Question
Which of the following polynomials can approximate the expression using the first and last terms only?
1)2x^3+ ... -4x^2
2)2x^3+ ... -12
3)2x^4+... -12
4)2x^3+ ... -6
1)2x^3+ ... -4x^2
2)2x^3+ ... -12
3)2x^4+... -12
4)2x^3+ ... -6
Answers
GPT-4o mini
To determine which polynomial can approximate an expression using only the first and last terms, we focus on the overall leading term (first term) and the constant term (last term).
Let's analyze the provided options:
1) **\(2x^3 + ... - 4x^2\)**
- Leading term: \(2x^3\)
- Constant term: No constant term in the given expression, but there's a last term: \(-4x^2\)
2) **\(2x^3 + ... - 12\)**
- Leading term: \(2x^3\)
- Constant term: \(-12\)
3) **\(2x^4 + ... - 12\)**
- Leading term: \(2x^4\)
- Constant term: \(-12\)
4) **\(2x^3 + ... - 6\)**
- Leading term: \(2x^3\)
- Constant term: \(-6\)
Now, if we want to focus on the first term and the constant term, we can summarize:
- The only polynomials with a clear leading term and a constant to provide a good approximation are options 2, 3, and 4.
- If we compare these regarding polynomial degree, options related to \(x^3\) and a constant suggest we consider options 2 and 4 since they both seem manageable when approximating.
However, option 3 contains a leading term of \(2x^4\) which suggests it might involve a broader scope than just approximating from \(x^3\) downward.
Thus, based on leading term considerations, **the appropriate choices for approximating the expression using the first and last terms only are**:
- **Option 2: \(2x^3 + ... - 12\)**
- **Option 4: \(2x^3 + ... - 6\)**
These two both follow the first term of \(x^3\), and thus can be simpler approximations. Hence, you would need to select between them based on the constant that may fit your expression.
If you just want which can approximate an expression correctly while adhering to the terms, both options 2 and 4 would be suitable.
Let's analyze the provided options:
1) **\(2x^3 + ... - 4x^2\)**
- Leading term: \(2x^3\)
- Constant term: No constant term in the given expression, but there's a last term: \(-4x^2\)
2) **\(2x^3 + ... - 12\)**
- Leading term: \(2x^3\)
- Constant term: \(-12\)
3) **\(2x^4 + ... - 12\)**
- Leading term: \(2x^4\)
- Constant term: \(-12\)
4) **\(2x^3 + ... - 6\)**
- Leading term: \(2x^3\)
- Constant term: \(-6\)
Now, if we want to focus on the first term and the constant term, we can summarize:
- The only polynomials with a clear leading term and a constant to provide a good approximation are options 2, 3, and 4.
- If we compare these regarding polynomial degree, options related to \(x^3\) and a constant suggest we consider options 2 and 4 since they both seem manageable when approximating.
However, option 3 contains a leading term of \(2x^4\) which suggests it might involve a broader scope than just approximating from \(x^3\) downward.
Thus, based on leading term considerations, **the appropriate choices for approximating the expression using the first and last terms only are**:
- **Option 2: \(2x^3 + ... - 12\)**
- **Option 4: \(2x^3 + ... - 6\)**
These two both follow the first term of \(x^3\), and thus can be simpler approximations. Hence, you would need to select between them based on the constant that may fit your expression.
If you just want which can approximate an expression correctly while adhering to the terms, both options 2 and 4 would be suitable.