# Polynomials

Polynomials are mathematical expressions involving a sum of terms, each term being a product of a constant coefficient and a variable raised to a non-negative integer power. Polynomials are fundamental in mathematics and have a wide range of applications in fields such as algebra, calculus, and number theory.

A polynomial \(P(x)\) in one variable \(x\) is typically written in the form: $$ P(x) = a_n x^n + a_{n-1} x^{n-1} + ⋯ + a_1 x + a_0 $$ Here, \(n\) is a non-negative integer called the degree of the polynomial, and \(a_i\) (for \(i = 0,1,...,n\) ) are the constant coefficients, where \(a_n \neq 0 \) for \( n \ge 1 \). The variable \(x\) is called the indeterminate of the polynomial, and each term \(a_i x^i \) is called a monomial.

Some examples of polynomials:

- \( P(x) = 5x^3 - 2x^2 + 4x-1 \) This is a polynomial of degree 3 (cubic polynomial).
- \( Q(x) = 7x^4 - 3x + 9 \) This is a polynomial of degree 4 (quartic polynomial).
- \( R(x) = 6x - 3 \) This is a polynomial of degree 1 (linear polynomial).

**Important properties and concepts related to polynomials include:**

**Roots or Zeros:** A root (or zero) of a polynomial \(P(x)\) is a value of x for which the polynomial evaluates to zero, i.e., \(P(x)=0\). The Fundamental Theorem of Algebra states that a non-constant polynomial of degree n will have exactly n (not necessarily distinct) complex roots, counting multiplicities.

**Polynomial addition and subtraction:** To add or subtract two polynomials, simply add or subtract their corresponding coefficients.

For example, if $$ P(x)=3x^2 +2x–1 $$ and $$ Q(x)=x^2 –x+4 $$ then $$ P(x)+Q(x)=(3+1) x^2+(2–1)x+ (-1+4)=4x^2+x+3 $$

**Polynomial multiplication:** To multiply two polynomials, apply the distributive law and combine like terms.

For example, if \( P(x)=2x^2+x–3 \) and \( Q(x)=x–1 \), then $$ P(x) \cdot Q(x)=(2x^2+x–3)(x–1)= 2x^3–2x^2+x^2–x–3x+3=2x^3–x^2–4x+3 $$

**Polynomial division:** Dividing one polynomial by another involves finding the quotient and remainder. The division algorithm for polynomials states that for any polynomials \( P(x) \) and \( D(x) \), where \(D(x) \neq 0 \), there exist unique polynomials \( Q(x) \) and \( R(x) \) such that $$ P(x)=D(x)Q(x)+R(x) $$ where either \(R(x)=0 \) or the degree of \(R(x)\) is less than the degree of \( D(x) \). This is analogous to the division of integers, where the quotient and remainder are uniquely determined.

Long division and synthetic division are two common methods used to perform polynomial division.

**Factoring:** Factoring a polynomial involves expressing it as a product of simpler polynomials.

For example, the polynomial \( x^2–5x+6 \) can be factored as \((x–2)(x–3) \). Factoring is an important tool in finding the roots of a polynomial, as a polynomial is equal to zero if and only if one of its factors is equal to zero.

**Greatest common divisor (GCD):** The GCD of two polynomials is the polynomial of the highest degree that divides both polynomials without leaving a remainder. The Euclidean algorithm can be used to find the GCD of two polynomials, similar to how it is used for integers.

**Polynomial functions:** A polynomial function is a function defined by a polynomial expression. These functions have many important properties and are widely used in mathematics and science. For example, polynomial functions are continuous and differentiable on their entire domain, which makes them useful in calculus and mathematical modeling.

**Interpolation:** Polynomials can be used to approximate or interpolate a given set of data points. One common method is Lagrange interpolation, which constructs a polynomial that passes through all the given data points.

**Polynomial equations:** A polynomial equation is an equation in which one side is a polynomial expression and the other side is either a constant or another polynomial expression. Solving polynomial equations involves finding the values of the variable for which the equation holds true. Techniques for solving polynomial equations include factoring, applying the Rational Root Theorem, and using numerical methods such as the Newton-Raphson method.