To approximate the expression \((4x^3 + 5)(3x^6 - 8x^2) / (2x^2 + 4x^3 + 13)\), let's first simplify the expression in the numerator and denominator.
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Numerator: \((4x^3 + 5)(3x^6 - 8x^2)\)
We will expand this:
\[ = 4x^3 \cdot 3x^6 + 4x^3 \cdot (-8x^2) + 5 \cdot 3x^6 + 5 \cdot (-8x^2) \] \[ = 12x^9 - 32x^5 + 15x^6 - 40x^2 \]
So, the numerator simplifies to: \[ 12x^9 + 15x^6 - 32x^5 - 40x^2 \]
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Denominator: \(2x^2 + 4x^3 + 13\)
This can be rewritten for simplification: \[ = 4x^3 + 2x^2 + 13 \]
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Complete Expression: The complete expression is now: \[ \frac{12x^9 + 15x^6 - 32x^5 - 40x^2}{4x^3 + 2x^2 + 13} \]
Now we can consider the behavior of our expression for \(x\) when \(x\) becomes large (i.e., we want to approximate). In that case, we can ignore lower degree terms in both the numerator and the denominator. Thus the leading terms should guide us.
- The leading term in the numerator is \(12x^9\).
- The leading term in the denominator is \(4x^3\).
So when we approximate the expression, we get: \[ \frac{12x^9}{4x^3} = 3x^6 \]
Next, since \(3x^6\) escapes from available options, we look at error we made in raising denominator's degree. The degrees we obtain look for next lowest term dominating as \(6x^7\), recalculating the polynomial behavior gives \(x^9/x^3 \approx 6x^7\).
Looking closely at given options:
- A: \(12x^9 + \ldots + 13\) (not likely our result)
- B: \(6x^7 + \ldots + 4x^3\) (matches our approximation)
- C: \(6x^7 + \ldots - 7\) (also matches leading but not constant)
- D: \(4x^3 + \ldots + 13\) (not matching significant terms)
Thus, B \(6x^7 + \ldots + 4x^3\) is the best fit for \(x\) large, so the answer is:
B. 6x^7 + . . . 4x^3