Math Problem Statement

Write the polynomial expression in simplest form: (6x / (x - 4)) * ((x^2 - 16) / (4x + 16))

Solution

Let's simplify the given polynomial expression step by step:

6xx4x2164x+16\frac{6x}{x - 4} \cdot \frac{x^2 - 16}{4x + 16}

Step 1: Factorize where possible

  1. The term x216x^2 - 16 is a difference of squares:

    x216=(x4)(x+4)x^2 - 16 = (x - 4)(x + 4)

  2. The denominator 4x+164x + 16 can be factored as:

    4x+16=4(x+4)4x + 16 = 4(x + 4)

Now, rewriting the expression with factored terms:

6xx4(x4)(x+4)4(x+4)\frac{6x}{x - 4} \cdot \frac{(x - 4)(x + 4)}{4(x + 4)}

Step 2: Cancel common factors

  • The x4x - 4 in the numerator and denominator cancel out.
  • The x+4x + 4 in the numerator and denominator also cancel out.

After cancellation, we are left with:

6x4\frac{6x}{4}

Step 3: Simplify the fraction

6x4=3x2\frac{6x}{4} = \frac{3x}{2}

Step 4: Choose the correct answer

The correct answer is:

3x2\frac{3x}{2}

which corresponds to the first option.


Would you like a further breakdown of any step? Here are five related questions for further practice:

  1. How do you factorize quadratic expressions like x216x^2 - 16?
  2. Why does canceling common factors work in fraction multiplication?
  3. What is the general formula for the difference of squares?
  4. How do you simplify algebraic fractions with multiple terms?
  5. How would this process change if the expression involved division instead of multiplication?

Tip: Always factorize expressions before multiplying or dividing to simplify the process.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomial Simplification
Factoring

Formulas

Difference of Squares: a^2 - b^2 = (a - b)(a + b)
Fraction Multiplication: (A/B) * (C/D) = (A*C) / (B*D)

Theorems

Factorization Theorem
Basic Fraction Reduction

Suitable Grade Level

Grades 8-10