Algebraic expressions are known as the combination of letters, signs and numbers in mathematical operations. Usually the letters represent unknown quantities and are called variables or unknowns. Algebraic expressions allow translations to the mathematical language expressions of ordinary language. Algebraic expressions arise from the obligation to translate unknown values into numbers that are represented by letters. The branch of mathematics responsible for the study of these expressions in which numbers and letters appear, as well as signs of mathematical operations, is Algebra.
What are algebraic expressions
Table of Contents
As mentioned previously, these operations are nothing more than the combination of letters, numbers and signs that are subsequently used in different mathematical operations. In algebraic expressions, letters have the behavior of numbers and when they take that course, between one and two letters are used.
Regardless of the expression you have, the first thing to do is simplify, this is achieved using the properties of the operation (s), which are equivalent to the numerical properties. To find the numerical value of an algebraic operation, the letter must be replaced by a certain number.
Many exercises can be done on these expressions and they will be done in this section to improve the understanding of the subject in question.
Algebraic expressions examples:
- (X + 5 / X + 2) + (4X + 5 / X + 2)
X + 5 + 4X + 5 / X + 2
5X + 10 / X + 2
5 (X + 2) / X + 2
5
- (3 / X + 1) - (1 / X + 2)
3 (X + 2) - X - 1 / (X + 1) * (X + 2)
2X - 5 / X ^ 2 + 3X + 2
Algebraic language
The algebraic language is one that uses symbols and letters to represent numbers. Its main function is to establish and structure a language that helps to generalize the different operations that take place within arithmetic where only numbers and their elementary arithmetic operations (+ -x%) occur.
The algebraic language aims to establish and design a language that helps to generalize the different operations that take place within arithmetic, where only numbers and their basic mathematical operations are used: addition (+), subtraction (-), multiplication (x) and division (/).
The algebraic language is characterized by its precision, since it is much more concrete than the numerical language. Through it, sentences can be expressed briefly. Example: the set of multiples of 3 is (3, 6, 9, 12…) is expressed 3n, where n = (1, 2, 3, 4…).
It allows you to express unknown numbers and perform mathematical operations with them. Example, the sum of two numbers is expressed like this: a + b. Supports the expression of general numerical properties and relationships.
Example: the commutative property is expressed like this: axb = bx a. When writing using this language, unknown quantities can be manipulated with simple symbols to write, allowing the simplification of theorems, formulation of equations and inequalities and the study of how to solve them.
Algebraic signs and symbols
In algebra, both symbols and signs are used in set theory and these constitute or represent equations, series, matrices, etc. The letters are expressed or named as variables, since the same letter is used in other problems and its value finds different variables. Among some of the classification algebraic expressions are the following:
Algebraic fractions
An algebraic fraction is known as one that is represented by the quotient of two polynomials that show a behavior similar to numerical fractions. In mathematics, you can operate with these fractions by doing multiplication and division. Therefore, it must be expressed that the algebraic fraction is represented by the quotient of two algebraic expressions where the numerator is the dividend and the denominator the divisor.
Among the properties of algebraic fractions, it can be highlighted that if the denominator is divided or multiplied by the same non-zero quantity, the fraction will not be altered. Simplifying an algebraic fraction consists of transforming it into a fraction that can no longer be reduced, being necessary to factor the polynomials that make up the numerator and denominator.
Classification algebraic expressions are reflected in the following types: equivalent, simple, correct, improper, composed of numerator or null denominator. Then we will see each of them.
Equivalents
You are facing this aspect when the cross product is the same, that is, when the result of the fractions is the same. For example, of these two algebraic fractions: 2/5 and 4/10 will be equivalent if 2 * 10 = 5 * 4.
Simple
They are those in which the numerator and denominator represent integer rational expressions.
Own
They are simple fractions in which the numerator is less than the denominator.
Improper
They are simple fractions in which the numerator is equal to or greater than the denominator.
Composite
They are formed by one or more fractions that can be located in the numerator, the denominator or both.
Null numerator or denominator
Occurs when the value is 0. In the case of having a 0/0 fraction, it will be indeterminate. When using algebraic fractions to perform mathematical operations, some characteristics of operations with numerical fractions must be taken into account, for example, to start the least common multiple must be found when the denominators are of different digits.
In both division and multiplication, operations are carried out and carried out the same as with numerical fractions, since these must be previously simplified whenever possible.
Monomials
Monomials are widely used algebraic expressions that have a constant called the coefficient and a literal part, which is represented by letters and can be raised to different powers. For example, the monomial 2x² has 2 as its coefficient and x² is the literal part.
On several occasions, the literal part can be made up of a multiplication of unknowns, for example in the case of 2xy. Each of these letters is called indeterminate or variable. A monomial is a type of polynomial with a single term, in addition, there is the possibility of being in front of similar monomials.
Elements of monomials
Given the monomial 5x ^ 3; The following elements are distinguished:
- Coefficient: 5
- Literal part: x ^ 3
The product of monomials is the coefficient, which refers to the number that appears by multiplying the literal part. It is usually placed at the beginning. If the product of monomials has a value of 1, it is not written, and it can never be zero, since the entire expression would have a value of zero. If there is one thing to know about monomial exercises, it is that:
- If a monomial lacks a coefficient, it is equal to one.
- If any term has no exponent, it is equal to one.
- If any literal part is not present, but is required, it is considered with an exponent of zero.
- If none of this concurs, then you are not dealing with monomial exercises, you could even say that the same rule exists with the exercises between polynomials and monomials.
Addition and subtraction of monomials
To be able to carry out sums between two linear monomials, it is necessary to keep the linear part and add the coefficients. In the subtractions of two linear monomials, the linear part must be kept, as in the sums, to be able to subtract the coefficients, then the coefficients are multiplied and the exponents are added with the same bases.
Multiplication of monomials
It is a monomial whose coefficient is the product or result of the coefficients, which have a literal part that has been obtained through the multiplication of powers that have exactly the same base.
Division of monomials
It is nothing more than another monomial whose coefficient is the quotient of the coefficients obtained that, in addition, have a literal part obtained from the divisions between the powers that have exactly the same base.
Polynomials
When we talk about polynomials, we refer to an algebraic operation of addition, subtraction, and ordered multiplication made of variables, constants, and exponents. In algebra, a polynomial can have more than one variable (x, y, z), constants (integers or fractions), and exponents (which can only be positive integers).
Polynomials are made up of finite terms, each term is an expression that contains one or more of the three elements with which they are made: variables, constants or exponents. For example: 9, 9x, 9xy are all terms. Another way to identify the terms is that they are separated by addition and subtraction.
To solve, simplify, add or subtract polynomials, you have to join the terms with the same variables as, for example, the terms with x, the terms with “y” and the terms that do not have variables. Also, it is important to look at the sign before the term that will determine whether to add, subtract or multiply. Terms with the same variables are grouped, added, or subtracted.
Types of polynomials
The number of terms that a polynomial has will indicate what type of polynomial it is, for example, if there is a single-term polynomial, then it is facing a monomial. A clear example of this is one of the polynomials exercises (8xy). There is also the two-term polynomial, which is called a binomial and is identified by the following example: 8xy - 2y.
Finally, the polynomial of three terms, which are known as trinomials and are identified by one of the polynomial exercises of 8xy - 2y + 4. Trinomials are a type of algebraic expression formed by the sum or difference of three terms or monomials (similar monomials).
It is also important to talk about the degree of the polynomial, because if it is a single variable it is the largest exponent. The degree of a polynomial with more than one variable is determined by the term with the greatest exponent.
Addition and subtraction of polynomials
The sum of polynomials involves combining terms. Similar terms refer to monomials that have the same variable or variables raised to the same power.
There are different ways to perform polynomial calculations, including the sum of polynomials, which can be done in two different ways: horizontally and vertically.
- Sum of polynomials horizontally: it is used to perform operations horizontally, redundancy is worth it, but first a polynomial is written and then it is followed on the same line. After that, the other polynomial that is going to be added or subtracted is written and finally, the similar terms are grouped.
- Vertical sum of polynomials: it is achieved by writing the first polynomial in an ordered way. If this is incomplete, it is important to leave the gaps of the missing terms free. Then, the next polynomial is written just below the previous one, in this way, the term similar to the one above will be below. Finally each column is added.
It is important to add that to add two polynomials, the coefficients of the terms of the same degree must be added. The result of adding two terms of the same degree is another term of the same degree. If any term is missing from any of the degrees, it can be completed with 0. And they are generally ordered from highest to lowest degree.
As mentioned above, to perform the sum of two polynomials, it is only necessary to add the terms of the same degree. The properties of this operation are made up of:
- Associative properties: in which the sum of two polynomials is solved by adding the coefficients that accompany the x's that rise to the same power.
- Commutative property: which alters the order of the addition and the result cannot be deduced. The neutral elements, which have all their coefficients equal to 0. When a polynomial is added to the neutral element, the result is equal to the first.
- Opposite property: formed by the polynomial that has all the inverse coefficients of the aggregate polynomial coefficients. thus, when performing the addition operation, the result is the null polynomial.
With regard to the subtraction of polynomials, (operations with polynomials) it is imperative to group monomials according to the characteristics they possess and begin with the simplification of those that were similar. The operations with polynomials are carried out by adding the opposite of the subtrahend to the minuend.
Another efficient way to proceed with subtracting polynomials is to write the opposite of each polynomial below the other. Thus, similar monomials remain in columns and we proceed to add them. It does not matter which technique is carried out, in the end, the result will always be the same, of course, if it is done correctly.
Multiplication of polynomials
Multiplication of monomials or exercises between polynomials and monomials, is an operation that is carried out to find the resulting product, between a monomial (algebraic expression based on the multiplication of a number and a letter raised to an integer and positive exponent) and another expression, if this is an independent term, another monomial, or even a polynomial (finite sum of monomials and independent terms).
However, as with almost all mathematical operations, the multiplication of polynomials also has a series of steps that must be followed when solving the proposed operation, which can be summarized in the following procedures:
The first thing to do is multiply the monomial by its expression (multiply the signs of each of its terms). After this, the coefficient values are multiplied and when the value is found in that operation, the literal of the monomials found in the terms is added. Then each result is written down in alphabetical order and, finally, each exponent is added, which are located in the base literals.
Polynomial Division
Also known as the Ruffini method. It allows us to divide a polynomial by a binomial and also allows us to locate the roots of a polynomial to factor it into binomials. In other words, this technique makes it possible to divide or decompose an algebraic polynomial of degree n, into an algebraic binomial, and then into another algebraic polynomial of degree n-1. And for this to be possible, it is necessary to know or know at least one of the roots of the unique polynomial, in order for the separation to be exact.
It is an efficient technique to divide a polynomial by a binomial of the form x - r. Ruffini's rule is a special case of synthetic division when the divisor is a linear factor. Ruffini's method was described by the Italian mathematician, professor and physician Paolo Ruffini in 1804, who in addition to inventing the famous method called Ruffini's rule, which helps to find the coefficients of the result of the fragmentation of a polynomial by the binomial; He also discovered and formulated this technique on the approximate calculation of the roots of equations.
As always, when it comes to an algebraic operation, Ruffini's Rule involves a series of steps that must be fulfilled to arrive at the desired result, in this case: find the quotient and remainder inherent in the division of any type of polynomial and a binomial of form x + r.
First of all, when starting the operation, the expressions must be reviewed to verify or determine if they are really treated as polynomials and binomials that respond to the expected form by the Ruffini Rule method.
Once these steps are verified, the polynomial is ordered (in descending order). Once this step is finished, only the coefficients of the polynomial terms (up to the independent one) are taken into account, placing them in a row from left to right. Some spaces are left for the terms that are needed (only in case of an incomplete polynomial). The galley sign is placed on the left of the row, which is made up of coefficients of the dividend polynomial.
In the left part of the gallery, we proceed to place the independent term of the binomial, which, now, is a divisor and its sign is inverse. The independent is multiplied by the first coefficient of the polynomial, thus registering in a second row below the first. Then the second coefficient and the product of the monomial independent term are subtracted by the first coefficient.
The independent term of the binomial is multiplied by the result of the previous subtraction. But also, it is placed in the second row, which corresponds to the fourth coefficient. The operation is repeated until all terms are reached. The third row that has been obtained based on these multiplications is taken as a quotient, with the exception of its last term, which will be considered as the remainder of the division.
The result is expressed, accompanying each coefficient of the variable and the degree that corresponds to it, beginning to express them with a lower degree than the one they originally had.
- The remainder theorem: it is a practical method used to divide a polynomial P (x) by another whose form is xa; in which only the value of the remainder is obtained. To apply this rule, the following steps are followed. The polynomial dividend is written without completing or ordering, then the variable x of the dividend is replaced with the opposite value of the independent term of the divisor. And finally, the operations are solved in combination.
The remainder theorem is a method by which we can obtain the remainder of an algebraic division but in which it is not necessary to do any division.
- Ruffini's method: Ruffini's method or rule is a method that allows us to divide a polynomial by a binomial and also allows us to locate the roots of a polynomial to factor in binomials. In other words, this technique makes it possible to divide or decompose an algebraic polynomial of degree n, into an algebraic binomial, and then into another algebraic polynomial of degree n-1. And for this to be possible, it is necessary to know or know at least one of the roots of the unique polynomial, in order for the separation to be exact.
- Roots of Polynomials: The roots of a polynomial are certain numbers that make a polynomial worth zero. We can also say that the complete roots of a polynomial of integer coefficients will be divisors of the independent term. When we solve a polynomial equal to zero, we obtain the roots of the polynomial as solutions. As properties of the roots and factors of polynomials we can say that the zeros or roots of a polynomial are by the divisors of the independent term that belongs to the polynomial.
This allows us to find out the remainder of the division of a polynomial p (x) by another of the form xa, for example. From this theorem it follows that a polynomial p (x) is divisible by xa only if a is a root of the polynomial, only if and only if p (a) = 0. If C (x) is the quotient and R (x) is the remainder of the division of any polynomial p (x) by a binomial that would be (xa) the numerical value of p (x), for x = a, it is equal to the remainder of its division by xa.
Then we will say that: nP (a) = C (a) • (a - a) + R (a) = R (a). In general, to obtain the remainder of a division by Xa, it is more convenient to apply Ruffini's rule than to replace x. Therefore, the remainder theorem is the most suitable method for solving problems.
In the mathematical world, Ruffini's rule is an efficient technique for dividing a polynomial by a binomial of the form x - r. Ruffini's rule is a special case of synthetic division when the divisor is a linear factor.
Ruffini's method was described by the Italian mathematician, professor and physician Paolo Ruffini in 1804, who, in addition to inventing the famous method called Ruffini's rule, which helps to find the coefficients of the result of the fragmentation of a polynomial by the binomial; He also discovered and formulated this technique on the approximate calculation of the roots of equations.
Then, for each root, for example, of the type x = a corresponds to a binomial of the type (xa). It is possible to express a polynomial in factors if we express it as a product or of all the binomials of the type (xa) that correspond to the roots, x = a, that result. It should be taken into account that the sum of the exponents of the binomials is equal to the degree of the polynomial, it should also be taken into account that any polynomial that does not have an independent term will admit as root x = 0, in another way, it will admit as a X Factor.
We will call a polynomial "prime" or "Irreducible" when there is no possibility of factoring it.
To delve into the subject we must be clear about the fundamental theorem of algebra, which states that it is enough that a polynomial in a non-constant variable and complex coefficients has as many roots as its degree, since the roots have their multiplicities. This confirms that any algebraic equation of degree n has n complex solutions. A polynomial of degree n has a maximum of n real roots.
Examples and exercises
In this section we will place some algebraic expressions solved exercises of each of the topics covered in this post.
Algebraic expressions exercises:
- X ^ 2 - 9 / 2X + 6
(X + 3) * (X - 3) / 2 * (X + 3)
X - 3/2
- X ^ 2 + 2X + 1 / X ^ 2 - 1
(X + 1) ^ 2 / (X + 1) * (X - 1)
X + 1 / X - 1
Sum of polynomials
- 2x + 3x + 5x = (2 + 3 + 5) x = 10 x
- P (x) = 2 × 2 + 5x-6
Q (x) = 3 × 2-6x + 3
P (x) + Q (x) = (2 × 2 + 5x-6) + (3 × 2-6x +3) = (2 × 2 + 3 × 2) + (5x-6x) + (-6 + 3) = 5 × 2-x-3
Subtraction of polynomials
P (x) = 2 × 2 + 5x-6
Q (x) = 3 × 2-6x + 3
P (x) -Q (x) = (2 × 2 + 5x-6) - (3 × 2-6x +3) = (2 × 2 + 5x-6) + (-3 × 2 + 6x-3) = (2 × 2-3 × 2) + (5x + 6x) + (-6-3) = -x2 + 11x-9
Polynomial Division
- 8 a / 2 a = (8/2). (A / a) = 4
- 15 ay / 3a = (15/3) (ay) / a = 5 and
- 12 bxy / -2 bxy = (12 / -2) (bxy) / (bxy.) = -6
- -6 v2.c. x / -3vc = (-6 / -3) (v2.c. x) / (v. c) = 2 v
Algebraic expressions (binomial squared)
(x + 3) 2 = x 2 + 2 • x • 3 + 32 = x 2 + 6 x + 9
(2x - 3) 2 = (2x) 2 - 2 • 2x • 3 + 32 = 4 × 2 - 12 x + 9
Remainder theorem
(x4 - 3 × 2 + 2):(x - 3)
R = P (3) = 34 - 3 • 32 + 2 = 81 - 27 + 2 = 56
Multiplication of monomials
axn bxm = (a b) xn + m
(5x²y³z) (2y²z²) = (2 · 5) x²y3 + 2z1 + 2 = 10x²y5z³
4x · (3x²y) = 12x³y
Division of monomials
8 a / 2 a = (8/2). (A / a) = 4
15 ay / 3a = (15/3) (ay) / a = 5 and
12 bxy / -2 bxy = (12 / -2) (bxy) / (bxy.) = -6
-6 v2. c. x / -3vc = (-6 / -3) (v2.c. x) / (v. c) = 2 v
Addition and subtraction of monomials
Exercise: 3 × 3 - 4x + 5 - 2 + 2 × 3 + 2 × 2
Solution: 3 × 3 - 4x + 5 - 2 + 2 × 3 + 2 × 2 = 3 × 3 + 2 × 3 + 2 × 2 - 4x + 5 -2 = 5 × 3 + 2 × 2 - 4x + 3
