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 , where .
Step 1: Calculate
Substitute into :
So, .
Step 2: Set up the difference
We now need to evaluate , which becomes:
Step 3: Simplify the expression
We need to simplify . Let's factor the quadratic expression .
The factoring process involves finding two numbers that multiply to and add up to . These numbers are and . So, we rewrite the middle term as : Now, factor by grouping: Factor out :
Step 4: Simplify the fraction
Now substitute the factored form into the expression: Cancel out the terms:
Thus, the simplified form of is:
Would you like further details or clarification?
Here are some related questions:
- How do you factor quadratic expressions in general?
- Why does factoring help simplify rational expressions?
- Can this method apply to any polynomial functions?
- How do you deal with cases where the denominator doesn't cancel?
- What is the significance of the result 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