Question
Which expressions are polynomials?
Select each correct answer.
Responses
4x+x√−1
4 x plus square root x end root minus 1
x5+2
fraction x over 5 end fraction plus 2
35x²
35 x ²
7x7−x−5
Select each correct answer.
Responses
4x+x√−1
4 x plus square root x end root minus 1
x5+2
fraction x over 5 end fraction plus 2
35x²
35 x ²
7x7−x−5
Answers
GPT-4o mini
To determine which expressions are polynomials, we need to consider the definition of a polynomial. A polynomial is an algebraic expression that consists of variables raised to non-negative integer powers and combined using addition, subtraction, and multiplication.
Let's analyze each expression given:
1. **4x + x√−1**
This expression includes \(x\sqrt{-1}\) (which is equivalent to \(xi\), where \(i\) is the imaginary unit). Since it contains an imaginary part, it is **not** a polynomial.
2. **x⁵ + 2**
This expression consists of a single term \(x^5\) (a non-negative integer power) and a constant \(2\). There are no variables with negative powers or imaginary components. Therefore, this is a polynomial.
3. **35x²**
This expression contains the term \(35x^2\), which again is a non-negative integer power of \(x\) and includes a constant coefficient. Thus, it is a polynomial.
4. **7x⁷ - x - 5**
This is a polynomial that includes terms with \(x^7\), \(x\) (which can be expressed as \(x^1\)), and the constant \(-5\). It follows the rules of polynomials, so this is also a polynomial.
Based on this analysis, the correct answers are:
- **x⁵ + 2**
- **35x²**
- **7x⁷ - x - 5**
Let's analyze each expression given:
1. **4x + x√−1**
This expression includes \(x\sqrt{-1}\) (which is equivalent to \(xi\), where \(i\) is the imaginary unit). Since it contains an imaginary part, it is **not** a polynomial.
2. **x⁵ + 2**
This expression consists of a single term \(x^5\) (a non-negative integer power) and a constant \(2\). There are no variables with negative powers or imaginary components. Therefore, this is a polynomial.
3. **35x²**
This expression contains the term \(35x^2\), which again is a non-negative integer power of \(x\) and includes a constant coefficient. Thus, it is a polynomial.
4. **7x⁷ - x - 5**
This is a polynomial that includes terms with \(x^7\), \(x\) (which can be expressed as \(x^1\)), and the constant \(-5\). It follows the rules of polynomials, so this is also a polynomial.
Based on this analysis, the correct answers are:
- **x⁵ + 2**
- **35x²**
- **7x⁷ - x - 5**