Some jargon

We start by defining some words used to describe interesting bits and pieces of a polynomial. Take a general polynomial function

\[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0, \]

where \(a_n, a_{n-1}, \dots, a_0\) are real numbers with \(a_n \ne 0\). We have already defined the degree \(n\) and the coefficients \(a_0, a_1, \dots, a_n\). The coefficient of \(x^k\) is \(a_k\). The highest degree term \(a_n x^n\) is called the leading term and its coefficient \(a_n\) is called the leading coefficient. If the leading coefficient is 1, the polynomial is called monic. The term \(a_0\) is called the constant term.

Polynomials of degree 1, 2, 3, 4, 5 are respectively called linear, quadratic, cubic, quartic and quintic. A degree-zero polynomial is just a constant function, such as \(f(x) = 3\).

A root or a zero of a polynomial \(f(x)\) is a number \(r\) such that \(f(r)=0\).


Let \(f(x) = 18x^4 - 11x^3 - 7 x + 12\). What is this polynomial's degree, leading term, leading coefficient, coefficient of \(x^3\), coefficient of \(x^2\), and constant term? Is \(f(x)\) monic?


The degree of \(f(x)\) is 4. The leading term is \(18x^4\) and the leading coefficient is 18. The coefficient of \(x^3\) is \(-11\). The coefficient of \(x^2\) is 0. The constant term is 12. As the leading coefficient is 18 (not 1), the polynomial \(f(x)\) is not monic.

A note of caution. Some of the coefficients may be zero. In the above example, the coefficient of \(x^2\) was zero; no \(x^2\) term is written. However the leading coefficient is always non-zero, as it's the coefficient of the highest power of \(x\) which actually appears.

Exercise 1

Show that 3 is a root of the polynomial \(2x^3 - 8x^2 + 7x - 3\).

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