To determine the type of polynomial given by \( P(x) = 4x - 2x^2 + 3 \), we need to follow these steps:
1. Identify the Terms and Their Degrees:
- In the polynomial \( P(x) = 4x - 2x^2 + 3 \), there are three terms:
- \( 4x \): This term has a degree of 1 because the exponent of \( x \) is 1.
- \( -2x^2 \): This term has a degree of 2 because the exponent of \( x \) is 2.
- \( 3 \): This term is a constant and has a degree of 0 because there is no \( x \).
2. Determine the Degree of the Polynomial:
- The degree of a polynomial is the highest degree among its terms.
- In \( P(x) = 4x - 2x^2 + 3 \), the term with the highest degree is \( -2x^2 \), which has a degree of 2.
3. Classify the Polynomial Based on Its Degree:
- A polynomial of degree 0 is called a constant polynomial.
- A polynomial of degree 1 is called a linear polynomial.
- A polynomial of degree 2 is called a quadratic polynomial.
- A polynomial of degree 3 is called a cubic polynomial.
- A polynomial of degree 4 is called a quartic polynomial.
- A polynomial of degree 5 is called a quintic polynomial.
Since \( P(x) = 4x - 2x^2 + 3 \) has its highest degree term as \( -2x^2 \), which is of degree 2, the polynomial is classified as a quadratic polynomial.
Thus, the answer is:
D. quadratic
