(x) = 2x^2 + 7x - 4$$ This is the simplified form of the function.