Question

Perform the indicated operations.

6a ²−5a−6/6a² ⋅ 9a³/9a²+27a+14
Options:

a) 1/2a
b) 2 +9a+5/3a ² −2a−2
c) 2 +3a+1/2a² −3a−3
d) a² +3a+2/a² −3a+2

Answer 1

The expression is simplified by factoring and canceling out common terms, leading to the final simplified form, option b) 2 + (9a+5) / (3a² - 2a - 2).

Step-by-step explanation:

The student asked to perform the indicated operations, which is to simplify a product and quotient of polynomial expressions. Starting with the initial expression:

\((6a^2 - 5a - 6) / 6a^2 \cdot 9a^3 / (9a^2+27a+14)\)

First, let's simplify each part of the expression:

• \(6a^2 - 5a - 6\) can be factored into \((2a+1)(3a-6)\).
• \(9a^2+27a+14\) can be factored into \((3a+2)(3a+7)\).

Now, the expression can be rewritten as:

\(((2a+1)(3a-6) / 6a^2) \cdot (9a^3 / (3a+2)(3a+7))\)

We then cancel out common terms:

• \(6a^2\) in the denominator cancels with \(3a\) in \(3a-6\) and \(3a^3\) in the numerator.

After canceling out, the expression simplifies down to:

\((2a+1)\cdot(3a / (3a+2)(3a+7))\)

The final answer, after simplification, is option b) \(2 + (9a+5) / (3a^2 - 2a - 2)\).