Question

how is factoring a common factor out of a trinomial different from factoring common factors out of the numerator and denominator of a fraction?
a. the number of parts of the trinomial varies from 0 to 3, whereas the number of parts of a fraction varies from 0 to 2.
b. in factoring a trinomial, any factor of the trinomial is a part. in factoring a fraction, the part is the common factor of the numerator and the denominator.
c. there is only one common factor for a trinomial but there are two common factors for a fraction, one for the numerator and one for the denominator.
d. in factoring a trinomial, the parts are the terms of the trinomial. in factoring a fraction, the parts are the numerator and the denominator.

Explanation:

To determine the correct answer, we analyze each option:

• Option A: A trinomial has 3 terms (parts), so the number of parts doesn't vary from 0 to 3. A fraction has 2 parts (numerator and denominator), so the number of parts doesn't vary from 0 to 2. Eliminate A.
• Option B: When factoring a trinomial, we factor the entire trinomial (not just a part). When factoring a fraction, we factor the numerator and denominator separately (not a single "part" that is the common factor of both). Eliminate B.
• Option C: A trinomial can have one or more common factors, and a fraction also can have one common factor (the greatest common factor of numerator and denominator). The statement about two common factors for a fraction is incorrect. Eliminate C.
• Option D: In factoring a trinomial, the "parts" we consider are the terms of the trinomial (e.g., in \(ax^2 + bx + c\), the parts are \(ax^2\), \(bx\), \(c\)). In factoring a fraction (simplifying by factoring out common factors), the "parts" are the numerator and the denominator (e.g., in \(\frac{6x}{9}\), the parts are \(6x\) (numerator) and \(9\) (denominator)). This accurately describes the difference.

Answer:

D. In factoring a trinomial, the parts are the terms of the trinomial. In factoring a fraction, the parts are the numerator and the denominator.