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. 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?