Math Problem Statement

Simplify the rational expression: (3x^2 - 4x + 1) / (x^2 - 16) ÷ (2x^2 - x - 3) / (x^2 + 4x + 4)

Solution

The image shows a complex rational expression involving division of two fractions. The given expression is:

3x24x+1x216÷2x2x3x2+4x+4\frac{3x^2 - 4x + 1}{x^2 - 16} \div \frac{2x^2 - x - 3}{x^2 + 4x + 4}

Steps to simplify:

  1. Division of Fractions Rule: To divide fractions, we multiply the first fraction by the reciprocal of the second fraction. So, we rewrite the expression as:

3x24x+1x216×x2+4x+42x2x3\frac{3x^2 - 4x + 1}{x^2 - 16} \times \frac{x^2 + 4x + 4}{2x^2 - x - 3}

  1. Factor the Numerators and Denominators:

    • Factor x216x^2 - 16 (difference of squares): x216=(x4)(x+4)x^2 - 16 = (x - 4)(x + 4)
    • Factor x2+4x+4x^2 + 4x + 4 (perfect square trinomial): x2+4x+4=(x+2)(x+2)x^2 + 4x + 4 = (x + 2)(x + 2)
    • Factor 2x2x32x^2 - x - 3: Find two numbers whose product is 6-6 and sum is 1-1. The factorization is: 2x2x3=(2x+3)(x1)2x^2 - x - 3 = (2x + 3)(x - 1)
  2. Multiply the Factored Expressions:

    Now we have: 3x24x+1(x4)(x+4)×(x+2)(x+2)(2x+3)(x1)\frac{3x^2 - 4x + 1}{(x - 4)(x + 4)} \times \frac{(x + 2)(x + 2)}{(2x + 3)(x - 1)}

    Next, check if the numerator 3x24x+13x^2 - 4x + 1 can be factored further. In this case, it cannot be factored nicely.

  3. Simplify the Expression:

    After multiplying, the expression becomes: (3x24x+1)(x+2)(x+2)(x4)(x+4)(2x+3)(x1)\frac{(3x^2 - 4x + 1)(x + 2)(x + 2)}{(x - 4)(x + 4)(2x + 3)(x - 1)}

This is the simplified form of the original expression.

Would you like further clarification on any step or details?

Related Questions:

  1. How do you find the reciprocal of a rational expression?
  2. What are the steps to factor quadratic expressions?
  3. Why do we multiply by the reciprocal when dividing fractions?
  4. Can you further simplify the expression if x=4x = 4 or x=4x = -4?
  5. How can you solve rational equations?

Tip: Always check for common factors to cancel out when simplifying rational expressions.

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Expressions
Factoring
Multiplication of Fractions

Formulas

(a^2 - b^2) = (a - b)(a + b)
Factoring Quadratic Trinomials

Theorems

Division of Fractions Rule

Suitable Grade Level

Grades 10-12