Name: 數學反常學 - 因式計算 (Math Antics - Factoring)
Uploaded: 2021-01-14T06:40:08.000Z
Duration: 6 min 18 s
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動名詞

In today’s lesson, we’re gonna learn all about factoring.

Factoring is a math operation.  It’s something you DO to a number.

Now with the other math operations you’ve done so far, you’re given two or more numbers to work with.

But with factoring, you only get one number.

That’s because factoring is like being given the answer to a multiplication problem and then having to figure out what that problem was.

So you can think of factoring as ‘UN-multiplying’.

When you multiply, you take two numbers and multiply them together to get one number.

And when you factor, you take one number, and figure out what two numbers you could multiply together to get that number.

For example, let’s say you’re asked to factor the number 10.

Now that means that you need to figure out what numbers you could multiply together to get 10.

If you know your multiplication table, you’ll remember that 2 times 5 equals 10.

That means that 2 and 5 are FACTORS of 10.

So factors are just the parts of a multiplication problem, and factoring is figuring out what those parts are.

I’m sure some of you are wondering, “Why would I ever want to factor a number?”

Now that’s a good question, and I have a good answer.

Factoring can make solving some math problems easier.

For example, factoring is really useful for simplifying fractions.

By breaking a number up into its factors, you can sometimes cancel out factors that aren’t really needed.

For now though, you just need to learn what factoring is and how to do it.

So let’s see another example.  Let’s factor the number 24.

For this one, I think I’ll use my multiplication table.

Let’s see here…  alright, well 4 times 6 is 24.  So that means that 4 and 6 are factors of 24.

Now hold on…  I’m sure some of you have seen that 3 times 8 is also 24.

And that’s true.  We could have decided to factor 24 into 3 times 8 instead.

So which of the factors is right?  Is is 4 and 6  or 3 and 8?

Actually, they’re both right.  There can be more than one way to factor a number.

That’s one of the things that might make factoring a little confusing at first.

You’re used to having just one right answer,

because when you add, subtract, multiply or divide, there is just one right answer.

But when you factor (or un-multiply) a number, you might find that there’s more than one correct way you can do it.

So we can see that the number 24 has quite a few factors.

4 is a factor, 6 is a factor, 3 is a factor and 8 is a factor.

The fact that each of these numbers is a factor of 24 means that each of them can divide evenly into 24.

And when I say “divide evenly”, I mean that it will divide in without a remainder.

For example, if we take our first factor (4) and divide it into 24 using a calculator, our answer will be 6.

4 divides into 24 six times with no remainder.

But what if we try to divide 24 by a number that ISN’T one of our factors; like the number 7?

If we try 24 divided by 7 on a calculator, we get 3.42857… blah, blah, blah … a long decimal number.

That didn’t divide in evenly because there’s a big remainder!

What we just did is called “testing for divisibility”.

Testing for divisibility is a way to find out if a number is a factor of another number.

With the test we just did, we confirmed that 4 is a factor of 24 but 7 is not.

Sometimes you may be asked to find ALL the factors of a number.

If that happens, you can use testing for divisibility to solve the problem.

To see how it works, let’s try to find ALL the factor or 24.

We already know four of them, but there’s a lot more numbers we can test.

Fortunately, we only need to test numbers that are less than half of the number we’re testing.

And since half of 24 is 12, we just need to test the numbers 1 thru 12.

To keep things organized, let’s list the numbers we are gonna test

and we'll circle the factors that we already know: 3, 4, 6, and 8.

We can also cross out the 7 since we already tested it and found out it wasn’t a factor.

Okay, now for the numbers we haven’t tested yet.  Let’s start with 1.

Well of course 1 is a factor, because 1 will divide evenly into ANY whole number.

And since 1 is a factor, then that means 24 is also a factor because 1 × 24 = 24.

It might seem weird that a number is always a factor of itself, but it’s true.

Knowing that helps you get started listing the factors,

because you can ALWAYS include 1 and the number itself.

So yes, 2 is a factor because it divided evenly.

That means that we can ALSO circle 12 as a factor of 24.

So, any time you do a divisibility test, and the number you are checking passes the test,

then the answer you get from dividing will also pass the test… it will ALSO be a factor!

That will speed things up because we know we don’t have to test 12.

Okay, let’s move on to the next number we haven’t checked: 5

Well that didn’t divide evenly because our answer is a decimal number.

Next we’ll try 9.  24 divided by 9 equals 2.66666…

Nope, that’s a decimal number, so 10 is not a factor.

It looks like the last one we have to try is 11.

24 divided by 11 equals 2.181818…  That’s not a factor either.

Alright, since we’ve tested all the numbers that are less than half of 24,

we’ve found ALL of its possible factors, and they are: 1, 2, 3, 4, 6, 8, 12, and 24.

Now I know that might seem like a lot of work, but fortunately you probably won’t have to do many of those problems.

The important thing is just to know what factoring is, and how you can use ‘testing for divisibility’ to help you find factors.

But since actually doing math is the best way to learn it, be sure to try the exercise problems for this section.

Thanks for watching and I’ll see you next time.