Question
Which expressions are polynomials?
Select each correct answer.
Responses
z + 1
z + 1
2x4−y
2 x being power 4 end power minus y
y2−y√3+4
y squared minus cube root y end root plus 4
6 + w
Select each correct answer.
Responses
z + 1
z + 1
2x4−y
2 x being power 4 end power minus y
y2−y√3+4
y squared minus cube root y end root plus 4
6 + w
Answers
GPT-4o mini
To determine which expressions are polynomials, we need to check if all terms in the expression meet the criteria for polynomial terms. A polynomial is defined as an expression that consists of variables raised to non-negative integer powers and multiplied by coefficients.
Let's analyze each expression:
1. **z + 1**: This is a polynomial because both terms (z and 1) are valid terms; z is raised to the power of 1 (which is a non-negative integer) and the constant 1 is also valid.
2. **2x^4 − y**: This is a polynomial because the term 2x^4 is a polynomial term (x raised to the power of 4, a non-negative integer) and y is considered as y^1 (also a non-negative integer). However, for a polynomial, it's generally acceptable to have terms with different variables.
3. **y^2 − y√3 + 4**: This expression contains the term -y√3. The term √3 can be expressed as 3^(1/2), which means that y is raised to the power of 1 (non-negative), but since the square root introduces a fractional power, the term is no longer a polynomial term. Thus, this is **not a polynomial**.
4. **6 + w**: This is a polynomial because both 6 and w are valid polynomial terms (w is considered w^1, which is a non-negative integer).
In conclusion, the expressions that are polynomials are:
1. **z + 1**
2. **2x^4 − y** (but note that this assumes y alone is fine in the context of polynomials with multiple variables)
3. **6 + w**
So, the correct answers are:
- z + 1
- 2x^4 − y
- 6 + w
Let's analyze each expression:
1. **z + 1**: This is a polynomial because both terms (z and 1) are valid terms; z is raised to the power of 1 (which is a non-negative integer) and the constant 1 is also valid.
2. **2x^4 − y**: This is a polynomial because the term 2x^4 is a polynomial term (x raised to the power of 4, a non-negative integer) and y is considered as y^1 (also a non-negative integer). However, for a polynomial, it's generally acceptable to have terms with different variables.
3. **y^2 − y√3 + 4**: This expression contains the term -y√3. The term √3 can be expressed as 3^(1/2), which means that y is raised to the power of 1 (non-negative), but since the square root introduces a fractional power, the term is no longer a polynomial term. Thus, this is **not a polynomial**.
4. **6 + w**: This is a polynomial because both 6 and w are valid polynomial terms (w is considered w^1, which is a non-negative integer).
In conclusion, the expressions that are polynomials are:
1. **z + 1**
2. **2x^4 − y** (but note that this assumes y alone is fine in the context of polynomials with multiple variables)
3. **6 + w**
So, the correct answers are:
- z + 1
- 2x^4 − y
- 6 + w