# How do you factor ?

May 18, 2018

1. Find two numbers that multiply to give ac (in other words a times c), and add to give b.
2. Rewrite the middle with those numbers.
3. Factor the first two and last two terms separately

#### Explanation:

${x}^{2} + 4 x - 12$
$6 \cdot - 2 = - 12$
6-2=4

Answer= $\left(x + 6\right) \left(x - 2\right)$

May 18, 2018

Factoring can involve many different things at different stages..

Firstly.. you'll be taught how to factorize when all the terms have something in common
For example
Factorize $12 + 36 x$
Prime factorize them
$12 = \textcolor{red}{2} \times \textcolor{red}{2} \times \textcolor{red}{3}$
$36 = \textcolor{red}{2} \times \textcolor{red}{2} \times \textcolor{red}{3} \times 3 \times x$
#12+36x=12(1+3x)

Then.. you'll be taught how to factorize using rules of factorising
Rules:
${\left(a + b\right)}^{2} = {a}^{2} + 2 a b + {b}^{2}$
${\left(a - b\right)}^{2} = {a}^{2} - 2 a b + {b}^{2}$
$\left(a + b\right) \left(a - b\right) = {a}^{2} - {b}^{2}$
$\left(x + a\right) \left(x + b\right) = {x}^{2} + \left(a + b\right) x + a b$

For example
Factorize

$16 - {a}^{2} + 2 a b - {b}^{2}$

$16 - \left(\textcolor{red}{{a}^{2} - 2 a b + {b}^{2}}\right)$

Remember the rule?

You get

$16 - {\left(a - b\right)}^{2}$

Now..

$16 = {4}^{2}$

You get

${4}^{2} - {\left(a - b\right)}^{2}$

remember the rule?
${a}^{2} - {b}^{2} = \left(a + b\right) \left(a - b\right)$

You get

$\left(4 + \left(a - b\right)\right) \left(4 - \left(a - b\right)\right)$

Expand

$\left(4 + a - b\right) \left(4 - a + b\right)$

Then will come splitting the middle term to factorize...

This video will teach you

Then will come harder laws(rules)
Like
${\left(a + b + c\right)}^{2} + {a}^{2} + {b}^{2} + {c}^{2} + 2 a b + 2 b c + 2 c a$