Let's analyze the polynomial [tex]\(3x^2 + 5x - 2\)[/tex] and determine which statement about it is correct step by step.
1. Identify the degree of the polynomial:
- The degree of a polynomial is the highest power of the variable [tex]\(x\)[/tex] in the polynomial.
- Here, the highest power of [tex]\(x\)[/tex] is 2 (since the highest term is [tex]\(3x^2\)[/tex]).
- Therefore, it is a second-degree polynomial.
2. Identify the leading coefficient:
- The leading coefficient is the coefficient of the term with the highest power of [tex]\(x\)[/tex].
- For the term [tex]\(3x^2\)[/tex], the coefficient is 3.
- Therefore, the leading coefficient is 3.
3. Identify the constant term:
- The constant term in a polynomial is the term that does not contain any variables.
- Here, the constant term is [tex]\(-2\)[/tex].
Now, let's evaluate each given statement based on our analysis:
1. (1) It is a third-degree polynomial with a constant term of -2.
- This statement is incorrect because the polynomial is a second-degree (not third-degree).
2. (2) It is a third-degree polynomial with a leading coefficient of 3.
- This statement is incorrect because the polynomial is a second-degree (not third-degree).
3. (3) It is a second-degree polynomial with a constant term of 2.
- This statement is incorrect because the constant term is [tex]\(-2\)[/tex] (not 2).
4. (4) It is a second-degree polynomial with a leading coefficient of 3.
- This statement is correct. We established that the polynomial is second-degree and the leading coefficient is 3.
Hence, the correct statement about the polynomial [tex]\(3x^2 + 5x - 2\)[/tex] is:
(4) It is a second-degree polynomial with a leading coefficient of 3.
