## Content

### 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 $$f(r)=0$$.

#### Example

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?

#### Solution

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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