The polynomial equations They are a statement that raises the equality of two expressions or members, where at least one of the terms that make up each side of the equality are polynomials P (x). These equations are named according to the degree of their variables.
In general, an equation is a statement that establishes the equality of two expressions, where in at least one of these there are unknown quantities, which are called variables or unknowns. Although there are many types of equations, they are generally classified into two types: algebraic and transcendental..
Polynomial equations only contain algebraic expressions, which can have one or more unknowns involved in the equation. According to the exponent (degree) they have, they can be classified into: first degree (linear), second degree (quadratic), third degree (cubic), fourth degree (quartic), degree greater than or equal to five and irrational.
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Polynomial equations are expressions that are formed by an equality between two polynomials; that is, by the finite sums of multiplications between values that are unknown (variables) and fixed numbers (coefficients), where the variables can have exponents, and their value can be a positive integer, including zero.
The exponents determine the degree or type of the equation. The term of the expression that has the highest value exponent will represent the absolute degree of the polynomial.
Polynomial equations are also known as algebraic, their coefficients can be real or complex numbers and the variables are unknown numbers represented by a letter, such as: "x".
If substituting a value for the variable "x" in P (x) the result is equal to zero (0), then that value is said to satisfy the equation (it is a solution), and is generally called the root of the polynomial.
When you develop a polynomial equation you want to find all the roots or solutions.
There are several types of polynomial equations, which are differentiated according to the number of variables, and also according to the degree of their exponent.
Thus, the polynomial equations -where its first term is a polynomial that has a single unknown, considering that its degree can be any natural number (n) and the second term is zero-, can be expressed as follows:
ton * xn + ton-1 * xn-1 +… + A1 * x1 + to0 * x0 = 0
Where:
- ton, ton-1 already0, are real coefficients (numbers).
- ton is different from zero.
- The exponent n is a positive integer that represents the degree of the equation.
- x is the variable or unknown to be searched.
The absolute or greater degree of a polynomial equation is the exponent with the highest value among all those that form the polynomial; thus, the equations are classified as:
First degree polynomial equations, also known as linear equations, are those in which the degree (the greatest exponent) is equal to 1, the polynomial is of the form P (x) = 0; and it is composed of a linear term and an independent one. It is written as follows:
ax + b = 0.
Where:
- a and b are real numbers and a ≠ 0.
- ax is the linear term.
- b is the independent term.
For example, the equation 13x - 18 = 4x.
To solve linear equations, all the terms that contain the unknown x must be passed to one side of the equality, and those that do not have they move to the other side, in order to solve it and obtain a solution:
13x - 18 = 4x
13x = 4x + 18
13x - 4x = 18
9x = 18
x = 18 ÷ 9
x = 2.
Thus, the given equation has only one solution or root, which is x = 2.
Second degree polynomial equations, also known as quadratic equations, are those in which the degree (the greatest exponent) is equal to 2, the polynomial is of the form P (x) = 0, and is composed of a quadratic term , one linear and one independent. It is expressed as follows:
axtwo + bx + c = 0.
Where:
- a, b and c are real numbers and a ≠ 0.
- axtwo is the quadratic term, and "a" is the coefficient of the quadratic term.
- bx is the linear term, and "b" is the coefficient of the linear term.
- c is the independent term.
Generally, the solution to this type of equations is given by clearing x from the equation, and it is as follows, which is called resolvent:
There, (btwo - 4ac) is called the discriminant of the equation and this expression determines the number of solutions that the equation can have:
- Yes (btwo - 4ac) = 0, the equation will have a single solution that is double; that is, it will have two equal solutions.
- Yes (btwo - 4ac)> 0, the equation will have two different real solutions.
- Yes (btwo - 4ac) < 0, la ecuación no tiene solución (tendrá dos soluciones complejas distintas).
For example, we have the equation 4xtwo + 10x - 6 = 0, to solve it, first identify the terms a, b and c, and then substitute it in the formula:
a = 4
b = 10
c = -6.
There are cases in which the second degree polynomial equations do not have all three terms, and that is why they are solved in different ways:
- In the case that the quadratic equations do not have the linear term (that is, b = 0), the equation will be expressed as axtwo + c = 0. To solve it, solve for xtwo and the square roots are applied to each member, remembering that the two possible signs that the unknown may have must be considered:
axtwo + c = 0.
xtwo = - c ÷ a
For example, 5 xtwo - 20 = 0.
5 xtwo = 20
xtwo = 20 ÷ 5
x = ± √4
x = ± 2
x1 = 2.
xtwo = -2.
- When the quadratic equation does not have an independent term (that is, c = 0), the equation will be expressed as axtwo + bx = 0. To solve it, we must take the common factor of the unknown x in the first member; Since the equation is equal to zero, it is true that at least one of the factors will be equal to 0:
axtwo + bx = 0.
x (ax + b) = 0.
Thus, you have to:
x = 0.
x = -b ÷ a.
For example: we have the equation 5xtwo + 30x = 0. First factor:
5xtwo + 30x = 0
x (5x + 30) = 0.
Two factors are generated which are x and (5x + 30). It is considered that one of these will be equal to zero and the other is given a solution:
x1 = 0.
5x + 30 = 0
5x = -30
x = -30 ÷ 5
xtwo = -6.
Polynomial equations of higher degree are those that go from third degree onwards, which can be expressed or solved with the general polynomial equation for any degree:
ton * xn + ton-1 * xn-1 +… + A1 * x1 + to0 * x0 = 0
This is used because an equation with a degree greater than two is the result of factoring a polynomial; that is, it is expressed as the multiplication of polynomials of degree one or greater, but without real roots.
The solution of this type of equations is direct, because the multiplication of two factors will be equal to zero if any of the factors is null (0); therefore, each of the polynomial equations found must be solved, setting each of their factors equal to zero.
For example, we have the third degree (cubic) equation x3 + xtwo +4x + 4 = 0. To solve it, the following steps must be followed:
- The terms are grouped:
x3 + xtwo +4x + 4 = 0
(x3 + xtwo ) + (4x + 4) = 0.
- The members are decomposed to get the common factor of the unknown:
xtwo (x + 1) + 4 (x + 1) = 0
(xtwo + 4)*(x + 1) = 0.
- In this way, two factors are obtained, which must be equal to zero:
(xtwo + 4) = 0
(x + 1) = 0.
- It can be seen that the factor (xtwo + 4) = 0 will not have a real solution, while the factor (x + 1) = 0 will. So the solution is:
(x + 1) = 0
x = -1.
Solve the following equations:
(2xtwo + 5)*(x - 3)*(1 + x) = 0.
In this case the equation is expressed as the multiplication of polynomials; that is, it is factored. To solve it, each factor must be set equal to zero:
- 2xtwo + 5 = 0, has no solution.
- x - 3 = 0
- x = 3.
- 1 + x = 0
- x = - 1.
Thus, the given equation has two solutions: x = 3 and x = -1.
x4 - 36 = 0.
A polynomial was given, which can be rewritten as a difference of squares to arrive at a faster solution. Thus, the equation is:
(xtwo + 6)*(xtwo - 6) = 0.
To find the solution of the equations, both factors are set equal to zero:
(xtwo + 6) = 0, has no solution.
(xtwo - 6) = 0
xtwo = 6
x = ± √6.
Thus, the initial equation has two solutions:
x = √6.
x = - √6.
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