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Simple Tips for Better Maths Results!
It is simple to follow the steps of solving a cubic equation which includes finding the linear factor by using calculator and the quadratic factor by long division or comparing coefficient.
However, when the question is asked in another manner in which the solutions are given and the polynomial equation is to be formed. Some students might be at a lost of how to start the question.
I share with you on how we approach this style of question below:
For many Secondary 3 A-Maths students, this topic called Factor - Remainder theorem or Polynomials is usually introduced in the first 3 months of the year.
In this topic, they are introduced to the method to find the factors of a cubic equation so that it can be solved. They also learnt when and how to use Remainder theorem to find the remainder when a dividend is divided by a divisor.
They also know to rewrite an expression into function f(x).
In the following two videos, I use less than 6 minutes to bring across the concepts of Factor theorem and Remainder theorem.
I hope through these videos, you will have a better understanding of these two concepts.
Also, I have included a page which contains all the O-Level Maths videos I have produced since 2007.
Solving a cubic equation is similar in some aspects to solving a quadratic equation, factorisation is in action.
In the two videos, I show you the step by step approach I took to solve a cubic equation either by synthetic division or comparing coefficients.
This concept is usually covered in Secondary 3 Additional Maths syllabus under Factor and Remainder Theorem.
Step 1 (same for both methods) : Guess and Check the first factor (with the help of the calculator).
Step 2: Find the Quadratic Factor.
Leave a comment to share with me which method do you prefer or usually use? I look forward to your comment!
Video 1: GCE O-Level A-Maths: Solve a Cubic Equation by Synthetic Division
Direct link: https://www.youtube.com/watch?v=CqCdD9a8q4c
Video 2: GCE O-Level A-Maths: Solve a Cubic Equation by Comparing Coefficients
Direct link: https://www.youtube.com/watch?v=6iqG74I07v4
In O level Additional Mathematics, there is a small section on Identity inside the topic of Factor & Remainder Theorem. Today I am going to share with you the 2 different approaches to solve this kind of questions.
I am going to use the question below to show you the step by step solutions of both methods.
Given that [pmath]3x^2+x-2=A(x-1)(x+2)+B(x-1)+C[/pmath] for all values of x, find the value of A, of B and of C.
Let x = 1,
[pmath]3+1-2 = C[/pmath]
[pmath]C=2[/pmath]
Let x = -2,
[pmath]3(4)-2-2 = B(-3) + 2[/pmath]
[pmath]3(4)-2-2 = B(-3) + 2[/pmath]
[pmath]B= -2[/pmath]
Let x = 0,
[pmath]-2 = -2A+ 2 + 2[/pmath]
[pmath]A = 3[/pmath]
Thus A = 3, B = -2 and C = 2
Concept behind the Subsitution method: The value of x choosen will cause one or more of the unknowns to be "cancel off", leaving just 1 unknown left. For example, when I choose x = 1 in the first subsituition, A & B are eliminated, allowing me to find 'C'.
Given that [pmath]3x^2+x-2=A(x-1)(x+2)+B(x-1)+C[/pmath] for all values of x, find the value of A, of B and of C.
By comparing coefficient of [pmath]x^2[/pmath]:
LHS: 3 = A => A = 3
By comparing coefficient of [pmath]x[/pmath]:
LHS: 1 = 2A - A + B => B = -2
By comparing coefficient of [pmath]x^0[/pmath]:
LHS: -2 = -2A - B + C => C = 2
Thus A = 3, B = -2 and C = 2
Concept behind the Comparing Coefficient method: Expansion is usually required on one side of the equation. It takes up time. The reason for the insignificant working shown is due to the fact that the expansion is done mentally instead of written. This method is highly recommended if there is more than 1 unknown other than x on the left hand side of the equation. For example, there's an unknown 'D' on the left hand side of the equation.
Which method do you usually use? And which method does your school teach you? Leave me your answer in the comment section below.
I was doing Factor, Remainder Theorem with one of my students. She's a brilliant girl but has yet to achieve the results she truly deserves. I think she knows the reasons clearly :-)
She likes to do things her way.
And I am totally fine with students having their own style. Afterall, my best method might not be their best method. However, if their style is not helping them in terms of scoring, time management..., this is where I step in to influence and get them to better understand how my strategy (usually a much shorter way as I am a lazy person :-)) might help them to understand and get the answer faster and easily.
I get them to buy my idea. And most of the time, my idea sells out :)
Today post will be on Identity covered with Factor-Remainder Theorem.
There are 2 main methods to solve the question. And for me, I choose the "lazy" method again. Simply because it is easy, fast and accurate!
Who else wants to be like me? "Lazy" strategy which is easy, fast and accurate!