Problem 18 - College Algebra Essentials

Question

Multiply or divide as indicated. $$\frac{x^{2}-4}{x^{2}-4 x+4} \cdot \frac{2 x-4}{x+2}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \[1\]

Step 1: Factoring of the polynomials

First, factor each polynomial in the expression. The polynomials can be factored as: \((x^{2}-4) = (x-2)(x+2)\), \((x^{2}-4x+4)= (x-2)^{2}\), and \((2x-4)= 2(x-2)\)

Step 2: Substitute the factors back into the equation

Substitute the factored form of the polynomials back into the equation. Therefore, the initial expression will be converted into: \((x-2)(x+2) / (x-2)^{2} \cdot 2(x-2)/(x+2)\)

Step 3: Cancelling out common factors

Now, cancel out all the factors that occur both in the numerators and denominators. This will leave \((x-2) / (x-2)\), which should be simplified to 1

Step 4: Multiplying the results

Since there's no other simplification can be made, proceed to multiply the remaining numbers. Thus, multiplying 1 by 1 yields 1