Question

Which of the following polynomials can approximate the expression (4x^3+5)(3x^6-8x^2)/2x^2

Answer 1

To simplify the expression \((4x^3 + 5)(3x^6 - 8x^2) / (2x^2)\), we will first expand the numerator and then divide by \(2x^2\).

1. **Expand the numerator:**

\[
(4x^3 + 5)(3x^6 - 8x^2) = 4x^3 \cdot 3x^6 + 4x^3 \cdot (-8x^2) + 5 \cdot 3x^6 + 5 \cdot (-8x^2)
\]

Calculating each term:
- \(4x^3 \cdot 3x^6 = 12x^{9}\)
- \(4x^3 \cdot -8x^2 = -32x^{5}\)
- \(5 \cdot 3x^6 = 15x^6\)
- \(5 \cdot -8x^2 = -40x^2\)

Now, combining these terms:
\[
12x^9 + 15x^6 - 32x^5 - 40x^2
\]

2. **Now divide the entire polynomial by \(2x^2\):**

\[
\frac{12x^9 + 15x^6 - 32x^5 - 40x^2}{2x^2} = \frac{12x^9}{2x^2} + \frac{15x^6}{2x^2} - \frac{32x^5}{2x^2} - \frac{40x^2}{2x^2}
\]

Performing the division term by term:
- \(\frac{12x^9}{2x^2} = 6x^{7}\)
- \(\frac{15x^6}{2x^2} = \frac{15}{2}x^{4}\)
- \(\frac{-32x^5}{2x^2} = -16x^{3}\)
- \(\frac{-40x^2}{2x^2} = -20\)

Putting it all together:
\[
6x^7 + \frac{15}{2}x^4 - 16x^3 - 20
\]

This is the polynomial that approximates the given expression:
\[
6x^7 - 16x^3 + \frac{15}{2}x^4 - 20
\]

If you want to specify or check among options for approximation, just let me know what those options are!