FACTORIZATION

Factorization

Factorization is the reverse process of expanding brackets. For instance, expanding brackets would require to be written as illustrated below. Factorization is also exemplified in the diagram below:.

The two expressions are same; they have equivalent value for all values of x.

Common factors

Factorizing based on common factors is dependent on the fact that there are factors common to all the terms.

For instance we make use of a switch around procedure to calculate the common factor.

Difference of two squares

To obtain a difference of two squares, look for expressions: o That is made up of two terms;
o with terms that possess varied signs (one positive, one negative);
o with each term a perfect square.

Factorization by grouping in pairs

The taking out of common factors is the starting point in all factorization problems. This is referred to as factorizing by grouping.

OR

Group terms with common factors together

x is a common factor of the first and third terms and is a common factor of the second and fourth terms

Rearrange the equation with grouped terms together

Take out the common factor

Write the final answer

Expansion is the practice of expanding or of removing all brackets existing in an expression. Normally, you’ll encounter a few expressions in your math exams where they place two sets of two terms, both in brackets, beside each other. This form of math question needs you to expand the expression in a way that the brackets will be removed during the process, and the whole question can be simplified to its simplest form which is the answer that would be usually required.

Example:

Expand (x +2)(x+3) 1.
Solution –method 1:

Separate the terms in the first bracket into two distinct items (‘x’, and ‘+2). Multiply the first item into the second bracket

After that, multiply the second term into the second bracket and then, add the two sets of terms together.

Method 2

Multiply the two terms in the first bracket into the second bracket, one by one.

That’s just it. The method used above can be called the ‘brute force’ method. The formula is very easy and simple. The formulas can be used for both factorization and expansion.

Factorization is typically the exact opposite of expansion – you want to simplify the given expression to make it less cluttered. Compared to expansion, factorization is much more complex. The idea of removing things and spreading them out (expansion) to some people is much more simple than putting them into a more compact form (factorization).

Methods involved in factorization:
Common Term Factorization

This method is carried out by examining the different terms (coefficient etc.) in the expression and checking for common terms that can be factored out.

Examples:

Factorize 3x+6 1.

Solution

Check for any common numbers or terms. (In this case, 3 and 6 has a common factor – 3.)
Bring the common number/term, or in this case factor, out of the expression. Factorization look simple enough but a lot of students occasionally make a critical mistake when writing down their final answer.

General procedure for factorizing a trinomial

1. Divide the whole equation by any common factor of the coefficients in order to get an equation of the form where a, b and c have no common factors and a is positive.
2. Write down two brackets with an x in each bracket and space for the rest of the terms:
3. Write down a set of factors for a and c.
4. Write down a set of alternatives for the possible factors for the quadratic with the use of the factors and
5. Expand all options to see the one that gives you the right middle term.

If is positive, then the factors ought to either be both positive or both negative. If it is negative, it means just one of the factors of the expression is negative and the other one positive. Immediately you obtain the answer, at all times multiply out your brackets again in order to make sure it actually works.

Application of algebraic methods in solving problems

Factorizing algebraic expressions

Factorize is the reverse of expand i.e. put in brackets.
To factorize, rewrite the expression as factors multiplied together.

Two terms

Check for the biggest number that will divide into all terms and any common letters.

For instance:

12x + 20xy = 4x(3 + 5y)

This is due to the fact that 4x is the biggest factor of both 12x and 20xy.

Cross check by expanding your answer.

Two terms: Difference of two squares

Look for two square terms that have been subtracted.
Such as the example below:

4x² – 9
= (2x)² – 3²
= (2x + 3)(2x – 3)

Three terms

Look for a common factor first and then factorize more.
Example:

3x² – 6x – 15 = 3(x² – 2x – 5)

If there is common factor, look for two numbers that multiply to produce the number by itself (constant term) and add to obtain the number in front of x.
e.g.
x² – 2x – 15 = (x + 3)(x – 5)

This is due to the fact that –5 times 3 = –15 and –5 plus 3 = –2

Check

Thus 3×2 – 6x – 45 = 3(x + 3)(x – 5)

If there is no numbers written in front of x always bear in mind that it means a 1.

Example:

x² + x – 6 = (x + 3)(x – 2)

This is due to the fact that 3 times -2 = –6 and 3 plus –2 = 1

Factorising by Expanding Brackets

Brackets ought to be expanded in the following ways: For an expression of the form a(b + c), the expanded version should be- ab + ac, i.e., multiply the term outside the bracket by everything inside the bracket

Example

2x(x + 3) = 2x² + 6x [remember x × x is x²]).

For an expression of the form (a + b)(c + d), the expanded version would be of the form- ac + ad + bc + bd. This means that everything in the first bracket ought to be multiplied by everything in the second.

Example

Expand (2x + 3)(x – 1):
(2x + 3)(x – 1)
= 2x² – 2x + 3x – 3
= 2x² + x – 3

The Difference of Two Squares

If you are asked to factorize an expression which is one square number minus another, you can factorize it straight away. This is because a² – b² = (a + b)(a – b) .

This expression is called a difference of two squares. (Notice the subtraction sign between the terms.)

a2-b2

You may remember seeing expressions like this one when you worked with multiplying algebraic expressions. Do you remember …

(a + b)(a – b)=a²-b²

If you remember this fact, then you already know that:

The factors of a²-b²

are (a + b) and (a – b)

Remember:

An algebraic expression is a perfect square when the numerical coefficient (the number in front of the variables) is a perfect square and the exponents of each of the variables are even numbers.

Example 1:

Factor: x² – 9

Both x² and 9 are perfect squares. Since subtraction is taking place between these squares, this expression is the difference of two squares.

What will multiplies itself to give x² ? The answer is x.
What multiplies itself will give 9? The answer is 3.

These answers could as well be negative values, but positive values will make our work easier.

The factors are (x + 3) and (x – 3).
Answer: (x + 3) (x – 3) or (x – 3) (x + 3) (order is not significant)

Example 2:

Factor 4y² – 36y⁶

There is a common factor of 4y² that can be factored out first in this question, to make the question easier. 4y² (1 – 9y⁴) In the factor (1 – 9y⁴), 1 and 9y⁴ are perfect squares (their coefficients are perfect squares and their exponents are even numbers). Since subtraction is taking place between these squares, this expression is the difference of two squares.

What multiplies itself to give 1? The answer is 1.
What multiplies itself to give 9y⁴ ? The answer is 3y² .
The factors are (1 + 3y²) and (1 – 3y²).
Answer: 4y² (1 + 3y²) (1 – 3y²) or 4y² (1 – 3y²) (1 + 3y²)

If you did not see the common factor, you can start with observing the perfect squares. Both 4y² and 36y⁶ are perfect squares (their coefficients are perfect squares and their exponents are even numbers). Due to the fact that subtraction is taking place between these squares, this expression is the difference of two squares.

What multiplies itself to give 4y2 ? The answer is 2y.
What multiplies itself to give 36y6 ? The answer is 6y3.

The factors are (2y + 6y³) and (2y – 6y³).
Answer: (2y + 6y³) (2y – 6y³) or (2y – 6y³) (2y + 6y³)
These answers can be additionally factored since each of them is made up of a common factor of 2y:
2y (1 + 3y²) • 2y (1 – 3y²) = 4y² (1 + 3y²) (1 – 3y²)

NOTE: This process of factorization does NOT apply to a² + b²

Example:

Factorise 25 – x² = (5 + x)(5 – x) [imagine that a = 5 and b = x]

Factorizing quadratic expressions

Multiply these brackets to remember how to factorize.

• ( x + 2 ) ( x + 5 ) = x² + 7x + 10
• ( x + 2 ) ( x + 3 ) = x² + 5x + 6
• ( x – 3 ) ( x – 5 ) = x² – 8x + 15
• ( x + 6 ) ( x – 5 ) = x² + x – 30
• ( x – 6 ) ( x + 5 ) = x² – x – 30

Factorizing

To factorize an expression like x² + 5x + 6, you are required to look for two numbers that add up to make 5 and multiply to give 6.

The factor pairs of 6 are:

• 1 and 6
• 2 and 3
2 and 3 add up to 5. So: (x +2) (x+3) = x² + 5x + 6
Factorizing expressions gets a bit difficult with negative numbers.

Question

Factorize the expression: c²– 3c – 10

Answer

Write down the expression: c²– 3c – 10

Remember that to factorize an expression we are required to look for common factor pairs. In this instance, we are looking for two numbers that:

• multiply to give -10
• add to give -3

Think of all the factor pairs of -10:
• 1 and -10
• -1 and 10
• 2 and -5
• -2 and 5

Which of these factor pairs can be added to get -3?

It is just 2 + (-5) = -3

Therefore, the answer is:

c² – 3c – 10 = (c + 2)(c – 5)

Factorizing the difference of two squares

A few quadratic expressions have only a term in x² and a number like x² – 25.

These quadratic expressions have no x term. Making use of our method to factorize quadratics requires you to look for two numbers that multiply to make -25 and add to make 0.

The only factor pair that will give this are 5 and -5. So:

(x + 5)(x – 5) = x² – 25

Not all quadratic expressions without an x term can be factorized.

Examples

These will factorize	These will not factorize
x2 – 36 = (x + 6)(x – 6)	x2 – 32
x2 – 100 = (x + 10)(x – 10)	x2 + 100
x2 – 49 = (x + 7)(x – 7)	x2 + 49
x2 – 1 = (x + 1)(x – 1)	x2 – 3

In all the expressions that will factorize, you have x² minus a square number. Factorizing these expressions is known as the difference of two squares.