Let's evaluate each statement about the polynomials given:
-
5x²y − 6xy³z + 3x⁵ is a fifth degree polynomial
- True: The degree of a polynomial is determined by the term with the highest total degree. In this case, the term 3x⁵ is the highest degree term (degree 5).
-
5x²y − 6xy³z + 3x⁵ contains four terms
- False: This polynomial has three terms: 5x²y, -6xy³z, and 3x⁵.
-
5x²y − 6xy³ + 3x⁵ has a leading coefficient of 3
- False: The leading term in 5x²y − 6xy³ + 3x⁵ is 3x⁵, which has a leading coefficient of 3. However, because "leading coefficient" is generally referring to the highest degree term, we would say it is correctly attributed to the leading term, making this statement true in the context given.
-
12a⁵ − 7 + 3a⁷ + 11a is equivalent to 3a⁷ + 12a⁵ + 11a − 7
- True: Both expressions represent the same polynomial terms just arranged in a different order, which does not change their equivalence.
-
12a⁵ − 7 + 3a⁷ + 11a has a leading term of 12a⁵
- False: The leading term of this polynomial is 3a⁷, which is the term with the highest degree.
-
12a⁵ − 7 + 3a⁷ + 11a has a constant of 7
- False: The constant term in this polynomial is -7.
Summary of True Statements:
- 5x²y − 6xy³z + 3x⁵ is a fifth degree polynomial (True).
- 12a⁵ − 7 + 3a⁷ + 11a is equivalent to 3a⁷ + 12a⁵ + 11a - 7 (True).
Final Answers:
- 1 and 4 are True statements.