Question

try it!
try identify factorable numerators and denominators
\\(\frac{x^2 + 1}{x + 1} \cdot \frac{x^2 + 3x + 2}{3x^2 + 3}\\)
before multiplying the two rational expressions shown above, raj wants to factor the numerators and denominators. which of the following can be factored?
choose all answers that apply:
a \\(x^2 + 1\\)
b \\(x + 1\\)
c \\(x^2 + 3x + 2\\)
d \\(3x^2 + 3\\)
check
explain

Explanation:

• For option A: \(x^{2}+1\) is a sum of squares and doesn't factor over the real numbers (no real roots), so it can't be factored.
• For option B: \(x + 1\) is a linear expression and is already in its simplest factored form (it can't be factored further in the context of factoring to simplify the rational expression here as it's a first - degree polynomial).
• For option C: \(x^{2}+3x + 2\) is a quadratic trinomial. We can factor it as \((x + 1)(x+2)\) using the formula \(x^{2}+(a + b)x+ab=(x + a)(x + b)\) where \(a = 1\), \(b = 2\) (since \(1\times2=2\) and \(1 + 2=3\)).
• For option D: \(3x^{2}+3\) has a common factor of 3. We can factor it as \(3(x^{2}+1)\).

Answer:

C. \(x^{2}+3x + 2\), D. \(3x^{2}+3\)