Six statements about polynomials are given. Select all the statements that are true(3 points)

Question

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

Answer 1

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.