# 4.1 Algebraic expressions

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## Sample set b

Identify the factors in each term.

$9{a}^{2}-6a-12$ contains three terms. Some of the factors in each term are

$\begin{array}{ll}\text{first}\text{\hspace{0.17em}}\text{term:}\hfill & 9\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}{a}^{2},\text{\hspace{0.17em}}\text{or},\text{\hspace{0.17em}}9\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}a\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}a\hfill \\ \text{second}\text{\hspace{0.17em}}\text{term:}\hfill & -6\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}a\hfill \\ \text{third}\text{\hspace{0.17em}}\text{term:}\hfill & -12\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}1,\text{\hspace{0.17em}}\text{or},\text{\hspace{0.17em}}12\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}-1\hfill \end{array}$

$14{x}^{5}y+{\left(a+3\right)}^{2}$ contains two terms. Some of the factors of these terms are

$\begin{array}{ll}\text{first}\text{\hspace{0.17em}}\text{term:}\hfill & 14,\text{\hspace{0.17em}}{x}^{5},\text{\hspace{0.17em}}y\hfill \\ \text{second}\text{\hspace{0.17em}}\text{term:}\hfill & \left(a+3\right)\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\left(a+3\right)\hfill \end{array}$

## Practice set b

In the expression $8{x}^{2}-5x+6$ , list the factors of the
first term:
second term:
third term:

8, $x$ , $x$ ; $-5$ , $x$ ; 6 and 1 or 3 and 2

In the expression $10+2\left(b+6\right){\left(b-18\right)}^{2}$ , list the factors of the
first term:
second term:

10 and 1 or 5 and 2; 2, $b+6$ , $b-18$ , $b-18$

## Common factors

Sometimes, when we observe an expression carefully, we will notice that some particular factor appears in every term. When we observe this, we say we are observing common factors . We use the phrase common factors since the particular factor we observe is common to all the terms in the expression. The factor appears in each and every term in the expression.

## Sample set c

Name the common factors in each expression.

$5{x}^{3}-7{x}^{3}+14{x}^{3}$ .

The factor ${x}^{3}$ appears in each and every term. The expression ${x}^{3}$ is a common factor.

$4{x}^{2}+7x$ .

The factor $x$ appears in each term. The term $4{x}^{2}$ is actually $4xx$ . Thus, $x$ is a common factor.

$12x{y}^{2}-9xy+15$ .

The only factor common to all three terms is the number 3. (Notice that $12=3\cdot 4,\text{\hspace{0.17em}}9=3\cdot 3,\text{\hspace{0.17em}}15=3\cdot 5$ .)

$3\left(x+5\right)-8\left(x+5\right)$ .

The factor $\left(x+5\right)$ appears in each term. So, $\left(x+5\right)$ is a common factor.

$45{x}^{3}{\left(x-7\right)}^{2}+15{x}^{2}\left(x-7\right)-20{x}^{2}{\left(x-7\right)}^{5}$ .

The number 5, the ${x}^{2}$ , and the $\left(x-7\right)$ appear in each term. Also, $5{x}^{2}\left(x-7\right)$ is a factor (since each of the individual quantities is joined by a multiplication sign). Thus, $5{x}^{2}\left(x-7\right)$ is a common factor.

$10{x}^{2}+9x-4$ .

There is no factor that appears in each and every term. Hence, there are no common factors in this expression.

## Practice set c

List, if any appear, the common factors in the following expressions.

${x}^{2}+5{x}^{2}-9{x}^{2}$

${x}^{2}$

$4{x}^{2}-8{x}^{3}+16{x}^{4}-24{x}^{5}$

$4{x}^{2}$

$4{\left(a+1\right)}^{3}+10\left(a+1\right)$

$2\left(a+1\right)$

$9ab\left(a-8\right)-15a{\left(a-8\right)}^{2}$

$3a\left(a-8\right)$

$14{a}^{2}{b}^{2}c\left(c-7\right)\left(2c+5\right)+28c\left(2c+5\right)$

$14c\left(2c+5\right)$

$6\left({x}^{2}-{y}^{2}\right)+19x\left({x}^{2}+{y}^{2}\right)$

no common factor

## Coefficient

In algebra, as we now know, a letter is often used to represent some quantity. Suppose we represent some quantity by the letter $x$ . The notation $5x$ means $x+x+x+x+x$ . We can now see that we have five of these quantities. In the expression $5x$ , the number 5 is called the numerical coefficient of the quantity $x$ . Often, the numerical coefficient is just called the coefficient. The coefficient of a quantity records how many of that quantity there are.

## Sample set d

$12x$ means there are $12x\text{'}\text{s}$ .

$4ab$ means there are four $ab\text{'}\text{s}$ .

$10\left(x-3\right)$ means there are ten $\left(x-3\right)\text{'}\text{s}$ .

$1y$ means there is one $y$ . We usually write just $y$ rather than $1y$ since it is clear just by looking that there is only one $y$ .

$7{a}^{3}$ means there are seven ${a}^{3\text{'}}\text{s}$ .

$5ax$ means there are five $ax\text{'}\text{s}$ . It could also mean there are $5ax\text{'}\text{s}$ . This example shows us that it is important for us to be very clear as to which quantity we are working with. When we see the expression $5ax$ we must ask ourselves "Are we working with the quantity $ax$ or the quantity $x$ ?".

$6{x}^{2}{y}^{9}$ means there are six ${x}^{2}{y}^{9\text{'}}\text{s}$ . It could also mean there are $6{x}^{2}{y}^{9\text{'}}\text{s}$ . It could even mean there are $6{y}^{9}{x}^{2\text{'}}\text{s}$ .

what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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
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
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Cied
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