1.8.1: Key Terms

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Key Terms

algebraic expression: constants and variables combined using addition, subtraction, multiplication, and division

associative property of addition: the sum of three numbers may be grouped differently without affecting the result; in symbols, $a + (b + c) = (a + b) + c$

associative property of multiplication: the product of three numbers may be grouped differently without affecting the result; in symbols, $a \cdot (b \cdot c) = (a \cdot b) \cdot c$

base: in exponential notation, the expression that is being multiplied

binomial: a polynomial containing two terms

coefficient: any real number $a_{i}$ in a polynomial in the form $a_{n} x^{n} + ... + a_{2} x^{2} + a_{1} x + a_{0}$

commutative property of addition: two numbers may be added in either order without affecting the result; in symbols, $a + b = b + a$

commutative property of multiplication: two numbers may be multiplied in any order without affecting the result; in symbols, $a \cdot b = b \cdot a$

constant: a quantity that does not change value

degree: the highest power of the variable that occurs in a polynomial

difference of squares: the binomial that results when a binomial is multiplied by a binomial with the same terms, but the opposite sign

distributive property: the product of a factor times a sum is the sum of the factor times each term in the sum; in symbols, $a \cdot (b + c) = a \cdot b + a \cdot c$

equation: a mathematical statement indicating that two expressions are equal

exponent: in exponential notation, the raised number or variable that indicates how many times the base is being multiplied

exponential notation: a shorthand method of writing products of the same factor

factor by grouping: a method for factoring a trinomial in the form $a x^{2} + b x + c$ by dividing the x term into the sum of two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire expression

formula: an equation expressing a relationship between constant and variable quantities

greatest common factor: the largest polynomial that divides evenly into each polynomial

identity property of addition: there is a unique number, called the additive identity, 0, which, when added to a number, results in the original number; in symbols, $a + 0 = a$

identity property of multiplication: there is a unique number, called the multiplicative identity, 1, which, when multiplied by a number, results in the original number; in symbols, $a \cdot 1 = a$

index: the number above the radical sign indicating the nth root

integers: the set consisting of the natural numbers, their opposites, and 0: ${\dots, −3, −2, −1, 0, 1, 2, 3,\dots}$

inverse property of addition: for every real number $a,$ there is a unique number, called the additive inverse (or opposite), denoted $- a,$ which, when added to the original number, results in the additive identity, 0; in symbols, $a + (- a) = 0$

inverse property of multiplication: for every non-zero real number $a,$ there is a unique number, called the multiplicative inverse (or reciprocal), denoted $\frac{1}{a},$ which, when multiplied by the original number, results in the multiplicative identity, 1; in symbols, $a \cdot \frac{1}{a} = 1$

irrational numbers: the set of all numbers that are not rational; they cannot be written as either a terminating or repeating decimal; they cannot be expressed as a fraction of two integers

leading coefficient: the coefficient of the leading term

leading term: the term containing the highest degree

least common denominator: the smallest multiple that two denominators have in common

monomial: a polynomial containing one term

natural numbers: the set of counting numbers: ${1, 2, 3,\dots}$

order of operations: a set of rules governing how mathematical expressions are to be evaluated, assigning priorities to operations

perfect square trinomial: the trinomial that results when a binomial is squared

polynomial: a sum of terms each consisting of a variable raised to a nonnegative integer power

principal nth root: the number with the same sign as $a$ that when raised to the nth power equals $a$

principal square root: the nonnegative square root of a number $a$ that, when multiplied by itself, equals $a$

radical: the symbol used to indicate a root

radical expression: an expression containing a radical symbol

radicand: the number under the radical symbol

rational expression: the quotient of two polynomial expressions

rational numbers: the set of all numbers of the form $\frac{m}{n},$ where $m$ and $n$ are integers and $n \neq 0.$ Any rational number may be written as a fraction or a terminating or repeating decimal.

real number line: a horizontal line used to represent the real numbers. An arbitrary fixed point is chosen to represent 0; positive numbers lie to the right of 0 and negative numbers to the left.

real numbers: the sets of rational numbers and irrational numbers taken together

scientific notation: a shorthand notation for writing very large or very small numbers in the form $a \times 10^{n}$ where $1 \leq | a | < 10$ and $n$ is an integer

term of a polynomial: any $a_{i} x^{i}$ of a polynomial in the form $a_{n} x^{n} + ... + a_{2} x^{2} + a_{1} x + a_{0}$

trinomial: a polynomial containing three terms

variable: a quantity that may change value

whole numbers: the set consisting of 0 plus the natural numbers: ${0, 1, 2, 3,\dots}$

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