13. Polynomials
Polynomials are mathematical functions that have a wide application in several areas, such as physics and engineering. In the syllabus of ENEM, polynomials are one of the topics covered in mathematics, being essential for candidates to understand this subject well in order to solve the questions of the test.
A polynomial is a mathematical expression involving variables and coefficients. Coefficients are real numbers and variables are raised to non-negative integer powers. The sum of these terms forms the polynomial. Each term of a polynomial is called a monomial. The degree of a polynomial is determined by the largest exponent of its variable.
Representation of a Polynomial
A polynomial P(x) of degree n in the variable x is represented by the expression:
P(x) = anxn + an-1xn-1 + an-2xn-2 + ... + a2x2 + a 1x + a0
Where, an, an-1, an-2, ..., a2, a1 and a0 are the coefficients of the polynomial and x is the variable. The coefficient an is called the leading coefficient and cannot be zero.
Operations with Polynomials
Operations with polynomials include addition, subtraction, multiplication, and division. In addition and subtraction, we combine like terms to get the result. In multiplication, each term of the first polynomial is multiplied by each term of the second polynomial. Dividing polynomials is a bit more complex and requires using the long division or synthetic division algorithm.
Numerical Value of a Polynomial
The numeric value of a polynomial is the result we get when we replace the polynomial variable with a number. For example, if we have the polynomial P(x) = 2x2 + 3x + 1 and we want to find the value of P(2), we substitute x for 2 in the polynomial expression to get P(2) = 2(2)2 + 3(2) + 1 = 11.
Identical Polynomials
Two polynomials are identical if they have the same coefficients and the same corresponding degrees in their variables. For example, the polynomials P(x) = 2x3 + 3x2 - x + 1 and Q(x) = 2x3 + 3x 2 - x + 1 are identical because they have the same coefficients (2, 3, -1, 1) and the same corresponding degrees (3, 2, 1, 0) in their variables. p>
Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra states that every polynomial of degree n, with n greater than zero, has exactly n complex roots. This means that we can solve any polynomial equation by finding its roots. The roots of a polynomial are the values of x that make the polynomial equal to zero.
In summary, polynomials are a fundamental part of mathematics and have many practical applications. In ENEM, understanding polynomials is essential to solve many mathematical problems. Therefore, it is important to study and understand this topic well.
