Algebra, Factorization, Expansion | O Level Mathematics 4024 & IGCSE Mathematics 0580 | Detailed Free Notes To Score An A Star (A*)

Basic Algebraic Concept

• In simple terms, Algebra means using variables and constant in different equations.
• A small letter in the equation shows a variable or a constant – usually a variable.
  • Variable is something which can take different values.

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Simple Algebraic Equations

• When considering simple algebraic equations, we can suggest something like:
• a= 2+x
• Here, a is a dependent variable – i.e. it’s value depends on the rest of the equation.
• x is the independent variable – we change the value of x which in turn changes the value of a.
• For example, if x is 2, then a is 2+2 = 4
• If there is no sign between two things, it is assumed they are being multiplied. For example ab means a multiplied by b.

Subject of the Formula

• One common algebraic objective is to make something the subject of the formula.
• In other words, it means isolating one variable on one side of the equation.
• We use basic DMAS rules for this purpose. Some common mistakes are as follows:
  • Do not multiply a divisor from one side to another unless it divides the complete equation on one side of the equation. For example

• • Here, we can not take b or c on the other side to multiply, because they are not dividing the complete equation on one side. First, we need to take LCM and make it as such that the entire equation on one side is divided by a common divisor.

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• • Here, bc is now a common divisor, so we can take it to the other side and multiply it with d.

 

• • Same rule goes for multiplication. Here, c is only multiplied with a and not the entire equation. So we can not take it on the other side and divide. Only something that is multiplied with the complete equation can be taken on the other side. For example.

 

• • Here, x is completely multiplied with one side of the equation. Therefore, we can take it on the other side to divide.

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Factorization and Expansion

• Some common expansions we must know about are as follows:
  • 
  • This equation is the same as (a+b)(a+b).
  • If we expand it, it goes as follows:
  • a*a +a*b + a*b + b*b
  • This can be further simplified as
  • Remember, ab and ba is the same thing – if the elements involved are the same, the setting does not matter.
• Another common factorization/ expansion is as follows
  • 
  • It is expanded as (a-b)(a-b)
  • It is expanded as a*a -a*b + a-b – b*-b
  • 
• The most common one in the exams will be as follows – make sure you don’t confuse this one with the upper one.
  • 
  • It is defined as (a+b)(a-b)
  • It is further expanded as
  • Therefore it comes back to
  • 

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• In your exam, this last one will be tested as follows:
  • What is the factorization of 1-121.
  • First, we can determine that both 1 and 121 are perfect squares. 1 is the perfect square of 1, and 121 is the perfect square of 121.
  • Therefore, we can simplify it as follows (1-11)(1+11)
  • A more difficult example of this can be seen as follows
    • 
    • Here, both are perfect squares
    • 1 is the square of 1 and
    • is the perfect square of
    • So we can factorize it as follows
    • 

Other Equations

• For any other equations, the rules of indices and DMAS will be applied for any expansion or factorization
• For expansion, for example, if it says:
  • 2(a+c)- 4a(b+c)
  • Simply use DMAS.
  • 2a + 2c – 4ac -4ac
• If there is any addition subtraction possible of similar ones, do it. Otherwise, leave the answer as it is if those are not possible.
• For factorization, in general (if not a quadratic equation), just take out the common items.
  • The steps are as follows
    • First check what elements are common in each individual item of one side of the equation.
    • Then, take it’s lowest present power, and divide the equation with it.

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• • • For example
    • 
    • Here, a is common in all, and the lowest power of a is 1. Secondly, 2 can divide all the numerical values perfectly. So, we can do the following
    • We can take out 2a, and divide the entire equation with it one by one.
    • It leaves us with the following
    • Remember, rules of indices apply when you are dividing or multiplying variables.