# 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 $6{y}^{9}{x}^{2\text{'}}\text{s}$ .

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what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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