Theory

Letters like numbers

Algebra is a branch of mathematics in which symbols, usually letters of the alphabet, represent numbers or members of a specified set and are used to represent quantities so that we can use letters for the arithmetical operations such as +, −, ×, ÷ and the power.

What do you do when you want to refer to a number that you do not know? Suppose you wanted to refer to the number of buildings in your town, but haven't counted them yet. You could say 'blank' number of buildings, or perhaps '?' number of buildings.

In mathematics, letters are often used to represent numbers that we do not know - so you could say 'x' number of buildings, or 'q' number of buildings. We call them variables.

Look at these examples:

      • The triple of a number: 3n
      • The triple of a number minus five units: 3n – 5
      • The following number to x: x + 1
      • The preceding number to y: y – 1
      • An even number: 2a
      • An non-even/odd number: 2z + 1 or 2z – 1

Polynomials

A polynomial is the addition or subtraction of two or more monomials.

      • If there are two monomials, it is called a binomial, for example x2 + x
      • If there are three monomials, it is called a trinomial, for example 2x2 - 3x + 1

The following are NOT polynomials:

The degree of the entire polynomial is the degree of the highest-degree term that it contains, so x2 + 4x - x is a second-degree trinomial, and x4 - 7x2 is a fourth-degree binomial.

The polynomial that follows is a second-degree polynomial, and there are three terms: 4x2 is the leading term, and (-7) is the constant term.

To evaluate a polynomial we only have to replace the value we have in the variable. For example, in x4 - 7x2 when x=-2 we will have (-2)4 - 7(-2)2 = 16 - 7·4 = 16 - 28 = -12

Adding / Subtracting polynomial

When adding polynomials you must add each like term of the polynomials, that is, monomials that have the same literal part, you use what you know about the addition of monomials. There are two ways of doing it. The format you use, horizontal or vertical, is a matter of preference (unless the instructions explicitly tell you otherwise). Given a choice, you should use whichever format you're more comfortable with.

Note that, for simple additions, horizontal addition (so you don't have to rewrite the problem) is probably the simplest, but, once the polynomials get complicated, vertical addition is probably the safest (so you don't "drop", or lose, terms and minus signs).

For example:

Take out parenthesis if it's necessary

Identify each of the monomial with it's same family ones:

Reorder the monomials:

Operate:

It can also be done vertically:

When we are going to subtract two or more polynomials we have to take out the parenthesis first. This negative symbol in front of the parenthesis will change the sign of all the monomial in the interior of the parenthesis.

For example:

We change the sing of each monomial in the interior of the second parenthesis:

It can also be done vertically:

Multiplying polynomial

If a monomial is multiplying a polynomial it will multiply term by term the monomials of the polynomial:

For example:

If we are multiplying two polynomials we will multiply each term of one polynomial with each term of the other polynomial, so we must be very careful not to forget any term.

For example:

Extracting the common factor

The common factor is something that a polynomial has repeated in each of its terms. In this case we can take it out one and write some braquets/parenthesis:

For example:

3 is repeated in each term of the polynomial

3x is repeated in each term of the polynomial

y is repeated in each term of the polynomial

First-degree equations

To solve an equation of first-degree we must follow some steps:

      1. Common denominator
      2. Take out all the brakets
      3. Erase the denominator
      4. Pass all the monomial of first-degree to one side and the ones without the variable to the other side (remember that when we do this the sign of the monomial will change)
      5. Isolate the variable

For example:

We want to solve this equation:

First we take out the parenthesis:

The LCM(6, 2, 3, 4) = 12 so we write the same denominator:

What is equal to this and we can take out the 12s:

What is equal to this and we can take out the 12s:

Now we solve the equation:

Second-degree equations

In a general equation of second degree each term of the variable is called by the letters a (the one for the second degree), b (the one for the first degree) and c (free one). There is a general formula that gives the two solutions (a equation of degree n will have n solutions) of this type of equations: