Equilateral and Equiangular Polygons

A polygon is a 2 dimensional geometric figure bound with straight sides. A polygon is called Equilateral if all of its sides are congruent. Common examples of equilateral polygons are a rhombus and regular polygons such as equilateral triangles and squares.  Now, a polygon is equiangular if all of its internal angles are congruent.  Some important facts to consider The only equiangular triangle is the equilateral triangle If P is an equilateral polygon that has more than three sides, it does not have to be equiangular. A rhombus with no right angle is an example of an equilateral but non-equiangular polygon.  Rectangles, including squares, are the only equiangular quadrilaterals Equiangular polygon theorem. Each angle of an equiangular n-gon is  \Bigg(\frac{n-2}{n}\Bigg)180^{\circ} = 180^{\circ} -   \frac{360^{\circ}}{n}
Viviani's theorem   Vincenzo Viviani (1622 – 1703) was a famous Italian mathematician. With his exceptional intelligence in math...

Polynomials: Key Vocabulary

Let us look at some key terms that pertain to Polynomials before we deep dive into the study of Polynomials. 

Constant: A number having a fixed numerical value
Example: 3, \cfrac{4}{5}, 4.2, 6.\overline{3}

Variable: A number which can take various numerical values
Example:   x,  y,  z

Algebraic Expression: A combination of constants and variables connected by arithmetic operators
Example: 2x^2 + 7, 5x^3 + 4xy + 2xy^2 + 7, etc  

Terms: Several parts of an algebraic expression separated  by + or - signs are called the terms of the expression. 
Example: In the expression 9x + 7y + 5, 9x, 7y, and 5 are terms.  

Coefficient of a Term: In the term 8x^2, 8 is the numerical coefficient of x^2 and x^2 is said to be the literal coefficient of 8. 

Like Terms: Terms having the same literal coefficients are called Like Terms. 
Example: 8xy, 9xy, and 10xy are Like Terms and 9x^2, -10x^2, and 5x^2 are Like Terms. 

Unlike Terms: Terms having different literal coefficients are called Unlike Terms. 
Example: 8xy, 9x^2, and 10xy^2 are Unlike Terms. 

Polynomial: An algebraic expression in which the variables involved have only non-negative integral powers. 
Example: 4x^2 + 2x + 1, 5x^3 + 16x^2 + 9

Note - The expression 3x^3 + 4x + \cfrac{5}{x} + 6x^ \frac {3}{2} is not a polynomial, since it has powers of x which are negative and fractions. 
      A polynomial that contains only one variable is known as a polynomial in that variable. 

Degree of a Polynomial in one variable: The highest power of the variable in a polynomial of one variable is called the degree of the polynomial. 
Example: (i)  2x^3 + 3x^2 + x + 8 is a polynomial of degree 3
               (ii)  8x^5 + 6x^3 + 7x is a polynomial of degree 5


Types of Polynomials with Respect to Degree

(1) Linear polynomial: A polynomial of degree one is called a linear polynomial. 
        Example: 2x + 5,   6y - 97z + 8, etc. 

(2) Quadratic polynomial: A polynomial of degree two is called a quadratic polynomial. 
        Example: 6x^2 + 5x + 4,   8y + 7y^2 + 9, etc. 

(3) Cubic polynomial: A polynomial of degree three is called a cubic polynomial. 
        Example: 7x^3 + 8x^2 + 5x + 3,   8z^3 + 9z + 3, etc. 


Types of Polynomials with Respect to Number of Terms

(1) Monomial: An expression containing only one term. 
        Example: 6x,   5x^37xyz^3, etc. 

(2) BinomialAn expression containing two terms
        Example: 6x^2 + 5y,   2xy + 5, etc. 

(3) TrinomialAn expression containing three terms
        Example: 7x + 2y - 4z ,   4z^3 + 5xy + 9, etc.

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