Multiplication & Division of Rational Expressions Quick Check

Question

2 of 52 of 5 Items
Question
Write the polynomial expression in simplest form: x2−9x2−4⋅4x−812x+36 .(1 point)
Responses

(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)
Start Fraction x minus 3 over 3 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

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Answer 1

To simplify the expression \(\frac{x^2 - 9}{x^2 - 4} \cdot \frac{4x - 8}{12x + 36}\), we'll start by factoring each polynomial where possible.

1. Factor \(x^2 - 9\): This is a difference of squares. \[ x^2 - 9 = (x - 3)(x + 3) \]

2. Factor \(x^2 - 4\): This is also a difference of squares. \[ x^2 - 4 = (x - 2)(x + 2) \]

3. Factor \(4x - 8\): This can be factored by taking out a common factor of 4. \[ 4x - 8 = 4(x - 2) \]

4. Factor \(12x + 36\): We can also factor this by taking out a common factor of 12. \[ 12x + 36 = 12(x + 3) \]

Now we can rewrite the expression with the factored forms: \[ \frac{(x - 3)(x + 3)}{(x - 2)(x + 2)} \cdot \frac{4(x - 2)}{12(x + 3)} \]

Next, combine the fractions: \[ \frac{(x - 3)(x + 3) \cdot 4(x - 2)}{(x - 2)(x + 2) \cdot 12(x + 3)} \]

Now let's cancel out common factors in the numerator and the denominator:

• The \(x - 2\) in the numerator and denominator cancels out.
• The \(4\) and \(12\) can be simplified to \(\frac{1}{3}\), so we can express this as: \[ \frac{(x - 3)(x + 3)}{3(x + 2)(x + 3)} \]

Now, we can cancel the \(x + 3\) from the numerator and the denominator: \[ \frac{x - 3}{3(x + 2)} \]

Thus, the expression in simplest form is: \[ \frac{x - 3}{3(x + 2)} \]

This matches better with the option you provided:

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