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Math with fun #1: Factorization in easy way

2020/02/15

Let’s see next formula.

\[ 6x^2 + 23x - 48 \]

If you are asked to factorize this formula, how will you do it? Because this is quadratic equation, it will be this form.

\[ (ax + b)(cx + d) \]

Now, we have to know what a, b, c and d are. This is the result of factorization of above formula.

\[ (3x + 16)(2x - 3) \]

There are some ways to solve this problem, but one popular way is writing this diagram.

3   16
  ×
2   -3

Multiply 3 and 2 is 6. Multiply 16 and -3 is -48.

Now, 2 numbers arranged diagonally (3 and -3, 2 and 16) can represent 23.

\[ 3 * -3 + 2 * 16 = 23 \]

To do factorization in this way, the steps are like this:

The problem of this solution is that we are not sure how many times we have to try. For example, divisors of 6 are 1, 2, 3, 6. So, this diagram can be this 2 patterns:

6

1

or

3

2

48 is \(2^4 * 3\) . So, it has more patterns. \((1, 48)\), \((2, 24)\), \((3, 16)\), \((4, 12)\), \((6, 8)\).

Because of above, this diagram can be 10 patterns (\(2 * 5\)). Also, we have to take care of - (minus). It becomes more. But we can simplify this method.

Don’t need to examine all the cases

To get straight to the point, we can find the answer by 2 times trial at most.

First, we need to check (6, 1) can be an answer.

6

1

Now, we don’t need to check all the cases for -48. Only possible answer is (1, 48).

6 +-1

1 -+48

We can ignore other cases like (2, 24). Why?

If (6, 1) and (2, 24) works, factorized answer will be like this:

\[ (6x \pm 2)(x \mp 24) \]

But, this can be still factorized:

\[ 2(3x \pm 1)(x \mp 24) \]

If this can be an answer, given formula must be able to be factorized by 2. However, it’s obviously impossible. In other words, left number and right number must be coprime. In this case, 6 and 2 is not coprime. So it never gets an answer.

6 +-1

1 -+48

This also looks not leading us to an answer. So, let’s see next case (3, 2).

3

2

48 is composed of \(2 * 2 * 2 * 2 * 3\). 6 is \(2 * 3\). To make them coprime, only possible answer is (16, 3).

3 +-16

2 -+3

Now, we only need to take care about sign.