Question

Which of the following statements are TRUE for the polynomial below?

p(x) = 2x^3 - 4x^2 + 2x - 15
a) The polynomial is of degree 3.
b) The coefficient of the linear term is 2.
c) The leading coefficient is -15.
d) The polynomial has three distinct real roots.

Answer 1

The polynomial is indeed of degree 3 with the coefficient of the linear term being 2. However, the leading coefficient is 2, not -15, and we cannot determine the number of distinct real roots without further analysis.

Step-by-step explanation:

Let's address the statements regarding the polynomial p(x) = 2x^3 - 4x^2 + 2x - 15 one by one:

• a) The polynomial is of degree 3. This statement is TRUE. The degree of a polynomial is the highest power of the variable x in the polynomial. In this case, the highest power is 3, as seen in the term 2x^3.
• b) The coefficient of the linear term is 2. This statement is TRUE. The linear term in the polynomial is 2x, and its coefficient is indeed 2.
• c) The leading coefficient is -15. This statement is FALSE. The leading coefficient is the coefficient of the term with the highest power, which is 2x^3. Therefore, the leading coefficient is 2, not -15.
• d) The polynomial has three distinct real roots. Determining whether this statement is true requires either the Rational Root Theorem, graphing, or numerical methods such as Newton's method. Without further analysis, we cannot definitively determine the nature of the roots.