Question
Six statements about polynomials are given. Select all the statements that are true(3 points)
Responses
5x^2y − 6xy^3z + 3x^5
is a fifth degree polynomial
5x^2y − 6xy^3z + 3x^5
contains four terms
5x^2y − 6xy^3 + 3x^5
has a leading coefficient of 3
12a^5 − 7 +3a^7+11^a
is equivalent to 3a^7 + 12a^5 + 11^a − 7
12a^5 − 7 +3a^7+11^a
has a leading term of 12 A to the 5th power
12a^5 − 7 +3a^7+11^a
has a contant of 7
Responses
5x^2y − 6xy^3z + 3x^5
is a fifth degree polynomial
5x^2y − 6xy^3z + 3x^5
contains four terms
5x^2y − 6xy^3 + 3x^5
has a leading coefficient of 3
12a^5 − 7 +3a^7+11^a
is equivalent to 3a^7 + 12a^5 + 11^a − 7
12a^5 − 7 +3a^7+11^a
has a leading term of 12 A to the 5th power
12a^5 − 7 +3a^7+11^a
has a contant of 7
Answers
There are no human answers yet.
Answered by
GPT-4o mini
Let's analyze each statement regarding the polynomials:
1. **5x²y − 6xy³z + 3x⁵ is a fifth-degree polynomial**:
The degree of a polynomial is determined by the term with the highest total exponent. Here, the term 3x⁵ has the highest degree (5).
**This statement is true.**
2. **5x²y − 6xy³z + 3x⁵ contains four terms**:
This polynomial has three distinct terms: 5x²y, -6xy³z, and 3x⁵.
**This statement is false.**
3. **5x²y − 6xy³ + 3x⁵ has a leading coefficient of 3**:
Among the terms, 3x⁵ is the term with the highest degree (5), and its leading coefficient is 3.
**This statement is true.**
4. **12a⁵ − 7 + 3a⁷ + 11^a is equivalent to 3a⁷ + 12a⁵ + 11^a − 7**:
The expression is equivalent because addition is commutative, so the order does not affect the sum.
**This statement is true.**
5. **12a⁵ − 7 + 3a⁷ + 11^a has a leading term of 12a⁵**:
The term with the highest degree in this polynomial is 3a⁷ (degree 7), not 12a⁵ (degree 5).
**This statement is false.**
6. **12a⁵ − 7 + 3a⁷ + 11^a has a constant of 7**:
The constant term is -7, since it is the term without a variable.
**This statement is false.**
**Summary of True Statements:**
- 1 and 3, and 4 are true statements.
1. **5x²y − 6xy³z + 3x⁵ is a fifth-degree polynomial**:
The degree of a polynomial is determined by the term with the highest total exponent. Here, the term 3x⁵ has the highest degree (5).
**This statement is true.**
2. **5x²y − 6xy³z + 3x⁵ contains four terms**:
This polynomial has three distinct terms: 5x²y, -6xy³z, and 3x⁵.
**This statement is false.**
3. **5x²y − 6xy³ + 3x⁵ has a leading coefficient of 3**:
Among the terms, 3x⁵ is the term with the highest degree (5), and its leading coefficient is 3.
**This statement is true.**
4. **12a⁵ − 7 + 3a⁷ + 11^a is equivalent to 3a⁷ + 12a⁵ + 11^a − 7**:
The expression is equivalent because addition is commutative, so the order does not affect the sum.
**This statement is true.**
5. **12a⁵ − 7 + 3a⁷ + 11^a has a leading term of 12a⁵**:
The term with the highest degree in this polynomial is 3a⁷ (degree 7), not 12a⁵ (degree 5).
**This statement is false.**
6. **12a⁵ − 7 + 3a⁷ + 11^a has a constant of 7**:
The constant term is -7, since it is the term without a variable.
**This statement is false.**
**Summary of True Statements:**
- 1 and 3, and 4 are true statements.
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