Date: Thu, 25 Mar 1999 18:45:51 -0800 (PST)
Sender: Maggie

I don't know anything about factoring would you plese help me.

3x4 - 48

54x6 + 16y3

125-8x3

12x2 - 36x + 27

9 - 81x2

a3 + b3c3

Hi Maggie

Remember that factoring is the process of unmultiplying. When you want to factor 6 you want to know what numbers can multiply to give 6. (You can have the obvious one [1x6] and another one [3x2].)
The other thing to remember is that, once you know one factor of a number, you can always get the other factor by division (which is unmultiplication). E.g. if you look at 994 you can see that it is even and so 2 is a factor. If you divide 994 by 2 you can see that the other factor is 497.
I use the following checklist to help me when I am factoring so that I can get a clue as to how to proceed.

### FACTORING BY NUMBERS

1. Count the number of terms.

2. Look for a common factor.

3. Look again!

4. If there are TWO terms then it is probably:
• difference of (two) squares
• difference of (two) cubes
• sum of (two) cubes.

5. If there are THREE terms then it is probably a trinomial (the exponent of the middle term will be one half of the exponent of the first and/or third term).

6. If there are FOUR OR MORE terms then it is probably grouping.

### FACTOR THEOREM

#### If f(x) is a polynomial and f(a) = 0 then x - a is a factor of f(x).

The Factor Theorem can be used almost any time.

Let's look at 3x4 - 48.

There are 2 terms.

Now we look to see if there is any number or variable that is common to both terms. We see that each term is divisible by 3 and so we know that 3 is a common factor. We can get the other factor by dividing (3x4 - 48) by 3 and we get 3(x4 - 16). You can verify that this is correct by multiplying the expression in the brackets by 3 to ensure that we get the original.

We have factored the expression and so can say that we are finished, however if we were asked to "factor completely" then we need to check to see if the expression in the brackets can be factored further.

(E.g. 30 = 2x15 = 2x3x5.)

So let's look at x4 - 16.

Start the process again. There are 2 terms and there is nothing common to the 2 terms. There are 2 terms separated by a - sign so it might be difference of squares or difference of cubes. Notice that both x4 and 16 are squares and so it is difference of squares.

Recall that a2 - b2 = (a - b)(a + b) so x4 - 16 = (x2 - 4)(x2 + 4).

You can verify this by multiplying.

Now we know that 3x4 - 48 = 3(x2 - 4)(x2 + 4).

If we were told to factor completely then we must check the two expressions in the brackets to see if they can be factored further.

Look at x2 - 4 and start again!

There are 2 terms and nothing is common.
It could be difference of squares or cubes.
It is difference of squares.

So x2 - 4 = (x - 2)(x + 2).

Now look at x2 + 4. There are 2 terms with nothing common and it is not the sum of cubes so we probably cannot factor it further.

Finally we know that

3x4 - 48 = 3(x2 + 4)(x - 2)(x + 2).

Now let's try 54x6 - 16y3

There are 2 terms and there is a common factor of 2.

Then the expression in the brackets is the difference of 2 cubes. Remember how to factor a3 - b3.

a3 - b3 = (a - b)(a2 + ab + b2).

125 - 8x3 can be done by going through your checklist.

Now 12x2 - 36x + 27

There are 3 terms and 3 is common.

So 12x2 - 36x + 27 = 3(4x2 - 12x + 9).

Now check the part in the brackets.

There are 3 terms and nothing is common and so it might be a trinomial. It is and can be factored by whatever method you use for factoring trinomials. (I don't want to give you a different method than the one that you are currently using.)

9 - 81x2 is like one of the earlier ones (common factor and difference of...)

a3 + b3c3

The key here is to realize that there are only 2 terms and there is nothing common and so it must be the sum of 2....

Factoring is both an art and a science. I find that the checklist helps with the science but you do need to look at each new expression with fresh eyes.

Good luck.
Jack

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