Merit of Grouping in Maths Solving

For those who do maths at above elementary level, you will encounter many terms involved in the already many steps to solve a mathematical question.

Example is the solution of Partial fraction, that is highly needed in calculus.

One of the steps needed is the comparing of coefficients to extract out equations to determine the numerators in the individual terms of the partial fractions.

An example is shown below.

x - 3 = A x² + B x (x + 1) + C (x - 2)

How do we solve for the unknown A, B and C?

One useful technique is to do "grouping" of relevant terms.

This is a simple yet powerful method that make the process of solving maths less confusing as it serves to gather common or liked term in the same boundaries.

What I mean is ....

From the above example, we can rewrite them as,

x - 3 = A x² + Bx (x + 1) + C (x - 2)
==> x - 3 = A x² + Bx² + Bx + Cx - 2C
==> x - 3 = (A + B)x² +(B + C)x - 2C

Here, you can see that the coefficient of the terms can be equated nicely to be :-

x² term: (A + B) = 0

x term: (B + C) = 1

Constant: 2C = 3

From herer A, B and C can be easily determined comfortably.

Thus, grouping has the ability to simplify the thinking steps due to clarity as reflected above.

Another example is in Indices simplification.

Take the example of 10ⁿ5^n/2 / 20^n/4.

We cannot see any way to combine the different base number (10, 5 and 20) unless we split them into their lowest factors. (Don't lost focus now, we are aiming for Grouping technique!)

10ⁿ = (5 x 2 )ⁿ
20ⁿ = (2 x 2 x 5)ⁿ

Rewriting to prepare for grouping,

(5 x 2 )ⁿ5^n/2 / (2 x 2 x 5)^n/4


==> 5^{(n + n/2 - n/4)} 2^{n- n/4 - n/4}

==> 5^5/4n 2^n/2

Here again, you can see the merit of grouping the common base number in order to perform the Indices operation.

Maths is "tricky" at times, but isn't this to train our mind to stay active and flexible to counter any challenges put forward. It actually enhances our self-esteem and confidence to handle problems in real life.

Happy grouping .....

:-)