User blog:Holomanga/Learn all of Maths I: Algebra and Functions

Welcome to the first of my series of blog posts, set up in such a way that after reading it, you ideally know all of maths. Clearly, that's an unfeasible goal, but it won't stop me from trying.

Prerequisites: Basic mathematical knowledge

Simplifying Expressions by Combining Like Terms
An expression is a mathematical statement. Most expressions you see will be the sum of a number of terms. A term is the product of a constant and a variable, in the form

$$kv$$

A constant is something that never changes, such as the number "2". No matter what you do to two, two will always equal two. A variable is something that can change, such as "x2". Depending on which value you choose for x, x2 can choose a different value.

An example expression is below:

$$4x + 2{x}^{2}{y}^{2} - 3sin(x) + 4xyz$$

Notice that this is the sum of four different terms:
 * $$4x$$
 * $$2{x}^{2}{y}^{2}$$
 * $$- 3sin(x)$$ (don't worry about what this means just yet)
 * $$4xyz$$

For the first, the constant is "4" and the variable is "x". For the second, the constant is 2 and the variable is "x2y2". For the third, the constant is -3 and the variable is "sin(x)". For the fourth, the constant is 4 and the variable is "xyz".

Expressions of this form (a sum of terms) can sometimes be simplified. If the variables of two terms are equal, then the two terms can be combined, as such:

$$kv + cv = (k+c)v$$

For example, 3x + 4x = 7x, and 7x2y3z - 5x2y3z = 2x2y3z. When all like terms have been combined in this way, the expression can be considered simplified.

Index Laws
Another type of expression comprises the product of multiple terms, containing indices (the results of the exponentiation function). These can also be simplified, using index laws. The main index laws are: For example, x3*x2 = x5.
 * $${v}^{0} = 1$$
 * $${v}^{n} \times {v}^{m} = {v}^{n+m}$$
 * $$\frac{{v}^{n}}{{v}^{m}} = {v}^{n-m}$$

When a term contains multiple variables or constants of the same type, they are multiplied together seperately. So, for example, x2y3 * xy2 = x3y5.

Expanding Expressions
Some expressions will be given in the form of two sums of terms, multiplied together using brackets. Two sums of two terms each will look something like this, where a, b, c and d are terms:

$$(a + b)(c + d)$$

In order to simplify this, it must first be transformed into a single expression. This can be done by a process called expanding. Each term in one set of brackets must be multiplied by every term in the other set of brackets, and then the totals added together. In the above example, this would be

$$ac + ad + bc + bd$$

Which, due to no terms sharing the same variable, can not be simplified further.

A geometrical interpretation of another expansion, that of (x+4)(x+2), is shown below:

https://www.learner.org/workshops/algebra/workshop5/images/chart4.gif

This is a rectangle with a side of length (x + 4) and a side of length (x + 2). The total area of the rectangle is therefore the product of the side lengths - that is, (x + 4)(x + 2) - as well as the sum of all the smaller rectangles - x*x + x*2 + 4*x + 4*2, which equals x2 + 6x + 8 when simplified using index laws and combining like terms.

When there are three or more brackets, multiply two of the brackets together, and then multiply that product by the third bracket, and so on.

Factorising Expressions
The opposite of expansion is factorisation. This is taking a single sum and transforming it into the product of two or more expressions. This section will cover factorising expressions into the form c(u + v + w...).

First, find the highest common factor of all terms in the expression. This is a variable or constant that every single term in the expression can be divided by without leaving any remainder. For example, y2 is a factor of 4xy3, because 4xy3 divided by y2 is 4xy.

Then, divide each term in this expression by the common factor, and take that factor out of the brackets. An example:

$$4{x}^{2}{y}^{3} + 12x{y}^{2} + 16{x}^{4}{y}^{2}$$

The highest common factor of all terms in the expressions is 4xy2 in this case, since it has the largest powers of x and y that can divide all terms in the expression and 4 is the largest number that divides all terms in the expression.

When all terms in the expression are divided by this, then it moved outside of the brackets, the following expression results.

$$4x{y}^{2}( xy + 4{x}^{3})$$

Factorising Quadratic Expressions
A quadratic expression, of the form

$$a{x}^{2} + b{x} + c$$

can also sometimes be factorised to give an equivalent expression in the form of the product of two linear expressions.

Fractional Index Laws
In the section of index laws, you learned the index laws for integer exponents. These can also apply for non-integer exponents; in this section, rational exponents will be covered.

For rational exponents, the same laws apply, with an additional rule linking indices to roots:
 * $${v}^{\frac{n}{m}} = \sqrt[m]{{v}^{n}}$$

This allows a greater range of expressions involving indices to be simplified.

Surds
There are some manipulations that can be done to work with square roots.
 * $$\sqrt{x}\sqrt{y} = \sqrt{xy}$$
 * $$\frac{\sqrt{x}}{\sqrt{y}} = \sqrt{\frac{x}{y}}$$ (this follows from the above rule)

From the first rule, there is a way of simplifying surds. The surd

$$\sqrt{{n}^{2}m}$$

Can be rewritten in the form

$$\sqrt{{n}^{2}}\sqrt{m}$$

using the inverse of the first rule, above. Since the square root of a square number is just that number, this can be simplified even further to

$$n\sqrt{m}$$

Rationalising the Denominator
If a rational expression has a surd in the denominator, then it can be simplified further by rationalising the denominator.