Question

which of the following polynomials can approximate the expression (4x^3+5)(3x^6-8x^2)/2x^2?
a. 6x^7+...4x^3
b. 12x^9+...13
c. 4x^3+...13
d. 6x^7+...-7

Answer 1

To approximate the expression \((4x^3 + 5)(3x^6 - 8x^2) / (2x^2)\), we will first simplify it.

1. **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:

\[
(4x^3 + 5)(3x^6 - 8x^2) = 12x^9 + 15x^6 - 32x^5 - 40x^2
\]

2. **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. \(6x^7 + ... - 7\)**.