Question
What must be true about this multiplication problem? ((9x ^ 3 + 7)/6)((x - 2)/5) (1 point) The result is not a closed system. The result is a polynomial, The result is unknown. The result is not a polynomial
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GPT-4o mini
Let's analyze the multiplication problem step-by-step:
The expression given is:
\[
\left(\frac{9x^3 + 7}{6}\right) \left(\frac{x - 2}{5}\right)
\]
To determine the nature of the result, we will look at each part:
1. **The first part**: \(\frac{9x^3 + 7}{6}\) consists of a polynomial \(9x^3 + 7\) divided by a constant \(6\). This division does not change the polynomial nature of the expression. Therefore, this part still represents a rational function (a polynomial divided by a number).
2. **The second part**: \(\frac{x - 2}{5}\) again consists of a polynomial \(x - 2\) divided by the constant \(5\). This part also represents a rational function.
Now when you multiply these two rational functions together:
\[
\left(\frac{9x^3 + 7}{6}\right) \left(\frac{x - 2}{5}\right) = \frac{(9x^3 + 7)(x - 2)}{30}
\]
The numerator \((9x^3 + 7)(x - 2)\) is a polynomial, and the denominator \(30\) is a constant. Thus, the overall expression is still a rational function.
Now, considering the options provided:
- "The result is not a closed system" - This is ambiguous without further context; it might not be relevant to the polynomial nature of the result.
- "The result is a polynomial" - This is not true because the overall expression is a rational function (a polynomial in the numerator divided by a constant).
- "The result is unknown" - This does not apply; we can analyze the nature of the result based on the input.
- "The result is not a polynomial" - This is true because the result is in the form of a rational function.
Thus, the correct statement from the provided options is:
**The result is not a polynomial.**
The expression given is:
\[
\left(\frac{9x^3 + 7}{6}\right) \left(\frac{x - 2}{5}\right)
\]
To determine the nature of the result, we will look at each part:
1. **The first part**: \(\frac{9x^3 + 7}{6}\) consists of a polynomial \(9x^3 + 7\) divided by a constant \(6\). This division does not change the polynomial nature of the expression. Therefore, this part still represents a rational function (a polynomial divided by a number).
2. **The second part**: \(\frac{x - 2}{5}\) again consists of a polynomial \(x - 2\) divided by the constant \(5\). This part also represents a rational function.
Now when you multiply these two rational functions together:
\[
\left(\frac{9x^3 + 7}{6}\right) \left(\frac{x - 2}{5}\right) = \frac{(9x^3 + 7)(x - 2)}{30}
\]
The numerator \((9x^3 + 7)(x - 2)\) is a polynomial, and the denominator \(30\) is a constant. Thus, the overall expression is still a rational function.
Now, considering the options provided:
- "The result is not a closed system" - This is ambiguous without further context; it might not be relevant to the polynomial nature of the result.
- "The result is a polynomial" - This is not true because the overall expression is a rational function (a polynomial in the numerator divided by a constant).
- "The result is unknown" - This does not apply; we can analyze the nature of the result based on the input.
- "The result is not a polynomial" - This is true because the result is in the form of a rational function.
Thus, the correct statement from the provided options is:
**The result is not a polynomial.**
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