4.8 Summary of key concepts

Elementary algebra Page 1 / 1

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".

Summary of key concepts

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 \div 0

is not meaningful.)

Terms ( [link] )

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 .

Coefficients ( [link] )

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 terms}{\underset{︸}{x + x + x}} = \underset{coefficient is 3}{3 x}

Exponents record the number of like factors in an expression

\underset{3 factors}{\underset{︸}{x \cdot x \cdot x}} = \underset{exponent is 3}{x^{3}}

Equation ( [link] )

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.

Polynomials ( [link] )

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.

Linear quadratic cubic polynomials ( [link] )

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} + 10 x - 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 (x - 3) = 7 x - 7 \cdot 3 = 7 x - 21

Simplifying $+ (a + b)$ And $- (a + b)$ ( [link] )

\begin{array}{l} + (a + b) = a + b \\ - (a + b) = - 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}{l} (x + 3) (x - 4) & = & x^{2} - 4 x + 3 x - 12 \\ = & x^{2} - x - 12 \end{array}

Special products ( [link] )

\begin{matrix} {(a + b)}^{2} & = & a^{2} + 2 a b + b^{2} & N o t e : & {(a + b)}^{2} & \neq & a^{2} + b^{2} \\ {(a - b)}^{2} & = & a^{2} - 2 a b + b^{2} & {(a - b)}^{2} & \neq & a^{2} - b^{2} \\ (a + b) (a - b) & = & a^{2} - b^{2} \end{matrix}

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 .

Domain ( [link] )

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.

Questions & Answers

how does Neisseria cause meningitis

Nyibol Reply

what is microbiologist

Muhammad Reply

what is errata

Muhammad

is the branch of biology that deals with the study of microorganisms.

Ntefuni Reply

What is microbiology

Mercy Reply

studies of microbes

Louisiaste

when we takee the specimen which lumbar,spin,

Ziyad Reply

How bacteria create energy to survive?

Muhamad Reply

Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments

_Adnan

But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?

Muhamad

they make spores

Louisiaste

what is sporadic nd endemic, epidemic

Aminu Reply

the significance of food webs for disease transmission

Abreham

food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.

Mark

explain assimilatory nitrate reduction

Esinniobiwa Reply

Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).

Elkana

This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products

Elkana

Examples of thermophilic organisms

Shu Reply

Give Examples of thermophilic organisms

Shu

advantages of normal Flora to the host

Micheal Reply

Prevent foreign microbes to the host

Abubakar

they provide healthier benefits to their hosts

ayesha

They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!

Mark

what is cell

faisal Reply

cell is the smallest unit of life

Fauziya

cell is the smallest unit of life

Akanni

Innocent

cell is the structural and functional unit of life

Hasan

is the fundamental units of Life

Musa

what are emergency diseases

Micheal Reply

There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️

_Adnan

define infection ,prevention and control

Innocent

I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life

Lubega

Heyy Lubega hussein where are u from?

_Adnan

en français

Adama

which site have a normal flora

ESTHER Reply

Many sites of the body have it Skin Nasal cavity Oral cavity Gastro intestinal tract

Safaa

skin

Asiina

skin,Oral,Nasal,GIt

Sadik

How can Commensal can Bacteria change into pathogen?

Sadik

How can Commensal Bacteria change into pathogen?

Sadik

all

Tesfaye

by fussion

Asiina

what are the advantages of normal Flora to the host

Micheal

what are the ways of control and prevention of nosocomial infection in the hospital

Micheal

what is inflammation

Shelly Reply

part of a tissue or an organ being wounded or bruised.

Wilfred

what term is used to name and classify microorganisms?

Micheal Reply

Binomial nomenclature

adeolu

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Source: OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3

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4.8 Summary of key concepts

Summary of key concepts

Algebraic expressions ( [link] )

Terms ( [link] )

Distinction between terms and factors ( [link] )

Common factors ( [link] )

Coefficients ( [link] )

Distinction between coefficients and exponents ( [link] )

Equation ( [link] )

Numerical evaluation ( [link] )

Polynomials ( [link] )

Degree of a polynomial ( [link] )

Linear quadratic cubic polynomials ( [link] )

Like terms ( [link] )

Descending order ( [link] )

Multiplying a polynomial by a monomial ( [link] )

Simplifying + ( a + b ) And − ( a + b ) ( [link] )

Multiplying a polynomial by a polynomial ( [link] )

Special products ( [link] )

Independent and dependent variables ( [link] )

Domain ( [link] )

Questions & Answers

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Simplifying $+ (a + b)$ And $- (a + b)$ ( [link] )