Math Problem Statement

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

Solution

We are tasked with finding and simplifying the expression f(x)f(2)x2\frac{f(x) - f(2)}{x - 2}, where f(x)=3x2x2f(x) = 3x^2 - x - 2.

Step 1: Calculate f(2)f(2)

Substitute x=2x = 2 into f(x)f(x): f(2)=3(2)2(2)2=3(4)22=124=8f(2) = 3(2)^2 - (2) - 2 = 3(4) - 2 - 2 = 12 - 4 = 8

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

Step 2: Set up the difference

We now need to evaluate f(x)f(2)x2\frac{f(x) - f(2)}{x - 2}, which becomes: f(x)8x2=3x2x28x2=3x2x10x2\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 3x2x10x2\frac{3x^2 - x - 10}{x - 2}. Let's factor the quadratic expression 3x2x103x^2 - x - 10.

The factoring process involves finding two numbers that multiply to 3×(10)=303 \times (-10) = -30 and add up to 1-1. These numbers are 55 and 6-6. So, we rewrite the middle term x-x as 5x6x5x - 6x: 3x2+5x6x103x^2 + 5x - 6x - 10 Now, factor by grouping: (3x2+5x)(6x+10)=x(3x+5)2(3x+5)(3x^2 + 5x) - (6x + 10) = x(3x + 5) - 2(3x + 5) Factor out (3x+5)(3x + 5): (x2)(3x+5)(x - 2)(3x + 5)

Step 4: Simplify the fraction

Now substitute the factored form into the expression: (x2)(3x+5)x2\frac{(x - 2)(3x + 5)}{x - 2} Cancel out the (x2)(x - 2) terms: 3x+53x + 5

Thus, the simplified form of f(x)f(2)x2\frac{f(x) - f(2)}{x - 2} is: 3x+5\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+53x + 5 in this context?

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

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Factoring
Simplification of Rational Expressions

Formulas

Quadratic equation: ax^2 + bx + c = 0
Factoring quadratics: (x - r1)(x - r2)

Theorems

Factorization Theorem
Rational Function Simplification

Suitable Grade Level

Grades 9-11