Polynomials

 Polynomials 

Algebraic expressions that contain monomials are considered polynomials. The study of these expressions is directly related to arithmetic operations.

 Polynomials are algebraic expressions that have monomials formed by a coefficient and a literal part. Polynomials are algebraic expressions that have monomials formed by a coefficient and a literal part. A polynomial is an algebraic expression formed by monomials and arithmetic operators. 

The monomial is structured by numbers (coefficients) and variables (literal part) in a product, and the arithmetic operators are: addition, subtraction, division, multiplication and power. To better understand what a polynomial is, see some examples:

5 Coefficient:

 5 Literal part: Any variable raised to zero, ie x⁰ = 1 → 5 . x⁰ Arithmetic Operators: Multiplication.

Two . x. and Coefficient: 2 Literal part: a. and Arithmetic Operators: Multiplication.

3 . x. y + (4 .x : 2 .x) Coefficient: 3, 4 and 2 Literal part: x .y ​​and x Arithmetic operators: Addition, multiplication and division.

{[(2 .x + 6 .x)2 – 5] + 3 . y - 1 . x}  Coefficient: 1, 2, 3, 5 and 6 Literal part: x and y Arithmetic operators: Addition, subtraction, multiplication and power.

Classification of Polynomials 

Polynomials can be classified according to their number of terms: 

Monomial: It has a single product with coefficient and literal part.

 Examples: ⇒ 2 . x. and ⇒ 6 ⇒ 12 . x²

Binomial: It is a polynomial that has only two monomials. 

Examples: ⇒ 4 . x. y + 5 . x ⇒ 34 . z + 12 . x ⇒ 105 . z + 25 . z²

Trinomial: It is a polynomial that has only three monomials. 

Examples: ⇒ 2 . x. y + 2x - y3  ⇒ x. z⁴ + 25 – z . x ⇒ 2 . w + 12 . x - 5 . w²                       3

Polynomial: has an infinite number of monomials. Its general expression is given by:

an xn+a(n-1) x(n-1)+...+a2 x2+a1 x+a

Degree of a Polynomial 

Degree of polynomial with one variable: When the polynomial has only one variable (unknown term), its degree is given by the highest value that the exponent of the variable assumes. Examples:

⇒ 2 . x2 + 3 . x

Variable: z Highest exponent in relation to variable z: 3 Degree: 3rd degree polynomial.

Degree of polynomial with more than one variable: When the polynomial has more than one variable, to know its degree, we must add the exponents of each monomial. The largest sum of exponents will determine the degree. Example:

3 + 12 . x . y – 2 . x . y2

Grau do monômio: x1 . Y1 → 1 + 1 = 2

Grau do monômio: x . y2 → 1 + 2 = 3