MATH

INTRODUCTION

MONOMIAL OPERATIONS

POLYNOMIA

FIRST-DEGREE EQUATIONS WITH AN INCOGNITE

EXPRESIONES ALGEBRAICAS

COMPLETE, ORDERED POLYNOMIA,
Homogeneous

ALGEBRA

DEFINITION OF MATH

The essence of mathematics is in the relationship between
quantities and qualities.

is the name we give to the collection of all
patterns and possible interrelationships.

Mathematics is not only about quantity, but also about the qualities and the relationships between the two. As stated María Moliner, mathematics is the “science that deals with relationships between quantities and magnitudes and operations that allow you to find something you want, knowing others ”.

Numerical analysis investigates the methods for performing the computer calculations. It is said that mathematics they cover three areas

Geometry, including Trigonometry and Conic Sections

Mathematical analysis, which uses letters and
symbols, and that includes algebra, geometry and calculus

Arithmetic

Algebra, on the other hand, can give a generalization
which always hold (a + b = c).

Classical algebra, which deals with solving equations, uses symbols (instead of specific numbers) and arithmetic operations to determine how to use these symbols.

Arithmetic only gives particular cases of relations (2 + 2
= 4).

Modern algebra has evolved from algebra classical by paying more attention to the structures math

is the branch of mathematics in which letters are used to
represent arithmetic relationships

For example, the expression 8a3 b2 c is an algebraic expression, in this case a monomial, which has the number 8 as the numerical part and the literal part a3 b2 c. Note that the exponents are considered literal part

Going a little deeper into what is mentioned above, there are basically two types of algebraic expressions, and they are

Algebraic expressions are all those that have a
numerical part and literal part.

Polynomials

Monomials

It is a single algebraic expression

They are two or more algebraic expressions (with different literal part) that are being added or subtracted

Polinomios Ordenados

Homogeneous Polynomials

Complete Polynomials

is complete with respect to a letter when it contains all consecutive exponents of a letter, from highest to lowest

6x3 -5x + 3x5 + x2 -x4 +5 as we see the exponents between 5 and 0 the numbers are consecutive we affirm that it is a complete polynomial

In the previous example we have seen the exponents of the polynomial are all consecutive numbers between 0 and 5, but they are in complete disorder

5a2 + 3a3 -a5 + a8 we see that the exponents go up when we see this it is an ordered polynomial

Remember that a polynomial is made up of two or more terms that are being added or subtracted.

3a2 b + 5ab2 -3abc

ADDITION AND SUBTRACTION OF MONOMIES

MONOMIAL MULTIPLICATION AND DIVISION

SIMILAR TERMS

MONOMIAL POTENTIATION AND RADICATION

Before going on to evaluate the different operations with monomials, it is worth looking at this concept, that of like terms

Let's look at the following pair of algebraic expressions: a) 4x2 y3 b) 2x2 y3

When the literal part in two monomials is equal, then we are talking about like terms

We see that in both expressions the literal part is repeated, in both monomials there are x 2, likewise, in both monomials there is y3

Monomial Radiation

Monomial Empowerment

First we will work the numerical part as we have always done, that is, applying the definition of power. Then we will work with the literal part, in which we will multiply the exponent of each letter by the exponent of the given power.

(3x2 y)4

First we will do the numerical part: 34 = 3 x 3 x 3 x 3 = 81

Finally, the answer will be: 81x8 y4

We are asked to raise the monomial 3x2 and to the 4th power

As in empowerment, in the case of filing we must work separately on the numerical part and the literal part. We will obtain the corresponding root for the numerical part; and in the numerical part we will divide the exponent of each letter by the degree of the radic

In the example, √ (〖16x〗 ^ 4 y ^ 6), we are asked to obtain the square root of the monomial 16x4 y6

Now we divide the exponents by 2 (the index of the radical): x4 ÷ 2 y 6 ÷ 2 = x2 y3

Finally, the answer will be: 4x2y3

We will start with the numerical part: √16 = 4

Monomial Division

Multiplication of Monomials

To divide polynomials it is not necessary that they be like terms

81a2 b3 c4 d5 between monomial 3b2 c2

81a2 b3 c4 d5 ÷ 3b2 c2

So the answer will be: 27a2 bc2 d5

for letter b we subtract 3 - 2 = 1
for letter c we subtract 4 - 2 = 2

First we will divide the numerical part as traditionally we do, that is: 81 ÷ 3 = 27

To multiply monomials it will not be necessary for them to be like terms. We can multiply any monomial between them.

a) 5x2 y 5 b) 2x3 y2 z

(5x2 y5) (2x3 y2 z) now we add the exponents of the letter y,
5 + 2 = 7

(5x2 y3) (2x3 y2 z) the letter z is not repeated so only the
I will place as is

(5x2 y5) (2x3 y2 z) first for the letter x, we add the exponents 2 + 3 = 5

The answer is 10x5 y7 z

In order to add or subtract monomials these must be
like terms

5x4 y3 -x4 y3

3m2 n + 6m2 n

now we will only subtract the numerical part, that is, 5 - 1, which means that two 4 5x4 y3 - 1x4 y3 = 4x4 y3

the monomial answer will be 4x4 y3

the monomial answer will be 9m2 n

but we will only add the numerical part, that is, 3 + 6, which gives us 9 3m2 n + 6m2 n = 9m2 n

POLYNOMIA MULTIPLICATION

PRODUCTOS NOTABLE

SUM OF POLYNOMIA

Notable Products (The Cube of a Binomial

The cube of a binomial is equal to the cube of the first term plus three times the square of the first term times the second term plus three times the first term times the square of the second term plus the cube of the second term.

(2a + 4b)3 = (2a)3 +3(2a)2(4b) +3(2a) (4b)2 + (4b)3

The triple of the first term times the square of the second term:
3 (2a) (4b1) 2 = 3 (2a) (16b2) = 96ab

The cube of the second term is: (4b1) 3 = 64b

The triple of the square of the first term by the second term:
3 (2a1) 2 (4b) = 3 (4a2) (4b) = 48a2

Finally, the answer will be: (2a + 4b) 3 = 8a3 + 48a2b +
96ab2 + 64b3

The cube of the first term is: (2a1) 3 = 8a3

There are some characteristic answers for some cases in which we must multiply, these are the remarkable products. They are types of multiplications whose results will follow certain identifiable patterns

Square of a Binomial

The Sum by the Difference of Two Quantities

The square of the sum of two terms is equal to the square of the first term plus the double product of both terms plus the square of the second term.

(5x+7)2 = (5x)2 + 2(5x) (7) + (7)

The double product of both terms is: 2 (5x) (7) = 70x

The square of the second term is: (7) 2 = 49

The square of the first term is: (5x1) 2 = 25x2

Finally, the answer will be: (5x +7) 2 = 25x2 + 70x + 4

The sum of two terms multiplied by their difference is equal
to the difference of the squares of both terms

(4a +7y3) (4a -7y3) = (4a)2 - (7y3)2 = 16a2- 49y6

The square of the second term is: (7y3) 2 = 49y

Finally, the answer will be: (4a + 7y3) (4a -7y3) = 16a2- 49y6

The square of the first term is: (4a1) 2 = 16a2

In the multiplication of polynomials we will have to multiply all the terms between them.

(5x2 y +3xy2) (3x3 -2x2y +xy2 -4y3

We do the same with the second term of the first polynomial (3xy2) (3x3) = 9x4 y2; (3xy2) (- 2x2y) = -6x3 y3; (3xy2) (xy2) = 3x2 y4; (3xy2) (- 4y3) = - 12xy5

Now we put together all the results of both multiplications 15x5y -10x4 y2 + 5x3 y3 -20x2 y4 + 9x4 y2 -6x3 y3 + 3x2 y4 + 12xy5

Now we multiply the first term of the first polynomial by each one of the terms of the second polynomial (5x2y) (3x3) = 15x5 y; (5x2y) (-2x2y) = -10x4y2; (5x2y) (xy2) = 5x3y3; (5x2y) (-4y3) = -20x2y4

We look for like terms and add or subtract them: 15x5 y -1x4 y2 -1x3 y3 -17x2 y4 -12xy

In a polynomial we can add or subtract only the like terms, everything else will be exactly the same

To perform a subtraction, the procedure is similar, but we must be very careful with the signs. Let's say that now we are going to subtract: 5x2 y + 3xy2 - (3x3 -2x2 y + xy2 -4y3

Let's add the polynomial 5x2 y + 3xy2 and the polynomial 3x3 -2x2y + xy2 -4y3

The parenthesis of the second polynomial is eliminated, changing the sign of its terms: 5x2y + 3xy2 -3x3 + 2x2 and -xy2 + 4y

Non-like terms will be included in the final answer as is

Now we just look for the like terms and carry out the corresponding operations: 5x2y + 2x2y + 3xy2 -xy2 -3x3

The answer is 7x2y + 2xy2 -3x3 + 4y

We introduce the partial results in our polynomial response: 3x2y + 4xy2 + 3x3 -4y3

We operate the terms with xy2: 3xy2 + 1xy2 = 4xy

We operate the terms with x2y: 5x2 y -2x2y = 3x2y

Now we must see if there are like terms and we add them between them

Then the sum will be: 5x2 y + 3xy2 + 3x3 -2x2 y + xy2 - 4y3

PROBLEM RESOLUTION

SOLVING EQUATIONS WITH GROUPING SIGNS

RESOLUTION OF EQUATIONS WITH INDICATED PRODUCTS

RESOLUTION OF EQUACIONE

To solve a problem, we must pose it mathematically and then perform the corresponding operations to find the value of the unknown (the data we want to know)

To solve this type of equations, first the indicated products are carried out and then the general procedure is followed (applying the criterion of inverse operations)

5(x -3) -(x -1) = (x +3) -1

5x -x -x = 3 -10 +15 We take the like terms to one side of the equality, and the independent terms to the other side (we use inverse operations.)

3x = 9 We reduce like terms on both sides of the equality

5x -15 -x -1 = x +3 -10 We solve the indicated product, and additionally we eliminate the parentheses

x = 9/3 = 3 We solve for x passing 3 to divide, then we simplify

To solve this type of equations, we must first suppress the grouping signs considering the law of signs, and if there are several groupings, we develop the operations from the inside out.

2x -[x -(x -50)] = x - (800 -3x

2x -50 = 4x -800 Ahora quitamos los corchetes.

2x -4x = -800 +50 We transpose the terms, using the criterion of inverse operations.

2x - [50] = 4x -800 We reduce like terms

-2x = -750 Again we reduce like terms

2x - [x -x +50] = x -800 + 3x First we remove the parentheses

x = -750 / -2 = 375 We solve for x going to divide to -2, then we simplify

An equation is an equality between two quantities or expressions

Solve the equation 2x - 3 = 53

Now we add and subtract the like and independent terms, leaving the equation like this: 2x = 5

Now we have the number 2 that is multiplying the variable or unknown x. So we transpose it (to number 2) to the other side of the equality. If the “2” was multiplying, it will happen by dividing (the inverse operation). The equation would look like this: X = 56/2

We must have the letters on one side and the numbers on the other side of the equality (=). To do this, we transpose the terms to the corresponding member (side). When transposing a term, it happens with the changed sign. For example: when transposing the -3 to the other side of the equality, it will be +3. We will then have 2x = 53 +3

We perform the division and we will obtain the following: x = 28. This is the answer and it tells us that the value of x is 28. The answer is also an equation since it tells us that the quantity "x" is equal to the quantity "28"