Question

Determine if the function is a polynomial function. If the function is a polynomial function, state the degree and the leading coefficient. If the function is not a polynomial, state why. 
 

Answer 1

• Polynomial function :

A polynomial function is a function such as a quadratic, a cubic, a quartic, and so on,

involving only non-negative integer powers of x. We can give a general defintion of a polynomial, and define its degree.

• A polynomial of degree n is a function of the form :

 

f(x) = a _n x ⁿ + a _{n - 1} x ^{n - 1} + . . .+ a ₂ x ² + a ₁ x + a ₀.

 

• The degree of a polynomial is the highest power of x in its expression. Constant (non - zero)

polynomials, linear polynomials, quadratics, cubics and quartics are polynomials of degree 0, 1,

2 , 3 and 4 respectively.

A polynomial can have any non-negative degree. In other words, the polynomial's degree n must be ≥ 0.

 

The function f(x) = 0 is also a polynomial, but we say that its degree is ‘undefined’.

 

• The leading term is the term with the highest power of x is the leading coefficient of the

polynomial function.

The function is f (x) = (π/4)x⁴ - x + 4.

The function f (x) = (π/4)x⁴ - x + 4 is a polynomial of degree 4 with a leading coefficient of π/4(0.7857).

Note that, the degree of the polynomial is the greatest power to which x is raised and the leading coefficient is the coefficient of that power regardless of the order in which the terms are written.

Comments

Sorry typo mistake.......

The function is f (x) = πx⁴ - x + 4.

The function f (x) = πx⁴ - x + 4 is a polynomial of degree 4 with a leading coefficient of π(3.14).

Note that, the degree of the polynomial is the greatest power to which x is raised and the leading coefficient is the coefficient of that power regardless of the order in which the terms are written.