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Algebra
45 flashcards
An equation is a statement of equality between two expressions, while an identity is an equation that is true for all values of the variables involved.
An equation is a mathematical statement that shows two expressions are equal using the equal sign (=). For example, 2x + 3 = 11.
A linear equation is an equation that can be written in the form ax + b = 0, where a and b are constants and a ≠ 0. For example, 3x - 2 = 0.
To solve a linear equation, isolate the variable term (containing the variable) on one side of the equation by performing inverse operations on both sides.
A quadratic equation is an equation that can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. For example, x^2 - 4x + 3 = 0.
The quadratic formula is given by: x = (-b ± √(b^2 - 4ac)) / (2a), and is used to find the roots of a quadratic equation.
An inequality is a mathematical statement that compares two expressions using inequality symbols such as <, >, ≤, or ≥. For example, 2x + 3 < 7.
A polynomial is an expression that consists of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. For example, 3x^2 + 2x - 1.
The degree of a polynomial is the highest exponent of the variable(s) in the polynomial. For example, the degree of 2x^3 - 4x^2 + 3x + 1 is 3.
A monomial is a polynomial with only one term, such as 5x^2 or -3y.
A binomial is a polynomial with two terms, such as x^2 + 3x or 5y - 2.
A trinomial is a polynomial with three terms, such as x^2 + 2x - 3 or 4y^3 - 2y^2 + 1.
An algebraic structure is a set of elements together with one or more operations defined on that set, satisfying certain axioms or rules.
A group is an algebraic structure with a set G and a binary operation * satisfying four axioms: closure, associativity, existence of an identity element, and existence of an inverse element for every element.
A ring is an algebraic structure with two binary operations, addition and multiplication, satisfying certain axioms. Examples include the integers and the polynomials.
A field is a ring where every non-zero element has a multiplicative inverse. Examples include the real numbers and the complex numbers.
The Fundamental Theorem of Algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.
Factoring is the process of rewriting a polynomial as a product of its factors.
The greatest common divisor (GCD) of two or more integers is the largest positive integer that divides each of the integers without a remainder.
The least common multiple (LCM) of two or more integers is the smallest positive integer that is divisible by each of the integers.
The Remainder Theorem states that if a polynomial P(x) is divided by (x - a), then the remainder is equal to P(a).
The Factor Theorem states that if a polynomial P(x) is divided by (x - a) and the remainder is 0, then (x - a) is a factor of P(x).
The Rational Root Theorem provides a method for finding possible rational roots of a polynomial equation with integer coefficients.
The Quadratic Formula is used to find the roots of a quadratic equation of the form ax^2 + bx + c = 0.
The Discriminant, represented by b^2 - 4ac in the Quadratic Formula, determines the nature of the roots of the quadratic equation.
A rational number is a number that can be expressed as a fraction of two integers, while an irrational number is a number that cannot be expressed as a fraction of two integers.
A linear equation is an equation where the highest exponent of the variable(s) is 1, while a non-linear equation has at least one term with a variable raised to a power greater than 1.
A consistent system of equations has at least one solution, while an inconsistent system of equations has no solution.
A dependent system of equations has at least one equation that is a combination of the other equations, while an independent system of equations has no equation that can be expressed as a combination of the others.
A homogeneous equation is an equation where all terms have the same degree, while a non-homogeneous equation has terms with different degrees.
A constant is a fixed value that does not change, while a variable is a symbol that represents a value that can change.
A coefficient is a numerical factor that multiplies a variable, while a constant is a fixed value that does not involve any variables.
A scalar is a quantity that has only magnitude, while a vector is a quantity that has both magnitude and direction.
A polynomial is an expression with variables and coefficients involving only addition, subtraction, multiplication, and non-negative integer exponents, while a rational function is a ratio of two polynomials.
A monomial is a polynomial with only one term, while a binomial is a polynomial with two terms.
A function is a relation that assigns exactly one output value to each input value, while an equation is a mathematical statement that shows two expressions are equal.
A linear function is a function whose graph is a straight line, while a quadratic function is a function whose graph is a parabola.
The domain of a function is the set of input values for which the function is defined, while the range is the set of output values that the function produces.
A root of a polynomial equation is a value of the variable that makes the equation true, while a factor is an expression that, when multiplied by another expression, gives the original polynomial.
A real number is a number that can be represented on the number line, while a complex number is a number that has both a real and an imaginary part.
A rational exponent is an exponent that can be expressed as a fraction of two integers, while an irrational exponent is an exponent that cannot be expressed as a fraction of two integers.
An algebraic expression is a combination of variables, constants, and operations, while an algebraic equation is a mathematical statement that shows two expressions are equal.
A symmetric equation is an equation that remains unchanged when the variables are interchanged, while an asymmetric equation is an equation that changes when the variables are interchanged.
A system of linear equations consists of two or more linear equations with the same variables, while a system of non-linear equations consists of two or more non-linear equations with the same variables.
Partial fraction decomposition is a method of breaking down a rational function into a sum of simpler fractions, while polynomial long division is a method of dividing one polynomial by another.