Unit 7: Patterns and Tricks

The middle vanishes

Multiply out (x + 3)(x - 3).

You get x^2 - 3x + 3x - 9, and the two middle terms cancel out completely.

So (x + 3)(x - 3) = x^2 - 9. Any time you see one square take away another square, it factors — always, with no hunting.

Now use it on plain numbers

Here is where it earns its keep. What is 99 × 101?

Those are 100 - 1 and 100 + 1. So the answer is 100^2 - 1^2 = 10000 - 1 = 9999.

No long multiplication. You spotted a pattern instead.

Squaring a bracket

(x + 4)^2 is not x^2 + 16. That is the most expensive mistake in algebra.

It means (x + 4)(x + 4), which is x^2 + 4x + 4x + 16 = x^2 + 8x + 16.

The pattern: square the first, double the product, square the last.

Spotting a perfect square backwards

If you meet x^2 + 14x + 49, check: is the end a square? Yes, 7^2. Is the middle twice 7x? Yes.

So it is (x + 7)^2, and you never had to hunt for factor pairs at all.

This is exactly the shape completing the square was building in unit 6.

Cubes have a pattern too

One worth knowing, because it turns up everywhere:

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

So x^3 - 8 is (x - 2)(x^2 + 2x + 4). Check the middle terms if you like — they all cancel, just as they did with the difference of squares.

It does not have to look like x²

4x^2 - 9 is a difference of squares too: 4x^2 is (2x)^2 and 9 is 3^2.

So it factors to (2x + 3)(2x - 3).

The pattern is about squares, not about the letter x. Once you look for squares rather than for a shape, they turn up everywhere.

Sometimes twice over

x^4 - 16 is (x^2)^2 - 4^2, so it factors to (x^2 + 4)(x^2 - 4).

But look at that second bracket — it is another difference of squares. So it goes further:

(x^2 + 4)(x + 2)(x - 2)

Always check whether a factor can still be broken down. The first bracket cannot: a sum of squares does not factor.

A sum of cubes factors too

A sum of *squares* refuses to factor. A sum of cubes does not:

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Notice the signs: the short bracket keeps the plus, and the long bracket has a minus in the middle. For the difference of cubes it is the other way round.

Remembering which sign goes where

The short bracket copies the sign in the question. The long bracket has the opposite sign in the middle, and a plus at the end.

x^3 - 27 = (x - 3)(x^2 + 3x + 9)

Multiply it out once and you can see why: everything in the middle cancels in a chain, exactly as it did for the difference of squares.

Factor first, then cancel

(x^2 - 9)/(x + 3) looks awkward until you factor the top:

(x + 3)(x - 3)/(x + 3)

The (x + 3) cancels, leaving x - 3. That is the whole method: factor everything, then cross out what matches.

You may only cancel whole factors

Here is a trap. In (x + 6)/6 you cannot cross out the sixes.

Cancelling means dividing top and bottom by the same thing, and the top is not 6 × anything — it is 6 added to something.

The rule: cancel factors, never terms. If it is not multiplied, it does not go.

Give the messy part a short name

(x + 1)^2 + 5(x + 1) + 6 looks horrible. But call x + 1 by the name u and it becomes:

u^2 + 5u + 6

That factors to (u + 2)(u + 3). Put x + 1 back and you have (x + 3)(x + 4).

Nothing was solved that you could not do already. Renaming just let you see it.

Symmetric expressions

When an expression is unchanged by swapping x and y, its answer usually depends only on x + y and xy.

x^2 + y^2 is (x + y)^2 - 2xy.
x^3 + y^3 is (x + y)^3 - 3xy(x + y).

So you never need the individual numbers — only their sum and their product. That turns many hard-looking problems into one substitution.

Take the common piece out first

Before anything else, ask what every term shares.

6x^2 + 9x has a 3x in both parts, so it is 3x(2x + 3).

Doing this first often turns a scary expression into an easy one. And it is the step people skip.

Rearranging is allowed

Sometimes nothing factors until you *move things around*.

What is 1/2 + 1/6 + 1/12 + 1/20? Each one is 1/(n(n+1)), which is 1/n - 1/(n+1).

Written that way the middle terms all cancel in a chain, and only the first and last survive. A page of adding becomes one subtraction.