Question
Which of the following polynomials can approximate the expression (4x^3+5) (3x^6-8x^2) /2x^2 + 4x^3 + 13 using first and last terms only A. 4x^3+ 13 B. 6x^7+ -7 C. 12x^9+13 D. 6x^7+ 4x^3
Answers
GPT-4o mini
To approximate the expression \((4x^3 + 5)(3x^6 - 8x^2) / (2x^2 + 4x^3 + 13)\) using the first and last terms only, we can follow these steps:
1. **Find the first term of the numerator**:
- The first terms of \(4x^3\) and \(3x^6\) are \(4x^3 \cdot 3x^6 = 12x^{9}\).
2. **Find the last term of the numerator**:
- The last terms of \(5\) and \(-8x^2\) are \(5 \cdot -8x^2 = -40x^2\), but since we are only interested in the leading term, we will focus on the first term.
3. **Find the first term of the denominator**:
- The first term of \(2x^2 + 4x^3 + 13\) is \(4x^3\).
4. **Simplify**:
- To get the overall leading term, we consider the leading term of the numerator (which is \(12x^9\)) and divide it by the leading term of the denominator (which is \(4x^3\)).
\[
\frac{12x^9}{4x^3} = 3x^6
\]
Now we approximate the expression using only the first and last terms. We also notice that the last term in the polynomial expression we need to keep track of is \(13\) from the denominator.
Thus, our approximation is:
\[
3x^6 + 13
\]
Now, let’s compare this approximation with the provided options:
A. \(4x^3 + 13\)
B. \(6x^7 - 7\)
C. \(12x^9 + 13\)
D. \(6x^7 + 4x^3\)
None of the answers match \(3x^6 + 13\) directly, but looking closely, we can see if the approximation can point towards a polynomial close to it.
The polynomial \(6x^7 - 7\) (B) does have leading order \(6x^7\) which is higher than our approximation and polynomial \(12x^9 + 13\) (C) which is also higher.
Thus the option that reflects a combination of leading highest and a constant while being the most reasonable consideration could arguably point towards:
**C. \(12x^9 + 13\)**, albeit it isn’t what we get from just the first and last approximation.
To summarize, none of the options is exactly a simplified form of the given expression, and if you had to pick one, option **C** might be the likeliest, although it's not exact to our consideration.
1. **Find the first term of the numerator**:
- The first terms of \(4x^3\) and \(3x^6\) are \(4x^3 \cdot 3x^6 = 12x^{9}\).
2. **Find the last term of the numerator**:
- The last terms of \(5\) and \(-8x^2\) are \(5 \cdot -8x^2 = -40x^2\), but since we are only interested in the leading term, we will focus on the first term.
3. **Find the first term of the denominator**:
- The first term of \(2x^2 + 4x^3 + 13\) is \(4x^3\).
4. **Simplify**:
- To get the overall leading term, we consider the leading term of the numerator (which is \(12x^9\)) and divide it by the leading term of the denominator (which is \(4x^3\)).
\[
\frac{12x^9}{4x^3} = 3x^6
\]
Now we approximate the expression using only the first and last terms. We also notice that the last term in the polynomial expression we need to keep track of is \(13\) from the denominator.
Thus, our approximation is:
\[
3x^6 + 13
\]
Now, let’s compare this approximation with the provided options:
A. \(4x^3 + 13\)
B. \(6x^7 - 7\)
C. \(12x^9 + 13\)
D. \(6x^7 + 4x^3\)
None of the answers match \(3x^6 + 13\) directly, but looking closely, we can see if the approximation can point towards a polynomial close to it.
The polynomial \(6x^7 - 7\) (B) does have leading order \(6x^7\) which is higher than our approximation and polynomial \(12x^9 + 13\) (C) which is also higher.
Thus the option that reflects a combination of leading highest and a constant while being the most reasonable consideration could arguably point towards:
**C. \(12x^9 + 13\)**, albeit it isn’t what we get from just the first and last approximation.
To summarize, none of the options is exactly a simplified form of the given expression, and if you had to pick one, option **C** might be the likeliest, although it's not exact to our consideration.