Question

what is the quotient?

\\(\frac{t + 3}{t + 4} \div (t^2 + 7t + 12)\\)

• \\((t + 3)^2\\)
• \\((t + 4)^2\\)
• \\(\frac{1}{(t + 4)^2}\\)
• \\(\frac{1}{(t + 3)^2}\\)

Explanation:

Factor the quadratic expression

$$ t^2 + 7t + 12 = (t + 3)(t + 4) $$

Rewrite the division as multiplication by the reciprocal

$$ \frac{t + 3}{t + 4} \div (t^2 + 7t + 12) = \frac{t + 3}{t + 4} \cdot \frac{1}{(t + 3)(t + 4)} $$

Simplify the expression

$$ \frac{t + 3}{(t + 4)(t + 3)(t + 4)} = \frac{1}{(t + 4)^2} \quad (t eq -3) $$

Answer:

• (A) \((t + 3)^2\)
• (B) \((t + 4)^2\)
• (C) \(\frac{1}{(t + 4)^2}\) (Correct answer)
• (D) \(\frac{1}{(t + 3)^2}\)