Chapter 4.CT, Problem 1CT

Determine the leading term, the leading coefficient, and the degree of the polynomial. Then classify the polynomial as constant, linear, quadratic, cubic, or quartic.

1. $f\left(x\right)=2x^3+6x^2-x^4+11$

To determine

To find:

The leading term, the leading coefficient, the degree and the type of given polynomial $f\left(x\right)=2x^3+6x^2-x^4+11$.

Answer to Problem 1CT

Solution:

The leading term is $-x^4$. The leading coefficient is $-1$. The degree of the polynomial is $4$. The type of the polynomial is quartic.

Explanation of Solution

Given:

The given polynomial is $f\left(x\right)=2x^3+6x^2-x^4+11$.

Concept:

            (a) The leading term is the first term of a polynomial arranged in descending order of their powers.

            (b) The leading coefficient is the coefficient of leading term.

            (c) The degree is the highest power of a polynomial.

            (d) The type of a polynomial is defined by its degree.

                            1 A polynomial with degree zero is the constant polynomial function.

                            2 A polynomial with degree one is the linear polynomial function.

                            3 A polynomial with degree two is the quadratic polynomial function.

                            4 A polynomial with degree three is the cubic polynomial function.

                            5 A polynomial with degree four is the quartic polynomial function.

The given polynomial $f\left(x\right)$ is in descending order,

$f\left(x\right)=-x^4+2x^3+6x^2+11$.

The first term of the polynomial is $-x^4$. So, the leading term is $-x^4$.

The coefficient of the leading term is $-1$. So, the leading coefficient is $-1$.

The leading term has the highest power of 4. So, the degree of the polynomial is 4.

The degree of the polynomial is 4. So, the polynomial is a quartic polynomial.