1.10 Polynomials - Avidemia

What we discuss here:

Meaning of the terms of an expression
What is a polynomial and what is not a polynomial
Degree of a polynomial
Like and unlike terms
How to add, subtract, and multiply polynomials

Terms of an expression

The parts of an expression connected by the signs + or – are called the terms of the expression. For example, $x^{2}$ , $2 x y$ , and $\sqrt{x}$ are the terms of the expression $x^{2} + 2 x y + \sqrt{x},$ and the expression $2 a - a b + c^{2}$ is made up of three terms: $2 a$ , $- a b$ , and $c^{2}$ .

Polynomials

A polynomial or more precisely a polynomial in $x$ is an algebraic expression consisting of terms in the form $a x^{k}$ where $k$ is a nonnegative integers (that is zero or a natural number 0, 1, 2, …) and $a$ is a real number called the coefficient of the term. For example,

$3, x, 7 x^{51}, - x^{3} + \frac{1}{\sqrt{14}} x - 3, and 7 x^{5} - π x^{4} - \sqrt{2} x^{3} - x^{2} + x - 3$

are all polynomials. In the last example, the coefficients are $7, - π, - \sqrt{2}, - 1, 1,$ and $- 3$ .

Note that any constant is also a polynomial because it can be written as $a x^{0}$ ; for example $3 = 3 x^{0}$ .

Examples of expressions that are not polynomials:

Here are some examples of expressions that are not polynomials

$7 x^{12} - 4 x^{- 3} + 8 x - 12, 3 x^{2} + \frac{1}{x} - 9, 4 \sqrt{x} + 5 x^{2} .$

The first example is not a polynomial because it has a negative exponent $- 3$ while all exponents must be nonnegative integers. The second example is not a polynomial because $1 / x = x^{- 1}$ and again all exponents must be nonnegative integers. Similarly $4 \sqrt{x} + 5 x^{2}$ is not a polynomial because $4 \sqrt{x} = 4 x^{1 / 2}$ and in this term the exponent of $x$ is not an integer.

Of course, instead of a polynomial in $x$ , we may have polynomials in $y$ or $z$ or any other letter. For example, $4 z^{3} - 0.5 z + 1$ is a polynomial in $z$ .

Monomials, binomials, and Trinomials

A polynomial that has only one term is called a monomial. For example, $4 x^{3}$ and $- \sqrt{7} x^{2}$ are two monomials.
A polynomial that has two terms is called a binomial. For example, $3 x + 2$ and $- 4 x^{3} + x$ are two binomials.
A polynomial that has three terms is called a trinomial. For example, $3 x + 8 x^{5} - 9$ is a trinomial.

Degree of a Polynomial

The largest exponent in a polynomial is called the degree of the polynomial. In $7 x^{5} - π x^{4} - \sqrt{2} x^{3} - x^{2} + x - 3,$ the largest power of $x$ and hence the degree of the polynomial is 5.

Polynomials of degree 0, 1, 2, and 3 have special names. If

a \neq 0

then

name	form	degree
Constant polynomial	$a$	0
Linear polynomial	$a x + b$	1
Quadratic polynomial	$a x^{2} + b x + c$	2
Cubic polynomial	$a x^{3} + b x^{2} + c x + d$	3

Basic Operations on Polynomial

When two or more terms differ only in their numerical coefficients, we say they are similar or like terms. For example, $4 x^{2}$ , $\frac{3}{2} x^{2}$ , and $- \sqrt{3} x^{2}$ are like terms but $3 x$ and $3 x^{2}$ are unlike terms.

Adding and subtracting polynomials: To add or subtract polynomials, we add or subtract the coefficients of like terms. For example:

$\begin{aligned} (7 x^{3} - 6 x^{2} + 4 x + 9) + (x^{3} + 4 x^{2} - 2) & = (7 + 1) x^{3} + (- 6 + 4) x^{2} + (4 + 0) x + (9 - 2) \\ = 8 x^{3} - 2 x^{2} + 4 x + 7 \end{aligned}$

and

$\begin{aligned} (- 3 x^{4} - 8 x) - (5 x^{3} - 3 x + \sqrt{2}) & = (- 3 - 0) x^{4} + (0 - 5) x^{3} + (- 8 - (- 3)) x + (0 - \sqrt{2}) \\ = - 3 x^{4} - 5 x^{3} - 5 x - \sqrt{2} \end{aligned}$
Multiplication of polynomials: We multiply polynomials like any other sums and simplify the result by using the exponent rule $a x^{n} \cdot b x^{m} = a b x^{m + n}$ and collect the like terms. For example,

$\begin{aligned} (4 x^{2} - 3 x + 5) (2 x^{3} - x) & = 4 x^{2} (2 x^{3} - x) - 3 x (2 x^{3} - x) + 5 (2 x^{3} - x) \\ = 8 x^{5} - 4 x^{3} - 6 x^{4} + 3 x^{2} + 10 x^{3} - 5 x \\ = 8 x^{5} - 6 x^{4} + (10 - 4) x^{3} + 3 x^{2} - 5 x \\ = 8 x^{5} - 6 x^{4} + 6 x^{3} + 3 x^{2} - 5 x \end{aligned}$

It is clear that if $A$ is a polynomial of degree $n$ and $B$ is a polynomial of degree $m$ , then the product $A \cdot B$ is a polynomial of degree $n + m$ . In the above example, the first one is a polynomial of degree 2 and the second one is a polynomial of degree 3, and the product is a polynomial of $5 = 2 + 3$ .
Division of polynomials: To divide one polynomial by another, we use long division; the procedure for such a division has been explained in detail in the next Section