Reflection on Factorisation and Expansion

Reflection

Initially, I was confused by expansion and factorisation however after learning more about these two chapters, i have learnt more about it. I have also learnt how to apply it in my daily life. Also, I now know how to differentiate between expansion and factorisation. I am quite amazed at how math has so many ways to go about a question. It is something that I have learnt in these two chapters.

Factorisation by Grouping (Chap 4.4)

Factorisation of Algebraic Expressions by Grouping

Expressions containing four or more terms can be factorised by the method of grouping. In this method, the terms are divided into groups such that the terms in each group have a common factor. 

ax + ay + bx + by = a(x + y) + b(x + y)

                  = (x+y)(a+b)

a) 3px - 9py - 4qx + 12 qy

         = 3p(x - 3y) - 4q(x - 3y)  

    (Extract common factor x+y)

=(x - 3y)(3p - 4q)

Factorisation Using Special Algebraic Identities (Chap 4.3)

Factorisation Using Special Algebraic Identities 

Since factorisation is the reverse of expansion, we have:

  a² + 2ab + b² = (a + b)²

a² - 2ab + b² = (a - b)²

                                                  a² - b² = (a + b)(a - b)

a) 4x² - 25y² = (2x)² - (5y)² 

= (2x + 5y)(2x - 5y)  

 (apply a² - b² = (a + b)(a - b), where a = 2x and b= 5y)

Expansion Using Special Algebraic Identities (Chap 4.2)

Expansion using Special Algebraic Identities

Now we will learn how to expand special products of algebraic expressions: 

Perfect Squares and Difference of Two Squares

Perfect squares: 

(a + b)² = a² + 2ab + b² 

(a - b)² = a² - 2ab + b² 

Difference of two squares:

(a + b)(a - b) = a² - b² 

• Perfect Squares Example: 

a)  (x + 5)² = x² + 2(x)(5) + 5²    <------   Use (a + b)²  = a²  + 2ab +b² 

                   = x²  + 10x + 25                    Here a = x and b = 5.

b) (3x - 4)² = (3x)² - 2(3x)(4) + 4²   <------  Use (a - b)² = a²  - 2ab + b² 

               = 9x² - 24x + 16

*Be careful when expanding expressions like (x + 5)².

(x + 5)² ≠ x² + 25 

(x + 5)² = (x + 5)(x + 5)

• Difference of two Squares:

a)  (x+1)(x-1) = x² - 1²    <------ Use (a + b)(a - b) = a² - b²

                  = x² - 1 

b)  (x + y + 2)(x + y - 2)

    = [(x + y) + 2)][(x + y) - 2]    <------- Use (a+b)(a-b) = a² - b²

    = (x + y)² - 2²                                   Here a = (x + y) and b = 2

    = x² + 2xy + y² - 4     <------- Use (a+b)² = a² + 2ab + b².

Expansion and Factorisation of Algebraic Expressions (Chap 4.1)

Expansion and Factorisation of Algebraic Expressions

Example:

Expand and simplify each of the following expressions:

a) -3x (y+z) - 4x(2y-5z)

=-3xy - 3xz -8xy + 20xz (Distributive law)

=-3xy - 8xy -3xz + 20xz (Group like terms)

=-11xy + 17xz

b) 2x (x + 2y) + 3x(2x - 3y) = 2x² + 4xy + 6x² - 9xy (Distributive law)

= 2x² + 6x² + 4xy -9xy (Group like terms)

= 8x² -5xy

Factorisation of Quadratic Expressions (Chap 3.3)

Factorisation of Quadratic Expressions using a Multiplication Frame

a) Example: x² + 8x + 12

Step 1: Write x² in the top-left corner and 12 in the bottom-right corner of the multiplication frame.

Step 2: Consider the factors of x²  and 12. Write them in the first column and the first row.

Step 3: Multiply them to complete the multiplication frame and check whether the result matches the given expression.

Therefore, x²  - 5x + 4 = (x-1)(x-4)

b) Example: 3x²  + 7x -6 

Therefore, 3x²  + 7x - 6 = (3x - 2)(x + 3)

c) Example: 4x² - 6x - 4 = 2(2x² - 3x -2)  

extract the common factor 2

Therefore, 4x² - 6x - 4 = 2(2x + 1)(x - 2)

Expansion and Simplification of Quadratic Expressions (Chap 3.2)

Expansion and Simplification of Quadratic Expressions

a) Example: -2 (3x -2) 

    *Multiply 3x-2 by 2 before changing its sign. 

    Therefore, -2(3x-2) = (- 2)(3x)+(- 2)(- 2)

                                    = -6x + 4

b) Example: -2x(3x-2) = (-2x)(3x) + (-2x)(-2) 

                                     = -6x² + 4x

(Prezi link):

http://prezi.com/uf3po-ri-vba/expanding-and-simplifying-quadratic-equations/

Quadratic Expression (Chap 3.1)

Quadratic Expression

*The general form of a quadratic expression in one variable is:

             ax² + bx +c
             where x is the variable, a v and c are constants and a≠0.

• Addition and Subtraction of Quadratic Expressions

       a) Example: 4x² + 2x² 
        Therefore,  4x² + 2x² =6x² 

       
       b) Example: 4x² + (-2x² )
  
        Therefore, 4x² + (-2x² ) =2x² 


       c) Example: -4x² -2x+3x² +x+2

        First, you would have to group like terms together
   
       = -4x² + 3x² -2x+x+2 

        Then, you solve it.

        = -x² -x+2

• Negative of a Quadratic Equation

                  -To find the negative of a quadratic expression, we flip all signs. 

a) Example: -(2x² +x-1)

Therefore, -(2x² +x-1) = -2x² -x+1

*We can also simplify quadratic expressions involving the negative of a quadratic expression. 

b) (2x² - 3x +1) - (x²-2x-3) 

Start by expanding the right part 

= 2x² - 3x +1 - x²+2x+3 

Then grouping like terms together

= 2x² - x² - 3x + 2x +1 + 3 

= x² - x + 4

• Expansion and Simplification of Simple Quadratic Expressions

a ) -x² -x +2 + 3(x² + x - 1) = -x² - x + 2 + 3x² +3x - 3 (expand)

                                          

                                             = -x² + 3x² - x + 3x + 2 - 3 (group like terms)

                                             = 2x² + 2x - 1

*Youtube Video:http://www.youtube.com/watch?v=zb7pBanmwr4

What is factorisation?

Factorisation

Factorisation is the process of writing an algebraic expression as a product of its factors.

*For example:

The common factor for the 2 terms are 5, hence 5 is taken out.

*Another example:

This is just one method of factorisation. There are many other factorisation techniques, including:

grouping, using algebraic identities and inspection.

What is expansion?

Expansion

Expansion is the process of removing brackets and multiplying them term by term before consolidating the terms together.

                                               *For example:

"Expanding" means removing the ( ) but you have to do it the right

Whatever is inside the ( ) needs to be treated as a "package".

• So when you multiply, you have to multiply by everything inside the "package".

Example: Expand 3 × (5+2)

3 × (5+2) = 3 × 5 + 3 × 2

It is now expanded.

You could also go on to calculate that it equals 15 + 6 = 21

In Algebra

In Algebra putting two things next to each other usually means to multiply.

So 3(a+b) means to multiply 3 by (a+b)

Conclusion

Multiply by everything inside the ().

• There are also special algebraic identities used in expansion which will be further elaborated upon on in my other posts.