# 4.8 Summary of key concepts

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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Operations with algebraic expressions and numerical evaluations are introduced in this chapter. Coefficients are described rather than merely defined. Special binomial products have both literal and symbolic explanations and since they occur so frequently in mathematics, we have been careful to help the student remember them. In each example problem, the student is "talked" through the symbolic form.This module contains a summary of the key concepts in the chapter "Algebraic Expressions and Equations".

## Algebraic expressions ( [link] )

An algebraic expression (often called simply an expression) is a number, a letter, or a collection of numbers and letters along with meaningful signs of operation. ( $5÷0$ is not meaningful.)

In an algebraic expression, the quantities joined by " $+$ " signs are terms .

## Distinction between terms and factors ( [link] )

Terms are parts of sums and are therefore separated by addition signs. Factors are parts of products and are therefore separated by multiplication signs.

## Common factors ( [link] )

In an algebraic expression, a factor that appears in every term, that is, a factor that is common to each term, is called a common factor .

The coefficient of a quantity records how many of that quantity there are. The coefficient of a group of factors is the remaining group of factors.

## Distinction between coefficients and exponents ( [link] )

Coefficients record the number of like terms in an expression. $\underset{3\text{\hspace{0.17em}}\text{terms}}{\underbrace{x+x+x}}=\underset{\text{coefficient}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}3}{3x}$
Exponents record the number of like factors in an expression $\underset{3\text{\hspace{0.17em}}\text{factors}}{\underbrace{x\cdot x\cdot x}}=\underset{\text{exponent}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}3}{{x}^{3}}$

An equation is a statement that two expressions are equal.

## Numerical evaluation ( [link] )

Numerical evaluation is the process of determining a value by substituting numbers for letters.

A polynomial is an algebraic expression that does not contain variables in the denominators of fractions and in which all exponents on variable quantities are whole numbers.

A monomial is a polynomial consisting of only one term.
A binomial is a polynomial consisting of two terms.
A trinomial is a polynomial consisting of three terms.

## Degree of a polynomial ( [link] )

The degree of a term containing one variable is the value of the exponent on the variable.
The degree of a term containing more than one variable is the sum of the exponents on the variables.
The degree of a polynomial is the degree of the term of the highest degree.

Polynomials of the first degree are linear polynomials.
Polynomials of the second degree are quadratic polynomials.
Polynomials of the third degree are cubic polynomials.

## Like terms ( [link] )

Like terms are terms in which the variable parts, including the exponents, are identical.

## Descending order ( [link] )

By convention, and when possible, the terms of an expression are placed in descending order with the highest degree term appearing first. $5{x}^{3}-2{x}^{2}+10x-15$ is in descending order.

## Multiplying a polynomial by a monomial ( [link] )

To multiply a polynomial by a monomial, multiply every term of the polynomial by the monomial and then add the resulting products together.
$7\left(x-3\right)=7x-7\cdot 3=7x-21$

## Simplifying $+\left(a+b\right)$ And $-\left(a+b\right)$ ( [link] )

$\begin{array}{l}+\left(a+b\right)=a+b\\ -\left(a+b\right)=-a-b\end{array}$

## Multiplying a polynomial by a polynomial ( [link] )

To multiply polynomials together, multiply every term of one polynomial by every term of the other polynomial. $\begin{array}{lll}\left(x+3\right)\left(x-4\right)\hfill & =\hfill & {x}^{2}-4x+3x-12\hfill \\ \hfill & =\hfill & {x}^{2}-x-12\hfill \end{array}$

## Special products ( [link] )

$\begin{array}{ccccccc}{\left(a+b\right)}^{2}& =& {a}^{2}+2ab+{b}^{2}& Note:& {\left(a+b\right)}^{2}& \ne & {a}^{2}+{b}^{2}\\ {\left(a-b\right)}^{2}& =& {a}^{2}-2ab+{b}^{2}& & {\left(a-b\right)}^{2}& \ne & {a}^{2}-{b}^{2}\\ \left(a+b\right)\left(a-b\right)& =& {a}^{2}-{b}^{2}& & & & \end{array}$

## Independent and dependent variables ( [link] )

In an equation, any variable whose value can be freely assigned is said to be an independent variable . Any variable whose value is determined once the other values have been assigned is said to be a dependent variable .

The collection of numbers that can be used as replacements for the independent variable in an expression or equation and yield a meaningful result is called the domain of the expression or equation.

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
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
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
hi
Loga
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
how did you get the value of 2000N.What calculations are needed to arrive at it
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