**Algebra** is a very broad term that covers many different operations and that can be applied in various different places in everyday life.

### Algebra lessons

Proof by contradiction (also known as indirect proof or the method of reductio ad absurdum) is a

Absolute value equalities and inequalities are equalities and inequalities in which the unknown or

For the given vector space $V$, we want more study the space $L (V, \mathbb{F})$, where $\mathbb{F}$

In this lesson we will examine in detail the procedure of joining the matrices to the vectors and

Definition. Let $V$ and $W$ be two vector spaces over the same field $\mathbb{F}$. A transformation

Let $$p(x) = ax^2 + bx + c, \quad a \neq 0$$ be the second degree polynomial. We can factorize it

Absolute value of a number represents its distance from zero on a number line. Since it represents a

Definition. Let $V$ be a vector space over the field $\mathbb{F}$. Any vector $v$ of the form $$v =

Euler’s phi function For arbitrarily chosen natural number $m$, we observe the following

A non- linear Diophantine equation is every Diophantine equation which is not linear. For instance,

Let $f$ be a polynomial with integer coefficients in one or more variables. An algebraic equation of

For every two nonnegative real numbers $a$ and $b$ the following inequality holds: $$ \frac{a+b}{2}

Let $a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}=0$, $a_{3}\neq 0$ be the cubic equation. By dividing the

Example 1. For a matrix $$\mathbf{A} = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$$ find

Linear system of equations Regarding the introduction of matrices, we can find their connection with

Determinant of a matrix The determinant is a real function such that each square matrix

Regular matrix A square matrix $\mathbf{A}$ of order $n$ is a regular (invertible) matrix if exists

Addition of matrices To define the addition of two matrices $\mathbf{A}=[a_{ij}]$ and

Let $n$ and $m$ be natural numbers. Every rectangular array $\mathbf{A} = \begin{bmatrix} a_{11}

Operations are divided into three degrees, exponents and roots, multiplication and division, addition ...

Function is injection if the equality of function values implies the equality of arguments.

The binomial theorem, also known as binomial expansion, explains the expansion of powers.

Here we’ll explain the method for rough sketching of polynomials.

Number of positive real roots of a polynomial is bounded by the number of changes of sign in its

The Irrational Root Theorem says if a + √b is also a root of observed polynomial.

The Rational Root Theorem says if a polynomial equation a_n x^n + ... + a0 = 0

If a polynomial equation a_n x^n + + ... + a_0 = 0 has an integer solution, a is not equal to 0

Polynomial equations are special type of equations that come in the one different form...

Factoring is an action where polynomial is represented as a product of simpler polynomials that no

Fundamental theorem of algebra says every polynomial with degree n ≥ 1 has exactly n zeros...

Radical equations are equations in which the unknown appears under a radical sign.

The problem of solving quadratic inequalities is very much connected to solving zeros.

One of the most important basics in theory of number is the definition of divisibility with rules.

Solve both of your inequalities individually and search for the intersection of their common solution

Vectors are directed line segments in which we can make difference between initial and terminal point.

Radicals is created with an opposite action from exponentiation. They are marked with specific symbol.

Polynomial is an expression consisting of variables and coefficients. It's made out of one or more terms.

Exponent is a number which tells us how many times a number has been multiplied by itself.

A linear function has a correspondence between two sets - one element is assigned to element.

They are similar to any other type of inequalities. The only difference is their absolute value.

Compound inequalities are a little bit more complicated... They are bounded by more than one condition.

The most difficult type of inequalities to solve. It includes operations with integers, decimals and

Inequalities that can be solved in maximum two steps are known as two-step inequalities.

The simplest form of inequalities are known as one-step inequalities. It can be solved in one step.

Inequalities represent relations between two objects. There are one-step, two-step and multi-step.

Quadratic equation are type of equations that have exactly two solutions (ax2 + bx + c = 0, a ≠ 0)

Find the unknown numbers in two or more linear equations using graphing, elimination or substitution.

More complex equations. Equations that needs to be solved in more than two steps.

Two-step equations are simple equations in which solution can be found in just two steps.

The simplest equations that require only one basic operation to find the solution.

Just like in everything else in your life, with algebra you have to start from the very beginning. It is implied that you already know the basic arithmetic. As you will notice, algebra will divide in two different parts. One part is where you can visualize the problems and the other part is algebra that is needed for even more complicated problems. You should know that everything here is somehow bounded which means that problems can be interlaced from different parts of algebra. This may seem a little bit complicated, but we’ll let you in on a secret. Try to be patient and do not rush. First look at the problem, think of a way you can solve it. The first solution that comes into your mind may be the best, but chances are that there are many different approaches that may be simpler, easier and quicker. In no time, you will learn how to use that kind of thinking in many different situations, even ones that do not include math. So before saying that you will never need something of this in your life, consider first how many logical skills you will be getting and you will never think of that again. Every part of algebra will let your brain expand beyond its thinking range. Your concluding will become quicker and you will feel more confident.

Also, algebra is a natural science. You can find its uses everywhere. From basic math operations to equations and inequalities, we will provide interesting and motivating real life problems. Every lecture we have is broken down into different levels which means that you can find everything you need.

## Starting with functions

Here you can also learn basic functions. You can learn what they really are and how we define them, how do we plot them and what do we use them for. If you read this lesson carefully and go through the examples, future function encounters will become much easier to deal with. There are many different functions that are used for different things and they are widespread through every part of math which is the reason why you should pay attention and really learn the basics. Another thing that causes many headaches is linear algebra. Linear algebra is used for cryptography, coding theory, economics, graph theory, geometry, genetics, networks, even in games. To come to these important uses, you must start with learning all there is to know about vectors. Vectors have many properties and there are many operations you can perform on them, but none of them is that hard to remember.

## Heating things up- for the patient ones

Another large part of algebra are polynomials. Operations with polynomials require a lot of patience and focus which is why they aren’t favourite with students. For those who master basic operations with polynomials and are sure of themselves can come and learn more. You can learn about their solutions and why do we try so hard to find them. We will also show you the basics of the number theory. Number theory is one of the most fun parts of math, as you will discover for yourself. Number theory is the base of modern cryptography. If you have trouble with proving mathematical statements, this is the best place to begin. The proofs require less writing and more thinking and this will help you start thinking logically and many proofs will come easier.

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