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

Discuss the differences between taste and flavor, including how other sensory inputs contribute to our perception of flavor.

John Reply

taste refers to your understanding of the flavor . while flavor one The other hand is refers to sort of just a blend things.

Faith

While taste primarily relies on our taste buds, flavor involves a complex interplay between taste and aroma

Kamara

which drugs can we use for ulcers

Ummi Reply

omeprazole

Kamara

what

Renee

what is this

Renee

is a drug

Kamara

of anti-ulcer

Kamara

Omeprazole Cimetidine / Tagament For the complicated once ulcer - kit

Patrick

what is the function of lymphatic system

Nency Reply

Not really sure

Eli

to drain extracellular fluid all over the body.

asegid

The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include: 1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of

asegid

to transport fluids fats proteins and lymphocytes to the blood stream as lymph

Adama

what is anatomy

Oyindarmola Reply

Anatomy is the identification and description of the structures of living things

Kamara

what's the difference between anatomy and physiology

Oyerinde Reply

Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.

AI-Robot

what is enzymes all about?

Mohammed Reply

Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems

Kamara

yes

Prince

how does the stomach protect itself from the damaging effects of HCl

Wulku Reply

little girl okay how does the stomach protect itself from the damaging effect of HCL

Wulku

it is because of the enzyme that the stomach produce that help the stomach from the damaging effect of HCL

Kamara

function of digestive system

Ali Reply

function of digestive

Ali

the diagram of the lungs

Adaeze Reply

what is the normal body temperature

Diya Reply

37 degrees selcius

Xolo

37°c

Stephanie

please why 37 degree selcius normal temperature

Mark

36.5

Simon

37°c

Iyogho

the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature

Stephanie

37A c

Wulku

what is anaemia

Diya Reply

anaemia is the decrease in RBC count hemoglobin count and PVC count

Eniola

what is the pH of the vagina

Diya Reply

how does Lysin attack pathogens

Diya

acid

Mary

I information on anatomy position and digestive system and there enzyme

Elisha Reply

anatomy of the female external genitalia

Muhammad Reply

Organ Systems Of The Human Body (Continued) Organ Systems Of The Human Body (Continued)

Theophilus Reply

what's lochia albra

Kizito

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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] )