Polynomials

Recall that algebraic terms are of the form: $$ \begin{equation}\begin{aligned} ax^b\\ \end{aligned}\end{equation} $$

where $a$ is the coefficient, $x$ is the base and $b$ is the power/exponent. Recall that algebraic terms are separated by $+$, $-$ and $=$ signs. The following algebraic expression thus has $3$ terms: $$ \begin{equation}\begin{aligned} x^2+5x+6\\ \end{aligned}\end{equation} $$ The following algebraic equation has $4$ terms involved: $$ \begin{equation}\begin{aligned} x-3=y+2\\ \end{aligned}\end{equation} $$

Challenge

How many terms are in the following expressions, equations and inequalities:

• $x^3-x+4$
• $\sin{x}+5$
• $4-x^2=1-\frac{1}{y}$
• $1-x\gt 4+y$

In an inequality, the same rule for distinguishing terms that applies to the $=$ sign, applied to the inequality sign. Thus algebraic terms in an algebraic inequality can be separated by $\gt$, $\ge$ , $\lt$ and $\le$ signs.

What is a polynomial?

Now that we’ve had a refresher on how to count the number of terms, we can define a polynomial. Polynomials are algebraic expressions with many terms.

Poly means “many” and nomial means “terms”.

More specifically, the powers of these terms must be whole numbers. The format for a polynomial is: $$ \begin{equation}\begin{aligned} a_n x^n+...+a_2 x^2+a_1 x^1+a_0\\ \end{aligned}\end{equation} $$

where $n$ is always an integer (positive and negative whole numbers and zero). The following is a polynomial: $$ \begin{equation}\begin{aligned} x^4-x+1\\ \end{aligned}\end{equation} $$ It is specifically referred to as a trinomial because it has $3$ terms. The number of terms in a polynomial can help us to give them special names:

• Polynomials with $1$ term are called monomials e.g. $x$
• Polynomials with $2$ term are called binomials e.g. $2x-1$
• Polynomials with $3$ term are called trinomials e.g. $x^2-5x+6$

The degree of a polynomial

This refers to the highest power of the variable. We have special classifications for polynomials with a particular degree. For example, the polynomial $x^2-5x+6$ has a term with the highest power being $2$. We refer to this polynomial as a quadratic polynomial. Classifications according to degree include:

• Constant polynomials have a degree of $0$
• Linear polynomials have a degree of $1$
• Quadratic polynomials have a degree of $2$
• Cubic polynomials have a degree of $3$
• Quartic polynomials have a degree of $4$
• Quintic polynomials have a degree of $5$

At this level, the highest degree we work with is $4$.

Challenge

What are the degrees of the following polynomials?

• $5x-12$
• $x^3+x^2+4x-1$
• $1-x^4$
• $x-x^5+1$
• $12$
• $y^2+2y-3$

What the degree tells us

The degree will tell us the maximum number of roots and the maximum number of stationary points (this concept will be explored when we cover calculus). In relation to the roots, consider the polynomial $x^2+5x+6$. It has a degree of $2$ which means that it can have at most $2$ roots. Indeed, when we solve the polynomial for its root, we get $x=2$ and $x=3$ (two solutions a.k.a. roots).

Notice that we say the degree tells us the maximum number of roots. Consider the polynomial $x^2+4x+4$. We should know that it factorizes to produce $(x+2)^2$. This means that when we solve for $x$, we will get $x=-2$ twice. Since $x=-2$ occurs twice, we only have one root. This shows that having a degree of $2$ does not mean we will get $2$ roots every time.

Challenge

What is the degree of the polynomial $x^2-3x+4$? How many roots does it have?