What is Expansion?
You probably would understand from the word itself, expanding! Expansion is done by removing the brackets from an equation by multiplying each other. This is done by taking the number outside the bracket to multiply the numbers inside the bracket.
Examples:
a(b + c) = ab + ac
-a(b + c) = -ab - bc
a(b - c) = ab - ac
When you want to expand an equation with two brackets, the terms in the first bracket must be multiplied by each term in the equation. This may sound confusing, so let me give you some examples!
Example:
(a + b)(c + d) = a(c + d) + b(c + d) = ac + ad + bc + bd
Take note whenever you deal with negative equations. Do remember to simplify the equation as well!
Example:
(x - 2)(x + 3)
= x(x + 3) - 2(x + 3)
= x² + 3x - 2x - 6
= x² + x - 6
Here are some of the formulas for expansion
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
(a + b)(a - b) = a² - b²
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What is Factorisation?
It is the process where of expression an algebraic expression as a product of two or more algebraic expressions. In simple terms, it means breaking down an expression into simpler terms. It is the reverse of expansion.
There are a few steps needed to factorise an equation.
Step 1: Identify the common factors from the expression
Step 2: Extract the common factors from the expression and write it down as a product of the common factor.
Step 3: Factorise!
Let me give you an example.
a + ab = a(1 + b)
3x² + 9xy = 3x(x + 3y)
To factorise a quadratic equation, we can use a multiplication frame.
For example, consider the expression 2x² - 5x - 3
Step 1: Write 2x² in the top-left corner and -3 in the bottom-right corner of the multiplication frame.
Step 2: Consider the factors of 2x² and -3. 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, 2x² - 5x - 3 = (2x + 1)(x - 3)
Another method that can we used in the cross factorisation. You can learn more by looking at some videos. http://www.youtube.com/watch?v=8IXICJBKUtg, http://www.youtube.com/watch?v=WHh1pWIMsbo and http://www.youtube.com/watch?v=vhBLGdELwhw.
Some formulas to take note when doing factorisation!
a² + 2ab + b² = (a + b)²
a² - 2ab + b² = (a - b)²
a² - b² = (a + b)(a - b)
In some cases, there is a need to factorise by grouping. For example, ax + bx + kay + kby. For this equation, this is how you solve it.
ax + bx + kay + kby
= x(a + b) + ky (a + b)
= (a + b)(x + ky)
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Difference between Expansion and Factorisation?
Expansion is opening up the brackets. Factorisation is is simplifying and placing brackets into the equation.
Expansion lengthens the equation while factorisation shortens it.
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Need More Help?
Still have some questions and want more information? Do visit this few websites listed below!
http://www.mathsisfun.com/algebra/expanding.html
http://www.mathsisfun.com/algebra/factoring-quadratics.html
http://www.youtube.com/watch?v=gJMNt9BAKqM
http://www.youtube.com/watch?v=eF6zYNzlZKQ
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Reflections
I have learnt more about expansion and factorisation. I had a better understanding on what these topics were. When I first started learning on expansion, I was a little confused about which numbers to multiply first. However, after my Mathematics teacher explained further, I understood expansion and know the steps well. One of my weaknesses in expansion is that I have a habit of not simplifying the expressions. I have to learn and remember to simplify the expression whenever I do these kind of questions.
As for factorisation, I took a longer time to get the hang of it. I was quite lost at first as I did not get how my teacher could get the final answer so quickly. I realised that we could find the equation by listing down some factors of the terms and the answer could be gotten at a shorter time. At first, I did not understand the use of the multiplication frame. However, my teacher taught us the cross factorisation method which makes factorisation easier for me to understand. My teacher also explained to us how to use the multiplication frame in the end and I could understand better, probably because I got the hang of doing factorisation.
During this period, I also learnt perseverance. There were times when I just could not understand how to do factorisation and just felt like giving up and not do Mathematics. However, one of my friends encouraged me and told me to continue. Good result comes with hard work and as long as i continue, I can do it! As the saying goes, "practise makes perfect", as long as i continue practising and working hard, I can get the hang of expansion and factorisation of quadratic expressions one day, which I did!
I found this chapter quite fun. I have learnt a lot and I can finally do quadratic expression questions!
Conclusion
All the best in Mathematics, especially expansion and factorisation. If you understand the steps and formulas, I am sure that you would enjoy this topic and it would be manageable for you!
Hope you enjoyed reading my blog! Hope you found useful information. Thank you.
