Write the polynomial expression in simplest form: x2−9x2−4⋅4x−812x+36.(1 point) Responses 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 Start Fraction 1 over 3 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 x−33(x+2)

1 answer

To simplify the expression \(\frac{x^2 - 9}{x^2 - 4} \cdot \frac{4x - 8}{12x + 36}\), we will first factor the polynomials in the numerator and the denominator.

  1. Factoring \(x^2 - 9\):

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

    • This is also a difference of squares: \(x^2 - 4 = (x - 2)(x + 2)\).
  3. Factoring \(4x - 8\):

    • Factor out a 4: \(4x - 8 = 4(x - 2)\).
  4. Factoring \(12x + 36\):

    • Factor out a 12: \(12x + 36 = 12(x + 3)\).

Now we can rewrite the expression using these factorizations:

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

Next, we can combine this into one fraction:

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

Now we'll cancel out common factors in the numerator and the denominator:

  1. \(x - 2\) in the numerator and denominator.
  2. \(x + 3\) in the numerator and denominator.

After canceling these common factors, we are left with:

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

We can simplify this further by reducing the coefficients:

\[ \frac{4}{12} = \frac{1}{3} \]

Thus, we end up with:

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

Therefore, the final expression in simplest form is:

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

From the given options, this corresponds to:

\[ \text{Start Fraction left parenthesis x minus 3 right parenthesis over 3 left parenthesis x plus 2 right parenthesis End Fraction} \]