Introduction to polynomials

In this lesson, we will discuss polynomials. Everyone studied them at school. But this time we will talk about them in more detail and find out what a polynomial is, what operations can be applied to a polynomial, what types of polynomials there are, and so on. They are used everywhere: from maths to physics, and chemistry to economics. Almost any area of science involves dealing with polynomials in one way or another.

What is a polynomial?

Polynomial is a name given to the following sum: $c_0+c_1x^1+ \dots + c_nx^n$, where $c_i$ are coefficients and $x$ is a variable in every power from $0$ to $n$. The sum could be written in a more compact way: $\sum_{k=0}^{n} c_kx^k$. Each term of a polynomial is called a monomial.

There also exist more complex polynomials that have other variables apart from $x$, but we will not touch them, rather we will consider polynomials with a singular variable and real coefficients.

Main operations on polynomials

Addition. In order to add two polynomials let’s remember school math: as you should remember, we can add coefficients of the same powers of $x$. Same thing here: add coefficients for the same powers.
If we had two polynomials: $2x^2+5x+6$ and $x^2+2x+1$ then their sum would have the following form:
$(2+1)x^2+(5+2)x+(6+1)=3x^2+7x+7$

It's okay if we’re missing some components, we can still apply the same concept. Assuming we had two such polynomials: $4x^4+3x+5$ and $2x^3+4x+2$. Their sum would be $(4+0)x^4+(0+2)x^3+(3+4)x+(5+2)=4x^4+2x^3+7x+7$.

Subtraction. The operation is performed exactly the same way as addition, except we subtract the coefficients. For example:$(2x^2-4x+3)-(x^2+x-4)=x^2-5x+7$.

Multiplication. Again, going back to school lessons. In order to multiply two polynomials, we need to multiply all possible pairs of monomials. So if we had two polynomials: $a_1x+a_0$ and $b_1x+b_0$, then the result of multiplication would be the polynomial $a_1b_1x^2+a_1b_0x+a_0b_1x+a_0b_0=a_1b_1x^2+(a_1b_0+a_0b_1)x+a_0b_0$.
Let’s take a look at an example with numbers this time: you’re given polynomials $x^2+5x+3$ and $3x-4$. As a result we get $3x^3-4x^2+15x^2-20x+9x-12=3x^3+11x^2-11x-12$.

Substitution. That’s a very simple operation. We just need to substitute $x$ with a number. So, for example, if we had a polynomials $P(x)=5x^4-2x^2+4x+11$, then after substituting $2$ we would get $P(2)=5\cdot2^4-2\cdot2^2+4\cdot2+11=91$.

Degree of a polynomial

There is another very important notion such as a degree of a polynomial. A degree of a polynomial is the highest power of $x$ among all monomials in this polynomial. For instance, the polynomial $3x^4+2x^2-x^2-14$ has the degree of $4$, since the monomial $3x^4$ is of such power and it’s the maximal among all others.

Also, there are two interesting notions related to a degree of a polynomial after addition or multiplication of two polynomials.

After addition of two polynomials with degrees $a$ and $b$, given that the corresponding monomials with the highest power do not cancel out, the result will have the degree of $max(a,b)$. Indeed, during addition, we can only add coefficients for the same powers of $x$, therefore the powers themselves won’t change, only their coefficients. To give you an example: let us have two polynomials $2x^2+5$ and $4x-6$ with the degrees $2$ and $1$ respectively. Let’s add them together and take a look at the degree of the result: $2x^2+4x-1$. It adds up.

The result of the multiplication of two polynomials of degrees $a$ and $b$ will be a polynomial of degree $a+b$. Let's check it. Multiplication involves multiplication of all pairs of monomials, which also includes monomials of highest power. Multiplication of monomials results in addition of their powers. To illustrate: assuming we have two polynomials $2x^2+5$ and $4x-6$. Multiplication results in $8x^3-12x^2+20x-30$. Degrees of polynomials were $2$ and $1$, and degree of the result is $3$, so everything fits.

Factorization

Any polynomial can be expressed as a product of linear (having a degree of one) factors, in this form, to be precise: $P(x)=a(x-c_1)(x-c_2) \dots (x-c_n)$ where $c_i$ are the roots of an equation.
For polynomials of degree of $2$ we can do it quite easily, for example, by using Vieta’s formulas, but for polynomials of higher degrees it is much tougher to do, since that requires finding all the roots, which isn’t always a simple task.

Roots of a polynomial

Roots of a polynomial are such numbers that result in $0$ after substitution. For instance, the polynomial $3x^3-2x^2+4x-5$ will have number $1$ as a root.
Polynomials can have more than one root. Actually, a polynomial of degree of $n$ can have no more than $n$ roots. Say, polynomial $4x^2+2x-6$ has only two roots: $1$ and $-1.5$, you can check it yourself.

It may happen that the root of polynomial leaves real number field and turns out to be a complex number. To give you an example, polynomial $x^2-x+3=0$ has two roots: $\frac{1-i\sqrt{11}}{2}$ and $\frac{1+i\sqrt{11}}{2}$, all of them are complex numbers.

A polynomial could also have repeated roots (multiple roots). They are numbers that have a power higher than one in the polynomial factorization. So, the polynomial $x^3-4x^2+5x-2$ after factorization will take the form $(x-2)(x-1)^2$ and turns out that the factor $(x-1)$ appears twice, which means that the root $1$ is repeated.

Conclusion

In this chapter we have tried to tell you as much as possible about polynomials. You probably had already known most of it from school mathematics, but, nonetheless, the repetition was warranted. We have examined main operations on polynomials and polynomial roots. Following that, you have extremely interesting exercises.