Question

skill #1: structure of polynomials study the four definitions the degree of the polynomial is the biggest exponent (the exponent carefully the constant is the number with no variable attached. it is the number (the little number at the top) the leading coefficient is the number in front of the term with the biggest exponent standard form is when the exponent go in descending order from highest to lowest 1 which statement is correct about the polynomial $3x^2 + 5x - 2$? (1) it is a third - degree polynomial with a constant term of -2. (2) it is a third - degree polynomial with a leading coefficient of 3. (3) it is a second - degree polynomial with a constant term of 2. (4) it is a second - degree polynomial with a leading coefficient of 3.

Explanation:

Step1: Analyze the polynomial \(3x^2 + 5x - 2\)

• Degree: The highest exponent of \(x\) is 2, so it's a second - degree polynomial.
• Constant term: The term without a variable is \(- 2\), not \(-2\) as a wrong constant in option (1), and the constant here is \(-2\), not 2 in option (3).
• Leading coefficient: The coefficient of the term with the highest exponent (\(x^2\)) is 3.
• Option (2) says it's a third - degree polynomial, which is wrong since the highest exponent is 2.
• Option (4) says it's a second - degree polynomial with a leading coefficient of 3. Let's check:
• Degree: Since the highest power of \(x\) is 2, it is a second - degree polynomial.
• Leading coefficient: The term with the highest power (\(x^2\)) has a coefficient of 3.

Step2: Evaluate each option

• Option (1): The polynomial is second - degree (highest exponent 2), not third - degree, and the constant term is \(-2\), but the option's description of constant term is wrong in context (and degree is wrong), so (1) is incorrect.
• Option (2): The degree is 2, not 3, so (2) is incorrect.
• Option (3): The constant term is \(-2\), not 2, so (3) is incorrect.
• Option (4): It is a second - degree polynomial (because the highest exponent of \(x\) is 2) and the leading coefficient (coefficient of \(x^2\)) is 3, so (4) is correct.

Answer:

(4) It is a second - degree polynomial with a leading coefficient of 3.