Factorizing

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 * Introduction:**

Factorizing is a very important part of math! Let's start by showing how to get to the format a//x//^2 + b//x// + c.

For example: (x + 3)(x - 4) equals x(x - 4) + 3(x - 4). (Distribution) This, therefore, equals x^2 - 4x + 3x - 12. After combining like terms, you get the result of x^2 - x - 12.

In the form a//x//^2 + b//x// + c, note that a = 1, b = -1, and c = -12.

**Practice:** Expand the following: 1. (5x + 3)(x + 4) 2. (f - 4)(f - 7)

*Answers are at the bottom of the page*


 * How To:**

Now, for how to get back to the original form.

For when **a equals 1**, it's relatively easy. You just find a pair of factors that multiply to c, and add to b.

EXAMPLE: x^2 + 2x - 8.

8 = 1*8 or 2*4

Since -8 is negative, we find two numbers that have a difference that is 2. And that is -2 and 4, right?

We insert these into our 2 by 2 (x + s)(x + t)

(s and t are random letters to symbolize our two factors)

So it's (x - 2)(x + 4)

Make sure you see why this works.

(//x// + s)(//x// + t) = //x//^2 + s//x// + t//x// + st, correct? Study this, and then you'll see why.

**Practice:** 3. Factorize s^2 - 5s + 6 4. Factorize r^2 + 7r - 18

*Answers are at the bottom of the page*

Now for when **a is a number greater than 1**:

This is going to be tricky, so make sure to read this over again if you don't understand the first time!

Find the product ac, and then find two factors that add to b.

Example: 3x^2 - 2x - 8

Start by finding the product ac, which equals -24.

Now, what two factors of -24 add to -2? -6 and 4, right?

So far, it's 3x^2 - 6x + 4x - 8.

We can change 3x^2 - 6x into 3x(x - 2), and 4x - 8 into 4(x - 2).

We can combine the like term of (x - 2), so now we have (x - 2)(3x + 4).

**Practice:** 5. Factorize 4//x//^2 + 6//x// + 2 6. Factorize 6//x//^2 - 13//x// + 6

**Answers:** 1. 5//x//^2 + 23//x// + 12 2. //f//^2 - 11//f// + 28 3. (//s// - 2)(//s// - 3) 4. (//r//^2 + 9)(//r//-2) 5. (//x// + 1)(//x// + 2) 6. (2//x// - 3)(3//x// - 2)