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Algebraic expression

An algebraic expression is a mathematical phrase that combines numbers, variables, and operators—such as addition, subtraction, multiplication, division, exponents, or roots—to represent a value or quantity without being set equal to another expression.^[1] Unlike an equation, which equates two expressions (e.g., $2x + 3 = 7), an algebraic expression stands alone and can be evaluated or simplified based on given values or rules.^[1] These expressions form the foundation of algebra, enabling the generalization of arithmetic operations and the modeling of real-world relationships.^[2] Key components of an algebraic expression include terms, which are individual parts separated by addition or subtraction; variables, such as x or y, representing unknown or variable quantities; constants, fixed numerical values like 5 or -2; and coefficients, the numerical factors multiplying variables (e.g., 3 in $3x).^[1] For instance, in the expression $4x^2 + 2xy - 7, the terms are $4x^2, $2xy, and -7, with coefficients 4 and 2, variable x and y, and constant -7.^[3] Expressions can be evaluated by substituting specific values for variables and applying the order of operations (parentheses, exponents, multiplication/division, addition/subtraction), yielding a numerical result—such as $2x + y becoming 14 when x = 4 and y = 6.^[1] Simplifying algebraic expressions involves combining like terms—those with identical variables raised to the same power—by adding or subtracting their coefficients, which reduces complexity while preserving equivalence.^[4] For example, $3x + 2x - x + 5 simplifies to $4x + 5.^[3] This process relies on algebraic properties like the commutative, associative, and distributive laws, which allow rearrangement and grouping for efficiency.^[5] Algebraic expressions underpin higher mathematics, including solving equations, polynomial manipulation, and applications in physics and engineering.^[6]

Definition and Terminology

Core Definition

An algebraic expression is a finite combination of variables, constants, and mathematical operation symbols, such as addition, subtraction, multiplication, division, exponents, and roots, that represents a quantity without assigning a specific value or including an equality relation.^[7] This symbolic representation allows for the generalization of arithmetic operations, enabling the description of relationships between unknown or varying quantities in a concise form.^[6] The concept of algebraic expressions traces its roots to ancient Babylonian mathematics around 2000 BCE, where problem-solving techniques akin to algebraic methods were recorded on clay tablets using rhetorical descriptions rather than symbols, followed by developments in Greek mathematics, such as Diophantus's syncopated notation in the 3rd century CE. The modern symbolic form of algebraic expressions emerged in the late 16th and early 17th centuries, formalized by French mathematicians François Viète, who introduced letters for unknowns and parameters in equations, and René Descartes, who advanced literal notation and coordinate methods to link algebra with geometry.^[8]^[9] Unlike an equation, which states that two expressions are equal using an equals sign (e.g., $2x + 3 = 7), an algebraic expression stands alone without such a relation, serving merely to denote a value or quantity.^[10] It also differs from a function, which specifies a mapping from input values to outputs, often using an expression like f(x) = 2x + 1 to define that relationship explicitly.^[11] A basic example is $2x + 3y - 5, where x and y are variables representing unknown quantities, 2, 3, and -5 are constants, and the operations of multiplication (implied in $2x and $3y) and addition/subtraction combine them to form the overall expression.^[7]

Key Components

An algebraic expression is composed of fundamental elements known as terms, which are separated by addition or subtraction operators. Each term typically consists of factors multiplied together, including coefficients, variables, and possibly constants. These components allow for the representation of mathematical relationships using symbols and numbers.^[2] Terms are the individual addends or subtrahends within an algebraic expression, connected by plus (+) or minus (-) signs. For instance, in the expression $3x^2 + 2x - 1, the terms are $3x^2, $2x, and -1. The constant term -1 has no variables, while the others include them. Terms provide the building blocks that are added or subtracted to form the full expression.^[2]^[12] Within each term, factors are the multiplicative components that form the product. For example, in the term $6xy, the factors are the coefficient 6, and the variables x and y. Factors can include numerical coefficients, variables raised to powers, or constants, and they are separated by multiplication, which may be implicit (juxtaposition) or explicit (using \times). This structure enables the breakdown of complex terms into simpler parts.^[2]^[13] Operators are the symbols that indicate the actions between terms or factors in an expression. The primary operators include addition (+), subtraction (-), multiplication (often implicit or denoted by \times), and division (denoted by / or a fraction bar). Addition and subtraction separate terms, while multiplication connects factors within a term. These operators define how the components interact to produce the expression's value.^[2]^[12] Like terms are terms within an expression that share identical variable parts, including the same variables raised to the same powers, allowing them to be grouped conceptually. For example, $2x and $3x are like terms because both involve x to the first power, whereas $2x and $3x^2 are not. Constant terms, such as +5 and -2, are also considered like terms since they lack variables. This similarity in structure is a key feature for analyzing expressions.^[2]^[14] Constants and variables serve as the base components from which terms are constructed. A constant is a fixed numerical value, such as 7 in the expression x + 7, which does not change. Variables, denoted by letters like x or y, represent unknown or varying quantities. Together, they form the core of algebraic notation, with constants providing numerical specificity and variables enabling generalization.^[13]^[12]

Notation and Conventions

Variables and Constants

In algebraic expressions, variables are symbols that represent unknown or varying quantities, allowing for generalization and abstraction in mathematical statements. Typically, these are denoted by lowercase letters such as x, y, or z, which stand in for numerical values that can change depending on the context or solution being sought.^[15]^[16] This convention enables the description of relationships that apply to a range of scenarios, distinguishing algebra from purely arithmetic computations.^[15] Constants, in contrast, are fixed values that do not vary within the expression, serving as unchanging parameters or numerical factors. These include integers, such as 5 in the term $5x, or specific irrational numbers like \pi (approximately 3.14159), which may appear in expressions blending algebraic and geometric contexts.^[15]^[17] For instance, in the expression $2\pi r, the symbols 2 and \pi function as constants, while r acts as the variable representing a varying length.^[15] The standard use of lowercase letters for variables was established by René Descartes in his 1637 work La Géométrie, where he designated x, y, z for unknowns to systematize algebraic notation and integrate it with geometry.^[9] In expressions like ax + b, a and b are constants that scale or shift the variable x, providing a framework for linear relationships applicable across mathematics.^[15] Earlier texts sometimes employed uppercase letters for constants or known quantities, a practice that persisted in some older European algebraic traditions before Descartes' standardization.^[18]

Exponents and Coefficients

In algebraic expressions, exponents denote repeated multiplication of a base, typically a variable or constant, and are represented using superscript notation to the right of the base. For instance, the expression x^2 signifies x \times x, where x is the base and 2 is the exponent indicating the number of times the base is multiplied by itself.^[19]^[20] Negative exponents represent reciprocals of the base raised to the corresponding positive power; for example, x^{-1} = \frac{1}{x}, extending the notation to express division in a compact form.^[21]^[22] Coefficients are numerical factors that multiply variables or terms in an algebraic expression, such as the 4 in $4x^3, which scales the variable term. When no explicit coefficient appears before a variable, it is implicitly 1, as in x^3 where the coefficient of x^3 is understood to be 1.^[2]^[13] Standard conventions for exponents include right-aligned superscript positioning for clarity in printed or digital formats, and fractional exponents to denote roots, where x^{1/2} = \sqrt{x}, combining exponentiation with radical operations.^[23] A key rule is that any non-zero base raised to the power of zero equals 1, so x^0 = 1 for x \neq 0, providing a consistent foundation for exponent properties.^[19]^[22] These elements apply to variables, which serve as the primary bases in expressions, modifying their scaling and repetition. For example, in the expression $3x^2 + 2xy, the coefficient 3 multiplies x^2 while the exponent 2 indicates the degree of x, and similarly, 2 is the coefficient of xy with exponents 1 for both variables (implicit for the single x).^[2]

Types of Expressions

Monomials and Polynomials

A monomial is an algebraic expression consisting of a single term formed by the product of a coefficient and zero or more variables raised to non-negative integer powers, such as $5x^2 y^3.^[24] The degree of a monomial is the sum of the exponents of its variables; for $5x^2 y^3, this degree is $2 + 3 = 5.^[25] A constant term, like 7, is a monomial of degree 0.^[26] A polynomial is an algebraic expression that is the sum of one or more monomials, where no variables appear in denominators and all exponents are non-negative integers, such as x^2 + 3x - 2.^[27] Polynomials are typically written in standard form, with terms arranged in descending order of degree, as in $2x^3 + x.^[28] The degree of a polynomial is the highest degree of its individual monomial terms after combining like terms; for $2x^3 + x, the degree is 3.^[29] Polynomials are classified by the number of terms: a binomial has two terms, like x^2 + 1, and a trinomial has three, like x^2 + 2x + 1.^[27] The leading coefficient of a polynomial is the coefficient of the term with the highest degree; in $3x^4 - 2x + 5, it is 3.^[28] Expanding expressions can yield polynomials; for instance, (x + 1)^2 = x^2 + 2x + 1, a trinomial of degree 2.^[25]

Rational Expressions

A rational expression is defined as the quotient of two polynomials, where the numerator and denominator are both polynomial expressions.^[30] For instance, the expression \frac{x^2 + 1}{x - 2} represents a rational expression, with x^2 + 1 as the numerator polynomial and x - 2 as the denominator polynomial.^[31] These expressions extend the concept of polynomials, which are sums of terms involving variables raised to non-negative integer powers, by incorporating division between such sums.^[30] A key restriction for rational expressions is that the denominator must not equal zero, as division by zero is undefined. In the example \frac{x^2 + 1}{x - 2}, this implies x \neq 2, excluding that value from the domain of the expression.^[32] This domain restriction ensures the expression is well-defined for all other real numbers where applicable. Rational expressions are classified as proper or improper based on the degrees of the numerator and denominator polynomials. A proper rational expression has a numerator degree strictly less than the denominator degree, such as \frac{x + 1}{x^2 + 2x + 1}, while an improper one has a numerator degree greater than or equal to the denominator degree, like \frac{x^2 + 3}{x + 1}.^[33] This distinction is useful in analysis and simplification processes.^[34] It is important to distinguish rational expressions from other algebraic fractions that are not rational, such as those involving radicals in the numerator or denominator, like \frac{\sqrt{x+1}}{x}, which are algebraic but not ratios of polynomials.^[30] Focusing on polynomial quotients maintains the rational nature. For example, the rational expression \frac{x^2 - 4}{x - 2} can be simplified by factoring the numerator as (x - 2)(x + 2) and canceling the common factor (x - 2), yielding x + 2 for x \neq 2.^[35] This cancellation highlights a basic property while preserving the restriction.^[30]

Operations and Simplification

Basic Arithmetic Operations

Algebraic expressions can be manipulated using the four basic arithmetic operations: addition, subtraction, multiplication, and division. These operations follow rules similar to those for numerical arithmetic but account for variables and their degrees. Addition and subtraction primarily involve combining like terms, where terms with identical variables and exponents are grouped together. For instance, the expression (2x + 3) + (x - 1) simplifies by adding the coefficients of like terms: $2x + x = 3x and $3 + (-1) = 2, resulting in $3x + 2.^[36] Subtraction works analogously, treating it as adding the negative of the subtrahend; for example, (4x^2 + 2x - 3) - (2x^2 - 5x - 3) becomes $4x^2 + 2x - 3 + (-2x^2 + 5x + 3) = 2x^2 + 7x.^[36] Multiplication of algebraic expressions relies on the distributive property, which states that a(b + c) = ab + ac. This property extends to multiplying polynomials by distributing each term of one factor across the other. A practical example is $4x(3x + 2), where $4x \cdot 3x = 12x^2 and $4x \cdot 2 = 8x, yielding $12x^2 + 8x.^[36] For binomials, the FOIL method (First, Outer, Inner, Last) provides a structured approach: in (x + 2)(x - 3), multiply the first terms (x \cdot x = x^2), outer terms (x \cdot (-3) = -3x), inner terms ($2 \cdot x = 2x), and last terms ($2 \cdot (-3) = -6), then combine like terms to get x^2 - x - 6.^[36] Division of algebraic expressions varies by form. For monomials, divide the coefficients and subtract the exponents of like variables; for example, \frac{24x^4 y^3}{3x y^2} = 8x^{4-1} y^{3-2} = 8x^3 y.^[36] When dividing a polynomial by a monomial, apply the division to each term individually, such as \frac{8a^5 - 6a^4}{2a^2} = \frac{8a^5}{2a^2} - \frac{6a^4}{2a^2} = 4a^3 - 3a^2.^[36] For polynomial division by another polynomial, long division is used, akin to numerical long division, to find quotient and remainder, though specifics depend on the degrees involved.^[36] When expressions involve multiple operations, the order of operations—often remembered as PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right)—ensures consistent evaluation. This convention applies to algebraic expressions with variables; for example, in $6a + [5a + (6 - 3a)], first resolve the innermost parentheses ($6 - 3a), then the outer brackets, and finally add the terms to obtain $8a + 6.^[37] BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) is an equivalent mnemonic used in some regions.^[37]

Simplification Techniques

Simplification of algebraic expressions involves reducing them to their simplest form by eliminating redundant terms and applying established rules, thereby making them easier to manipulate in further calculations. This process is essential in algebra to avoid unnecessary complexity and to reveal underlying structures in expressions. Techniques such as combining like terms, factoring, canceling common factors in rational expressions, and applying exponent rules are fundamental to this endeavor.^[38] Combining like terms is a primary method where terms with identical variables and exponents are identified and their coefficients are added or subtracted. For instance, in the expression $5x + 2x - 3y + y, the like terms $5x

and &#36;2x

combine to $7x, while -3yandycombine to-2y, yielding 7x - 2y$. This technique applies after performing basic arithmetic operations on polynomials, ensuring the expression contains no similar terms that can be merged further.^[39] Factoring reverses the process of multiplication by expressing an expression as a product of simpler factors, often revealing common structures. To factor out the greatest common factor, each term is divided by that factor; for example, $2x + 4 = 2(x + 2). Special factoring formulas include the difference of squares, where x^2 - 9 = (x - 3)(x + 3). These methods simplify polynomials by breaking them into manageable components.^[40] In rational expressions, simplification occurs by factoring the numerator and denominator and then canceling common factors, provided they are not zero. For example, \frac{x^2 - 1}{x - 1} factors as \frac{(x + 1)(x - 1)}{x - 1}, and canceling (x - 1) (with x \neq 1) results in x + 1. This reduction ensures the expression is in lowest terms, preventing errors in subsequent operations.^[41] Exponent rules facilitate simplification when multiplying or dividing powers with the same base. The product rule states that x^a \cdot x^b = x^{a+b}, allowing terms like x^3 \cdot x^2 to simplify to x^5. Similarly, the quotient rule gives x^a / x^b = x^{a-b}, as in x^5 / x^2 = x^3. These rules apply to both integer and rational exponents in algebraic contexts.^[42] A notable example in radical simplification is \sqrt{x^2} = |x| for real numbers x, where the principal (non-negative) square root requires the absolute value to ensure the result is non-negative regardless of the sign of x. This highlights the importance of domain considerations in simplification.^[43]

Applications in Mathematics

Role in Equations and Inequalities

Algebraic expressions serve as the fundamental components of equations, where two such expressions are set equal to each other to form statements like $2x + 3 = 7, allowing for the determination of variable values that satisfy the equality.^[44] Solving these equations typically involves applying inverse operations to isolate the variable, such as subtracting 3 from both sides and then dividing by 2 in the example above to find x = 2.^[45] This process relies on the structure of the expressions to maintain balance and equivalence throughout manipulations. In the context of linear equations, which involve expressions of the form ax + b = c where a, b, and c are constants and a \neq 0, solutions are straightforward and yield a single value for x.^[44] For instance, solving $4x - 2 = 10 requires adding 2 to both sides to get $4x = 12, followed by dividing by 4, resulting in x = 3; verification confirms that substituting x = 3 back into the original expressions yields equality on both sides.^[44] Quadratic equations, built from expressions like ax^2 + bx + c = 0 with a \neq 0, introduce a squared term and may have zero, one, or two real solutions, requiring techniques such as factoring or the quadratic formula for resolution.^[46] Algebraic expressions also underpin inequalities, where relational symbols like <, >, \leq, or \geq connect them, as in x - 1 > 0, which simplifies to x > 1 by adding 1 to both sides.^[47] Unlike equations, solving inequalities demands attention to the direction of the inequality sign when multiplying or dividing by negative numbers to preserve the truth of the statement.^[48] A key operation involving algebraic expressions is evaluation through substitution, where specific values replace variables to compute numerical results; for example, substituting x = 2 into $3x^2 - x gives $3(2)^2 - 2 = 3(4) - 2 = 10.^[49] This method is essential for verifying solutions in equations and inequalities, as well as for modeling real-world scenarios prior to solving. Simplification techniques, such as combining like terms, often precede evaluation or solving to streamline the expressions.^[45]

Connection to Polynomial Roots

In the context of algebraic expressions, particularly polynomials, a root is a value r that makes the polynomial evaluate to zero when substituted for the variable, satisfying p(r) = 0.^[50] For instance, the quadratic polynomial x^2 - 5x + 6 has roots x = 2 and x = 3, as it factors into (x - 2)(x - 3), and substituting these values yields zero.^[51] The factor theorem establishes a direct connection between roots and factorization: if r is a root of a polynomial p(x), then x - r is a factor of p(x).^[52] This theorem follows from the polynomial remainder theorem, which states that the remainder of dividing p(x) by x - r is p(r); if p(r) = 0, the remainder is zero, implying exact division.^[53] To apply this for higher-degree polynomials, synthetic division provides an efficient method to factor by dividing by linear terms. For a cubic polynomial like x^3 + 2x^2 - 5x - 6 with suspected root r = 2, synthetic division yields:

\begin{array}{r|r} 2 & 1 & 2 & -5 & -6 \\ & & 2 & 8 & 6 \\ \hline & 1 & 4 & 3 & 0 \\ \end{array}

The quotient x^2 + 4x + 3 confirms x - 2 as a factor, with no remainder.^[54] The rational root theorem aids in identifying possible rational roots for polynomials with integer coefficients: any rational root, expressed in lowest terms p/q, has p as a factor of the constant term and q as a factor of the leading coefficient.^[55] This narrows testing to a finite list, such as for $2x^3 - 3x^2 + 4x - 1, where possible roots are \pm1, \pm1/2.^[56] Polynomials may have multiple roots, where a root r has multiplicity k > 1 if (x - r)^k divides the polynomial but (x - r)^{k+1} does not.^[50] For example, (x - 1)^2 = x^2 - 2x + 1 has root x = 1 with multiplicity 2, as the factor appears squared.^[51] Multiplicity affects the graph's behavior, such as touching the x-axis without crossing for even values. As an illustration, the cubic polynomial x^3 - 6x^2 + 11x - 6 factors completely as (x - 1)(x - 2)(x - 3), with roots 1, 2, and 3, each of multiplicity 1.^[57] Using the rational root theorem, possible candidates include \pm1, \pm2, \pm3, \pm6; testing confirms these three as the roots.^[58]

Comparisons with Other Expressions

Algebraic vs. Arithmetic Expressions

Arithmetic expressions consist solely of numbers and arithmetic operations, such as addition, subtraction, multiplication, and division, and can be evaluated directly to yield a numerical result. For instance, the expression $2 + 3 \times 4 follows the order of operations to compute $2 + 12 = 14.^[59]^[6] In contrast, algebraic expressions incorporate variables alongside numbers and operations, allowing for generalization beyond specific numerical cases. The key difference lies in this symbolic representation: while arithmetic expressions are concrete and fixed, algebraic ones use placeholders like a, b, or c to represent unknown or variable quantities, enabling broader applicability. For example, a + b \times c generalizes the arithmetic computation by allowing substitution of any values for a, b, and c.^[60]^[6] Evaluation processes further highlight the distinction: arithmetic expressions always simplify to a single number through direct computation, whereas algebraic expressions are typically simplified by combining like terms or factoring but retain variables until specific values are substituted, at which point they yield a numerical result. Consider the arithmetic expression $5 + 7, which evaluates immediately to 12; the algebraic counterpart $5 + x remains symbolic until x = 7 is substituted, resulting in 12.^[60]^[6] This shift from arithmetic to algebraic methods traces back to ancient civilizations, where texts from Egypt and Babylon emphasized numerical computations and geometric problem-solving without symbolic notation, treating mathematics as practical arithmetic for specific instances. The introduction of algebraic symbolism emerged during the Renaissance, particularly through François Viète's work in 1591, which used letters systematically to denote unknowns and constants, transforming mathematics from verbal, number-bound descriptions to a more abstract, symbolic framework.^[61]^[62]

Algebraic vs. Other Mathematical Forms

Algebraic expressions are fundamentally composed of finite combinations of variables, constants, and algebraic operations—including addition, subtraction, multiplication, division, exponents, and roots—such as polynomials, rational expressions, or radical expressions. In contrast, transcendental expressions incorporate functions that cannot be captured by such finite algebraic forms, including trigonometric functions like \sin(x), exponential functions like e^x, and logarithmic functions. These transcendental functions "transcend" algebra by not being solvable through polynomial equations of finite degree.^[63]^[64] Geometric expressions often blend algebraic elements with transcendental constants or functions, distinguishing them from purely algebraic forms. For instance, the area of a circle given by \pi r^2 relies on the transcendental constant \pi, which is not the root of any non-zero polynomial with rational coefficients, rendering the expression non-algebraic overall. This integration of transcendental elements in geometric contexts highlights how algebraic expressions are limited to rational or root-based constructions, while geometric ones frequently require irrational or transcendental values for precision.^[65] Logical expressions, such as Boolean forms like A \land B or A \lor B, operate within a different framework using connectives that model truth values rather than numerical magnitudes, diverging from the arithmetic operators central to algebraic expressions. Although Boolean algebra provides a structured way to manipulate these logical forms analogous to numerical algebra, the operators (AND, OR, NOT) do not align with algebraic addition or multiplication, emphasizing the discrete, truth-based nature of logic over the continuous, quantitative focus of algebra.^[66] A key limitation of algebraic expressions lies in their closure properties: the set of algebraic numbers is closed under addition, subtraction, multiplication, and division, meaning operations between algebraic numbers yield another algebraic number, as seen in the irrational \sqrt{2} remaining algebraic despite not being rational. Transcendental or other non-algebraic forms lack this closure; for example, sums or products involving transcendentals like \pi or e typically remain transcendental. This closure ensures algebraic expressions maintain solvability within polynomial frameworks, whereas others may introduce irresolvable complexities.^[67] To illustrate these differences, consider the algebraic equation x^2 + 1 = 0, which has solutions in the complex numbers (x = \pm i), extending the real algebraic domain predictably. In contrast, the transcendental equation e^x = 0 has no real solutions, as the exponential function is strictly positive for all real x, demonstrating how transcendental expressions can evade solutions even in extended domains where algebraic ones succeed.^[68]^[69]

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### Summary of Order of Operations (PEMDAS/BODMAS) and Basic Operations on Expressions
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[PDF] Simplifying Variable Expressions by Combining Like Terms
To combine like terms, we must first identify those terms which are alike. We then combine them by adding the numerical coefficients. The variable part stays ...
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Appendix G: Algebra Review – Physics 131 - Open Books
We can simplify an expression by combining the like terms. What do you think 3x+6x would simplify to? If you thought 9x , you would be right!
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[PDF] Factoring Methods
Factoring methods include finding the greatest common factor, grouping, slide and divide, difference of perfect squares, sum and difference of cubes, and ...
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[PDF] MODULE 3 - RATIONAL EXPRESSIONS Simplifying Rational ...
Given a rational expression, simplify it. 1. Factor the numerator and denominator. 2. Cancel any common factors. Example 1.
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MFG Exponents
x−n=1xn for x≠0. x − n = 1 x n for x ≠ 0. An expression is completely simplified if it does not contain any negative exponents.
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2.4 - Solving Equations Algebraically
The square root of both sides is then taken. Remember that the square root of x2 is the absolute value of x. When you solve an equation involving an ...
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Algebra - Solving Equations and Inequalities
Jun 6, 2018 · We will cover a wide variety of solving topics in this chapter that should cover most of the basic equations/inequalities/techniques that are involved in ...Missing: explanation | Show results with:explanation
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[PDF] Linear equations and inequalities in one variable
In the last chapter, we considered algebraic expressions: expressions formed by combining numbers and variables with the operations of addition, subtraction ...
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Quadratic Equations - College Algebra - West Texas A&M University
Dec 17, 2009 · If a quadratic equation factors, it will factor into either one linear factor squared or two distinct linear factors. So, the equations found ...
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Solving Equations and Inequalities - UTSA
Nov 13, 2021 · Solving Inequalities. Solving algebraic inequalities is more or less identical to solving algebraic equations. Consider the following example ...Missing: explanation | Show results with:explanation
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ORCCA Modeling with Equations and Inequalities
Subsection 1.8. 4 Translating Phrases into Algebraic Expressions and Equations/Inequalities. Void of context, there are certain short phrases and expressions ...
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[PDF] Evaluating Algebraic Expressions - Palm Beach State College
Variable terms have two parts – a numerical part (the number), called the coefficient, and a literal part (the letter or variable). The term 3y is read "3 times ...
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Roots of Polynomials
The multiplicity of a root of a polynomial is the number of times it is repeated. From the formula, we can tell the multiplicity by the power on the factor. 🔗 ...
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Algebra - Zeroes/Roots of Polynomials - Pauls Online Math Notes
Nov 16, 2022 · If r r is a zero of a polynomial and the exponent on the term that produced the root is k k then we say that r r has multiplicity k k . Zeroes ...
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[PDF] Factoring Polynomials
Jun 19, 2006 · Corollary 2 Factor Theorem: The number a is a root of f if and only if x − a is a factor of f(x). Proof: The number a is a root of f if and ...
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[PDF] Some Polynomial Theorems
Factor Theorem is a factor of the polynomial if and only if. %& = ;'. $%&'. $%;' - *3. Factor Theorem Proof: Assume is a factor of . Then we know divides evenly ...
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Synthetic Division and the Remainder and Factor Theorems
Mar 15, 2012 · Another use is finding factors and zeros. The Factor Theorem states that if the functional value is 0 at some value c, then x - c is a factor ...Missing: definition | Show results with:definition
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[PDF] R-VI. Polynomials
Jul 1, 2004 · Rational Root Theorem Let p(x) = anxn +an−1xn−1 +···+a0 be a polynomial with integer coefficients. Then any rational solution r/s (expressed in ...
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Algebra - Finding Zeroes of Polynomials - Pauls Online Math Notes
Nov 16, 2022 · Process for Finding Rational Zeroes. Use the rational root theorem to list all possible rational zeroes of the polynomial P(x) P ( x ) .
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[PDF] Recurrence Relations
The characteristic equation is x3 − 6x2 + 11x − 6 = 0. The roots are x = 1,2,3, which are distinct, so the general solution is: an = α1(1)n + α2(2)n + α3(3)n.
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Rational Zero Theorem and Descartes' Rule of
Mar 15, 2012 · Rational Zero (or Root) Theorem ... We can use this theorem to help us find all of the POSSIBLE rational zeros or roots of a polynomial function.
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[PDF] Notes on Proving Arithmetic Equations 1 Expressions and Values
Feb 25, 2003 · Definition 1.1. Arithmetic expressions (ae's) are defined inductively as follows: • The numerals 0 and 1 are ae's. • Any variable, x, is an ...
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[PDF] From arithmetic to algebra - UC Berkeley math
Feb 20, 2009 · (A) Arithmetic is about computation of specific numbers. Algebra is about what is true in general for all numbers, all whole numbers, all ...
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Genres of Math: Arithmetic, Algebra, and Algorithms in Ancient ...
Feb 17, 2020 · The process of abstraction manipulated the geometry or arithmetic of ancient math into algebra, a way of examining mathematical problems that ...
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Francois Viète: Father of Modern Algebraic Notation
He was "the first to apply algebraic transformation to trigonometry, particularly to the multisection of angles" [2, p. 138]. He established "the earliest ...
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[PDF] Transcendental Functions
Transcendental functions are useful functions other than algebraic ones, including trigonometric, inverse trigonometric, exponential, and logarithmic functions.
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Basic Classes of Functions
These functions are known as transcendental functions because they are said to “transcend,” or go beyond, algebra.
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[PDF] The Life of π - MIT Mathematics
• 1882: Lindemann proves π is transcendental – π can't be expressed as the solution to an algebraic equation. Page 23. You can't square a circle. • Given a ...
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1.1 Logical Operations - Whitman College
Complicated sentences and formulas are put together from simpler ones, using a small number of logical operations.<|control11|><|separator|>
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[PDF] 1 Basic facts - Berkeley Math Circle
1. The set of algebraic numbers is closed under addition, subtraction, multiplication and division. The set of algebraic integers is closed under addition, ...
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[PDF] Complex Numbers and the Complex Exponential
The equation x2 + 1 = 0 has no solutions, because for any real number x the square x2 is nonnegative, and so x2 + 1 can never be less than 1.
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Functions and Their Inverses - Worked Examples
(x)(ex)(3+x)=0⇒x=0,ex=0,or3+x=0. But ex≠0 for any x∈R. Consequently, the second equation yields no solution. Therefore, our only solutions are x=0 and x=−3. 3.