# 4.1 Factoring  (Page 2/2)

 Page 2 / 2
$\left(x+a\right)\left(x-a\right)={x}^{2}-{a}^{2}$

You can run this formula in reverse whenever you are subtracting two perfect squares . For instance, if we see ${x}^{2}-\text{25}$ , we recognize that both ${x}^{2}$ and 25 are perfect squares. We can therefore factor it as $\left(x+5\right)\left(x-5\right)$ . Other examples include:

• ${x}^{2}-\text{64}$ $=\left(x+8\right)\left(x-8\right)$
• $\text{16}{y}^{2}-\text{49}$ $=\left(4y+7\right)\left(4y-7\right)$
• ${2x}^{2}-\text{18}$ $=2\left({x}^{2}-9\right)$ $=2\left(x+3\right)\left(x-3\right)$

And so on. Note that, in the last example, we begin by pulling out a 2, and we are then left with two perfect squares. This is an example of the rule that you should always begin by pulling out common factors before you try anything else!

It is also important to note that you cannot factor the sum of two squares. ${x}^{2}+4$ is a perfectly good function, but it cannot be factored.

## Brute force, old-fashioned, bare-knuckle, no-holds-barred factoring

In this case, the multiplication that we are reversing is just FOIL. For instance, consider:

$\left(x+3\right)\left(x+7\right)={x}^{2}+3x+7x+\text{21}={x}^{2}+\text{10}x+\text{21}$

What happened? The 3 and 7 added to yield the middle term (10), and multiplied to yield the final term $\left(\text{21}\right)$ . We can generalize this as: $\left(x+a\right)\left(x+b\right)={x}^{2}+\left(a+b\right)x+\text{ab}$ .

The point is, if you are given a problem such as ${x}^{2}+\text{10}x+\text{21}$ to factor, you look for two numbers that add up to 10, and multiply to 21. And how do you find them? There are a lot of pairs of numbers that add up to 10, but relatively few that multiply to 21. So you start by looking for factors of 21.

Below is a series of examples. Each example showcases a different aspect of the factoring process, so I would encourage you not to skip over any of them: try each problem yourself, then take a look at what I did.

If you are uncomfortable with factoring, the best practice you can get is to multiply things out . In each case, look at the final answer I arrive at, and multiply it with FOIL. See that you get the problem I started with. Then look back at the steps I took and see how they led me to that answer. The steps will make a lot more sense if you have done the multiplication already.

Factor ${x}^{2}+\text{11}x+\text{18}$

$\left(x+\text{__}\right)\left(x+\text{__}\right)$

What multiplies to 18? $1\cdot \text{18}$ , or $2\cdot 9$ , or $3\cdot 6$ .

Which of those adds to 11? $2+9$ .

$\left(x+2\right)\left(x+9\right)$

Start by listing all factors of the third term. Then see which ones add to give you the middle term you want.

Factor ${x}^{2}-\text{13}x+\text{12}$

$\left(x+\text{__}\right)\left(x+\text{__}\right)$

What multiplies to 12? $1\cdot \text{12}$ , or $2\cdot 6$ , or $3\cdot 4$

Which of those adds to 13? $1+\text{12}$

$\left(x-1\right)\left(x-\text{12}\right)$

If the middle term is negative, it doesn’t change much: it just makes both numbers negative. If this had been ${x}^{2}+\text{13}x+\text{12}$ , the process would have been the same, and the answer would have been $\left(x+1\right)\left(x+\text{12}\right)$ .

Factor ${x}^{2}+\text{12}x+\text{24}$

$\left(x+\text{__}\right)\left(x+\text{__}\right)$

What multiplies to 24? $1\cdot \text{24}$ , or $2\cdot \text{12}$ , or $3\cdot 8$ , or $4\cdot 6$

Which of those adds to 12? None of them.

It can’t be factored. It is “prime.”

Some things can’t be factored. Many students spend a long time fighting with such problems, but it really doesn’t have to take long. Try all the possibilities, and if none of them works, it can’t be factored.

Factor ${x}^{2}+2x-\text{15}$

$\left(x+\text{__}\right)\left(x+\text{__}\right)$

What multiplies to 15? $1\cdot \text{15}$ , or $3\cdot 5$

Which of those subtracts to 2? 5–3

$\left(x+5\right)\left(x-3\right)$

If the last term is negative, that changes things! In order to multiply to –15, the two numbers will have to have different signs—one negative, one positive—which means they will subtract to give the middle term. Note that if the middle term were negative, that wouldn’t change the process: the final answer would be reversed, $\left(x+5\right)\left(x-3\right)$ . This fits the rule that we saw earlier—changing the sign of the middle term changes the answer a bit, but not the process.

Factor ${2x}^{2}+\text{24}x+\text{72}$

$2\left({x}^{2}+\text{12}x+\text{36}\right)$

$2{\left(x+6\right)}^{2}$

Never forget, always start by looking for common factors to pull out. Then look to see if it fits one of our formulae. Only after trying all that do you begin the FOIL approach.

Factor ${3x}^{2}+\text{14}x+\text{16}$

$\left(3x+\text{__}\right)\left(x+\text{__}\right)$

What multiplies to 16? $1\cdot \text{16}$ , or $2\cdot 8$ , or $4\cdot 4$

Which of those adds to 14 after tripling one number ? $8+3\cdot 2$

$\left(3x+8\right)\left(x+2\right)$

If the ${x}^{2}$ has a coefficient, and if you can’t pull it out, the problem is trickier. In this case, we know that the factored form will look like $\left(3x+\text{__}\right)\left(x+\text{__}\right)$ so we can see that, when we multiply it back, one of those numbers—the one on the right—will be tripled, before they add up to the middle term! So you have to check the number pairs to see if any work that way.

There are two different ways to check your answer after factoring: multiplying back, and trying numbers.

1. Problem : Factor $\text{40}{x}^{3}-\text{250}x$
• $\text{10}x\left(4x-\text{25}\right)$ First, pull out the common factor
• $\text{10}x\left(2x+5\right)\left(2x-5\right)$ Difference between two squares
2. So, does $\text{40}{x}^{3}-\text{250}x=\text{10}x\left(2x+5\right)\left(2x-5\right)$ ? First let’s check by multiplying back.
• $\text{10}x\left(2x+5\right)\left(2x-5\right)$
• $=\left(\text{20}{x}^{2}+\text{50}x\right)\left(2x-5\right)$ Distributive property
• $=\text{40}{x}^{3}-\text{100}{x}^{2}+\text{100}{x}^{2}-\text{250}x$ FOIL
3. Check by trying a number. This should work for any number. I’ll use $x=7$ and a calculator.
• $\text{40}{x}^{3}-\text{250}x\stackrel{?}{=}\text{10}x\left(2x+5\right)\left(2x-5\right)$
• $\text{40}{\left(7\right)}^{3}-\text{250}\left(7\right)\stackrel{?}{=}\text{10}\left(7\right)\left(2\cdot 7+5\right)\left(2\cdot 7-5\right)$

I stress these methods of checking answers, not just because checking answers is a generally good idea, but because they reinforce key concepts. The first method reinforces the idea that factoring is multiplication done backward . The second method reinforces the idea of algebraic generalizations.

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Got questions? Join the online conversation and get instant answers!    By By Anonymous User By Lakeima Roberts    By