Question

which of the following polynomials can approximate the expression 3x^5+4x^4-2x^3/x^3+2x+1 +8x^2-4x+6
11x^2+6
11x^+4
11x^2+8
3x^2+6

Answer 1

To approximate the expression \( \frac{3x^5 + 4x^4 - 2x^3}{x^3 + 2x + 1} + 8x^2 - 4x + 6 \), we can start by simplifying it when \( x \) is large. The dominant term in the numerator is \( 3x^5 \), and the dominant term in the denominator is \( x^3 \).

1. We simplify the fraction:
\[
\frac{3x^5 + 4x^4 - 2x^3}{x^3 + 2x + 1} \approx \frac{3x^5}{x^3} = 3x^2 \quad \text{(for large } x\text{)}
\]

2. Now, we add \( 8x^2 - 4x + 6 \):
\[
\text{Approximate expression } \approx 3x^2 + 8x^2 - 4x + 6 = 11x^2 - 4x + 6
\]

For large \( x \), the \( -4x + 6 \) terms become less significant compared to \( 11x^2 \).

Now, let's review the given polynomial options:
- \( 11x^2 + 6 \)
- \( 11x^4 \)
- \( 11x^2 + 8 \)
- \( 3x^2 + 6 \)

Among these options, \( 11x^2 + 6 \) is the best approximation for the dominant term \( 11x^2 \) when \( x \) is large since we often consider the leading order term for approximation.

Thus, the polynomial that can approximate the given expression is:
\[
\boxed{11x^2 + 6}
\]