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

differentiate between demand and supply giving examples

Lambiv Reply

differentiated between demand and supply using examples

Lambiv

what is labour ?

Lambiv

how will I do?

Venny Reply

how is the graph works?I don't fully understand

Rezat Reply

information

Eliyee

devaluation

Eliyee

WARKISA

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Lambiv

multiple choice question

Aster Reply

appreciation

Eliyee

explain perfect market

Lindiwe Reply

In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,

Ezea

What is ceteris paribus?

Shukri Reply

other things being equal

AI-Robot

When MP₁ becomes negative, TP start to decline. Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab

Kelo

Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)

Kelo

yes,thank you

Shukri

Can I ask you other question?

Shukri

what is monopoly mean?

Habtamu Reply

What is different between quantity demand and demand?

Shukri Reply

Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded

Ezea

Shukri

how do you save a country economic situation when it's falling apart

Lilia Reply

what is the difference between economic growth and development

Fiker Reply

Economic growth as an increase in the production and consumption of goods and services within an economy.but Economic development as a broader concept that encompasses not only economic growth but also social & human well being.

Shukri

production function means

Jabir

What do you think is more important to focus on when considering inequality ?

Abdisa Reply

any question about economics?

Awais Reply

sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏

Asui

it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs

Awais

thank you so much 👍 sir

Asui

In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p

Cornelius

Suppose a consumer consuming two commodities X and Y has The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50. A,Calculate quantities of x and y which maximize utility. B,Calculate value of Lagrange multiplier. C,Calculate quantities of X and Y consumed with a given price. D,alculate optimum level of output .

Feyisa Reply

Answer

Feyisa

Jabir

the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema

Gsbwnw Reply

suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product

Abdureman

types of unemployment

Yomi Reply

What is the difference between perfect competition and monopolistic competition?

Mohammed

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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] )