Question

Which of the following multiplies and reduces the polynomial expression x2−9/x2−4⋅4x−8/12x+36 to its simplest form?(1 point) Responses x−33(x+2) Start Fraction x minus 3 over 3 left parenthesis x plus 2 right parenthesis End Fraction (x−3)(x−2)3(x+2)(x+2) Start Fraction left parenthesis x minus 3 right parenthesis left parenthesis x minus 2 right parenthesis over 3 left parenthesis x plus 2 right parenthesis left parenthesis x plus 2 right parenthesis End Fraction 2(x−3)6(x+2) Start Fraction 2 left parenthesis x minus 3 right parenthesis over 6 left parenthesis x plus 2 right parenthesis End Fraction 13

Answer 1

To simplify the given expression \((x^2 - 9)/(x^2 - 4) \cdot (4x - 8)/(12x + 36)\), we will first factor all the polynomials involved.

1. Factor \(x^2 - 9\) and \(x^2 - 4\):

   • \(x^2 - 9\) is a difference of squares: \(x^2 - 9 = (x - 3)(x + 3)\)
   • \(x^2 - 4\) is also a difference of squares: \(x^2 - 4 = (x - 2)(x + 2)\)

2. Factor \(4x - 8\) and \(12x + 36\):

   • \(4x - 8\) can be factored out as \(4(x - 2)\)
   • \(12x + 36\) can be factored out as \(12(x + 3)\)

Now substituting back into the expression, we have:

\[ \frac{(x - 3)(x + 3)}{(x - 2)(x + 2)} \cdot \frac{4(x - 2)}{12(x + 3)} \]

3. Combine the fractions:

\[ \frac{(x - 3)(x + 3) \cdot 4(x - 2)}{(x - 2)(x + 2) \cdot 12(x + 3)} \]

4. Cancel out common factors:

• The \(x - 2\) in the numerator and denominator can cancel.
• The \(x + 3\) in the numerator and denominator can cancel.

This results in:

\[ \frac{(x - 3) \cdot 4}{12(x + 2)} = \frac{4(x - 3)}{12(x + 2)} = \frac{(x - 3)}{3(x + 2)} \]

Thus, after simplification, the expression reduces to:

\[ \frac{x - 3}{3(x + 2)} \]

So the simplest form of the expression is:

\(\frac{x - 3}{3(x + 2)}\)

This corresponds to the option provided:

\( \frac{x - 3}{3(x + 2)} \).