Let's evaluate each of the statements about polynomials:
-
"5x²y − 6xy³z + 3x⁵ is a fifth degree polynomial."
True. The degree of a polynomial is determined by the term with the highest sum of exponents. Here, the term \(3x^5\) has a degree of 5, which is greater than the other terms. -
"5x²y − 6xy³z + 3x⁵ contains four terms."
False. This polynomial contains three terms: \(5x²y\), \(-6xy³z\), and \(3x⁵\). -
"5x²y − 6xy³ + 3x⁵ has a leading coefficient of 3."
True. In the polynomial \(5x²y − 6xy³ + 3x⁵\), the leading term is \(3x⁵\), and the leading coefficient is 3. -
"12a⁵ − 7 + 3a⁷ + 11a is equivalent to 3a⁷ + 12a⁵ + 11a − 7."
True. The terms are the same; the order does not affect the equivalency of the polynomial expressions. -
"12a⁵ − 7 + 3a⁷ + 11a has a leading term of 12a⁵."
True. The leading term is the term with the highest degree, and here it is \(12a⁵\). -
"12a⁵ − 7 + 3a⁷ + 11a has a constant of 7."
False. The constant term here is \(-7\), not \(7\).
In summary, the true statements are:
- 1 (5x²y − 6xy³z + 3x⁵ is a fifth degree polynomial)
- 3 (5x²y − 6xy³ + 3x⁵ has a leading coefficient of 3)
- 4 (12a⁵ − 7 + 3a⁷ + 11a is equivalent to 3a⁷ + 12a⁵ + 11a − 7)
- 5 (12a⁵ − 7 + 3a⁷ + 11a has a leading term of 12a⁵)
Final selections: Statements 1, 3, 4, and 5 are true.