What must be true about this multiplication problem?

Question

$(\frac {9x^{3}+7}{6})(\frac {x-2}{5})$
(1 point)
The result is unknown
The result is a polynomial
The result is not a closed system
The result is not a polynomial

Answer 1

To analyze the multiplication problem \((\frac {9x^{3}+7}{6})(\frac {x-2}{5})\), let's break it down:

1. The first factor \(\frac{9x^3 + 7}{6}\) is a rational expression involving a polynomial in the numerator \(9x^3 + 7\) divided by the constant 6.

2. The second factor \(\frac{x - 2}{5}\) is also a rational expression with the polynomial \(x - 2\) in the numerator divided by the constant 5.

Now, let's look at the resulting expression when these two factors are multiplied:

\[ (\frac {9x^{3}+7}{6})(\frac {x-2}{5}) = \frac{(9x^3 + 7)(x - 2)}{6 \cdot 5} = \frac{(9x^3 + 7)(x - 2)}{30} \]

The numerator \((9x^3 + 7)(x - 2)\) involves the multiplication of a polynomial of degree 3 by a polynomial of degree 1, which gives a polynomial of degree \(3 + 1 = 4\).

As for the denominator, since it is just a constant (30), it does not affect the polynomial nature of the numerator.

Thus, the overall expression \(\frac{(9x^3 + 7)(x - 2)}{30}\) is indeed a rational function, and importantly, the numerator (though divided by a non-zero constant) remains a polynomial.

Now we can evaluate the options provided:

1. The result is unknown: This is not true. We have expressed the result, albeit still not simplified completely.
2. The result is a polynomial: This is false; the result is a rational function, not a polynomial, because it includes division by a constant.
3. The result is not a closed system: This is irrelevant to the context provided.
4. The result is not a polynomial: This is true because the final answer can be expressed as a rational expression, not just a standalone polynomial.

Therefore, the correct assertion is:

The result is not a polynomial.