# Polynomials

A polynomial expression is a fundamental concept in algebra, which consists of a combination of monomial that are added or subtracted to form the expression. A monomial is essentially a single term that comprises a constant, a variable, or a product of both, for example, \(ax^n\), of the form:

\[a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{2}x^{2}+a_{1}x+a_{0}\]

- \(ax^n\) is a monomial.
- \(a\) is a real numbers called the coefficient of the polynomial.
- \(n\), the exponent of the variable, is a non-negative integer.

The degree of a polynomial is the highest power of the \(x\) in its expression. For example \(x^5+2x+1\) is a polynomial of degree 5.

A polynomial function \(P(x)\) is considered a zero polynomial when all its coefficients are equal to zero. In other words, \(P(x)\) can be expressed as \(a_n x^n + a_{n-1} x^{n-1} + … + a_1 x + a_0 = 0\), where \(a_n, a_{n-1}, …, a_1\), and \(a_0\) are all equal to zero. For instance, the polynomial \(0x^3 + 0x^2 + 0x + 0\) is a zero polynomial, as all coefficients are zero.

The fundamental theorem of algebra states that every non-constant polynomial with coefficients over the field of complex numbers \(\mathbb{C}\) can be factored into a product of linear factors.

## Monomials, binomials, trinomials

A monomial is a polynomial expression comprising only one term, a constant, a single variable, or a combination of constants and variables raised to non-negative integer powers. For instance, \(3x^2\) and \(-5y\) are both monomials.

A binomial is a polynomial expression consisting of two terms: constants, variables, or the product of constants and variables raised to non-negative integer powers. For example, \(3x + 7\) and \(-2y^2 + 5y\) are both binomials.

A trinomial is a polynomial expression consisting of three terms, which can also be constants, variables, or the product of constants and variables raised to non-negative integer powers. For instance, \(x^2-2x + 4\) and \(3y^3 + 2y^2- y\) are both trinomials.

## Sum or difference of two polynomials

The sum or difference of two polynomials of the same degree results in a polynomial of the same degree. For example, if we have two polynomials of degree \(n\), say \(P(x)\) and \(Q(x)\), then their sum or difference, denoted by \(P(x) ± Q(x)\), is also a polynomial of degree \(\leq n\).

The sum or difference of the two polynomials is obtained by adding or subtracting the corresponding coefficients of the like terms.

\[ \begin{align*} P(x) + Q(x) &= (ax^n + bx^{n-1} + \ldots + z) + (px^n + qx^{n-1} + \ldots + w) \\[0.6em] &= (a+p)x^n + (b+q)x^{n-1} + \ldots + (z+w) \end{align*} \]

\[ \begin{align*} P(x)-Q(x) &= (ax^n + bx^{n-1} + \ldots + z) – (px^n + qx^{n-1} + \ldots + w) \\[0.6em] &= (a-p)x^n + (b-q)x^{n-1} + \ldots + (z-w) \end{align*} \]

## Example

Given two polynomial \(P(x)\) and \(Q(x)\), calculate the sum \(P(x)\) + \(Q(x)\):

\[ P(x) = x^2 + 3x-1 \] \[ Q(x) = 2x^2-x + 5 \]

Their sum is given by:

\[ P(x) + Q(x) = \left( x^2 + 3x-1 \right) + (2x^2-x + 5) \]

Let us proceed with the simplification of the given equation by eliminating the parentheses and grouping the terms that share the same degree.

\[ = x^2 + 3x-1 + 2x^2-x + 5 \] \[ = (x^2 + 2x^2) + (3x-x) + (-1 + 5) \] \[ = 3x^2 + 2x + 4 \]

Hence, the resultant of the two polynomials \(P(x) + Q(x)\) is expressed as:

\[3x^2 + 2x + 4 \]

As we mentioned above, let’s consider two polynomials of degree \(n\). Their sum or difference is a polynomial of, to most degree, \(n\). Let’s illustrate this with an example:

\[ P(x) = 2x^2+3x-1 \] \[ Q(x) = 2x^2-x+5 \]

The difference \( P(x)-Q(x) \) is:

\[ P(x)-Q(x) = \left( 2x^2+3x-1 \right)-\left( 2x^2-x+5 \right) \]

Now let’s simplify:

\begin{align*} P(x)-Q(x) &= 2x^2+3x-1- 2x^2+x-5 \\ &= (2x^2-2x^2)+(3x+x)+(-1-5) \\ &= 4x-6 \end{align*}

It can be observed that subtracting the monomials of degree \(n=2\) from the polynomial produces a first degree \(n-1\) polynomial. \( 4x – 6 \).

## Factoring polynomials

The manipulation of polynomials, comprehension of their properties, and application of methods such as factoring are indispensable for solving various mathematical problems. A solid grasp of the manipulation of polynomials, their properties, and the multiple techniques for solving them is essential for tackling a wide range of mathematical problems. We suggest exploring the following topics to expand your knowledge in this field.

## Polynomial equations

A polynomial or algebraical equation that takes the form of a sum of terms, each consisting of a constant multiplied by one variable raised to a non-negative integer power:

\[a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{2}x^{2}+a_{1}x+a_{0} = 0\]

Polynomial equations are classified according to the degree of the monomial with the highest degree within the expression. Depending on their degree, they can be categorized as linear, quadratic, cubic of degree greater than three.