To approximate the expression \((4x^3+5)(3x^6-8x^2)(3x^6-8x^2) + 4x^2 + 13\) using only the first and last terms, we can focus on the leading term and the constant term.
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First Terms:
- The first term of \((4x^3 + 5)\) is \(4x^3\).
- The first term of \((3x^6 - 8x^2)\) is \(3x^6\). Since \((3x^6 - 8x^2)\) appears twice, we consider it squared: \[ (3x^6)^2 = 9x^{12} \]
Therefore, the leading term from the product \((4x^3)(3x^6)(3x^6)\) is: \[ 4x^3 \cdot 9x^{12} = 36x^{15} \]
But since we are only interested in the first term of \((4x^3 + 5)(3x^6 - 8x^2)(3x^6 - 8x^2)\), which contributes to the highest degree, we primarily focus on the leading contributions: \[ (4x^3)(9x^{12}) = 36x^{15} \]
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Constant Terms:
- The constant from \(4x^2 + 13\) is \(13\).
- The constant term from \(5\) (from \(4x^3 + 5\)) is \(5\).
- The constant term from \((3x^6 - 8x^2)\) is \(-8\). But because they are squared, we will ignore contributing squares, thus results in \(64\).
Combining all, we closely see the leading polynomial with high-degree terms:
From our working approximation:
- The constant term will simply be \(13\) derived from \(4x^2 + 13\).
Thus, matching possible answers:
- Option 1: \(8x^5 + \ldots + 18\)
- Option 2: \(12x^9 + \ldots + 13\)
- Option 3: \(4x^3 + \dots + 18\)
- Option 4: \(4x^3 + \dots + 13\)
After capturing leading and constant contributions closer to options, the polynomial that approximates our expression considering leading (first) term and last (constant) term aligns closely with Option 4: \(4x^3 + \ldots + 13\) as we derive that approach using highest degrees and constants.
So, the correct answer is: 4x³ + ... + 13.