Question

Identify the leading coefficient, and classify the polynomial by degree and by number of terms. $$16-4 x+3 x^{2}$$

Short answer

The leading coefficient of the polynomial \(3x^2 - 4x + 16\), is \(3\). The polynomial is of degree \(2\) (quadratic polynomial) and has \(3\) terms making it a trinomial.

Step-by-step solution

Identify the Polynomial's terms and order it in descending order

First, write down all terms of the polynomial and put them in descending order (degrees from highest to lowest). The given polynomial was \(16-4 x+3 x^{2}\) . In descending order, this will be: \(3x^2 - 4x + 16\)

Identify the Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term of the highest degree. In our polynomial, the term of the highest degree is \(3x^2\). So, the leading coefficient of our polynomial is \(3\).

Classify the Polynomial by degree

The degree of a polynomial is determined by the power of the term with the highest exponent. In our polynomial, the term with the highest exponent is \(3x^2\), so the degree of our polynomial is \(2\). This implies our polynomial is known as a quadratic polynomial.

Classify the Polynomial by number of terms

The number of terms in a polynomial is determined by counting how many parts are being added or subtracted. In our polynomial \(3x^2 - 4x + 16\), we have three terms. Therefore, our polynomial is called a 'trinomial'.