Polynomials

An algebraic term follows the format: $$ \begin{equation}\begin{aligned} ax^b\\ \end{aligned}\end{equation} $$

where $a$ is the coefficient, $x$ is the variable and $b$ is the power/exponent. As an easy rule of thumb, we can say that algebraic terms are separated by $+$, $-$ and $=$ signs. Consider the following expression: $$ \begin{equation}\begin{aligned} 5x+1\\ \end{aligned}\end{equation} $$

How many terms are there? Because, there is one $+$ sign, acting as a boundary between the terms, we conclude that there are $2$ algebraic terms present. Consider the equation: $$ \begin{equation}\begin{aligned} 5x+1=3-y\\ \end{aligned}\end{equation} $$

There are now $3$ boundaries (a $+$, an $=$ and a $-$). Therefore, we have $4$ algebraic terms involved in this equation.

What is a polynomial?

A polynomial means “many terms”. Generally, a polynomial follows the format: $$ \begin{equation}\begin{aligned} a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0\\ \end{aligned}\end{equation} $$

where $a_0$ to $a_n$ are real numbers and $n$ is an integer. Thus all of the following are polynomials:

• $x+1$
• $2x^-4$
• $5-x^3+2x$
• $12$

Classifying polynomials

One way in which we can classify polynomials is by the number of terms:

• 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. $x^2-5x+6$
• Polynomials with $4$ term are called quadrinomials e.g. $x^3-x^2-5x+6$

Challenge

Classify the following polynomials according to the number of terms:

• $x-16$
• $5x^3-2x^2+5x-7$
• $12x$
• $1-x^2$
• $11x+6x+5$

The degree of a polynomial

The degree of a polynomial is the highest power of $x$ in the polynomial. 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 $5$

Challenge

Classify the following polynomials by their degree:

• $2x-13$
• $x^3+x^2+4x-1$
• $1-x^4$
• $x-x^5+1$
• $12$
• $y^2+2y-3$

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 in 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?