To approximate the expression \((4x^3 + 5)(3x^6 - 8x^2) / (2x^2)\), we will first simplify it.
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Multiply the polynomials in the numerator:
\[ (4x^3 + 5)(3x^6 - 8x^2) = 4x^3 \cdot 3x^6 + 4x^3 \cdot (-8x^2) + 5 \cdot 3x^6 + 5 \cdot (-8x^2) \]
This expands to:
\[ 12x^9 - 32x^5 + 15x^6 - 40x^2 \]
So, we have:
\[ (4x^3 + 5)(3x^6 - 8x^2) = 12x^9 + 15x^6 - 32x^5 - 40x^2 \]
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Now divide by \(2x^2\):
\[ \frac{12x^9 + 15x^6 - 32x^5 - 40x^2}{2x^2} \]
This simplifies to:
\[ \frac{12x^9}{2x^2} + \frac{15x^6}{2x^2} - \frac{32x^5}{2x^2} - \frac{40x^2}{2x^2} \]
Simplifying each term gives:
\[ 6x^7 + \frac{15}{2}x^4 - 16x^3 - 20 \]
Thus, the simplified expression is:
\[ 6x^7 + \frac{15}{2}x^4 - 16x^3 - 20 \]
Now we can compare this with the options given:
- a. \(6x^7 + ... + 4x^3\)
- b. \(12x^9 + ... + 13\)
- c. \(4x^3 + ... + 13\)
- d. \(6x^7 + ... - 7\)
Among these, the term \(6x^7\) correctly matches with option d which also includes higher degree terms and a constant.
Therefore, the polynomial that can approximate the expression is:
d. \(6x^7 + ... - 7\).