for f(x)= 3x^2-x-2, find and simplify f(x)-f(2)/(x-2)

We are tasked with finding and simplifying the expression \(\frac{f(x) - f(2)}{x - 2}\), where \(f(x) = 3x^2 - x - 2\).

### Step 1: Calculate \(f(2)\)
Substitute \(x = 2\) into \(f(x)\):
\[
f(2) = 3(2)^2 - (2) - 2 = 3(4) - 2 - 2 = 12 - 4 = 8
\]

So, \(f(2) = 8\).

### Step 2: Set up the difference
We now need to evaluate \(\frac{f(x) - f(2)}{x - 2}\), which becomes:
\[
\frac{f(x) - 8}{x - 2} = \frac{3x^2 - x - 2 - 8}{x - 2} = \frac{3x^2 - x - 10}{x - 2}
\]

### Step 3: Simplify the expression
We need to simplify \(\frac{3x^2 - x - 10}{x - 2}\). Let's factor the quadratic expression \(3x^2 - x - 10\).

The factoring process involves finding two numbers that multiply to \(3 \times (-10) = -30\) and add up to \(-1\). These numbers are \(5\) and \(-6\). So, we rewrite the middle term \(-x\) as \(5x - 6x\):
\[
3x^2 + 5x - 6x - 10
\]
Now, factor by grouping:
\[
(3x^2 + 5x) - (6x + 10) = x(3x + 5) - 2(3x + 5)
\]
Factor out \((3x + 5)\):
\[
(x - 2)(3x + 5)
\]

### Step 4: Simplify the fraction
Now substitute the factored form into the expression:
\[
\frac{(x - 2)(3x + 5)}{x - 2}
\]
Cancel out the \((x - 2)\) terms:
\[
3x + 5
\]

Thus, the simplified form of \(\frac{f(x) - f(2)}{x - 2}\) is:
\[
\boxed{3x + 5}
\]

Would you like further details or clarification?

Here are some related questions:
1. How do you factor quadratic expressions in general?
2. Why does factoring help simplify rational expressions?
3. Can this method apply to any polynomial functions?
4. How do you deal with cases where the denominator doesn't cancel?
5. What is the significance of the result \(3x + 5\) in this context?

**Tip:** Always check if the denominator can be canceled after factoring, especially when simplifying expressions involving division.

For f(x) = 3x^2 - x - 2, find and simplify f(x) - f(2) / (x - 2)

Solve and simplify the expression f(x) - f(2) / (x - 2), where f(x) = 3x^2 - x - 2. This problem covers factoring and simplifying rational expressions. Learn step-by-step with detailed algebraic methods. Suitable for Grades 9-11.

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