Download Coefficient Term Polynomial Monomial Variable

Resources cited from http://www.mathwords.com
Coefficient
The number multiplied times a product of variables or powers of variables in a term.
For example, 123 is the coefficient in the term 123x3y.
Term
Parts of an expression or series separated by + or – signs, or the parts of a sequence
separated by commas.
Expression
Terms
5a3 – 2xy + 3
5a3, 2xy, and 3
p, 2q, a2, and b
Polynomial
The sum or difference of terms which have variables raised to positive integer powers
and which have coefficients that may be real or complex.
The following are all polynomials: 5x3 – 2x2 + x – 13, x2y3 + xy, and (1 + i)a2 + ib2.
Note: Even though the prefix poly- means many, we use the word polynomial to refer to
polynomials with 1 term (monomials), 2 terms (binomials), 3 terms, (trinomials), etc.
Standard form for a polynomial in one variable:
anxn + an–1xn–1 + ··· + a2x2 + a1x + a0
Monomial
A polynomial with one term. The following are all monomials: 5x3, 8, and 4xy.
Variable
A quantity that can change or that may take on different values. Variable also refers to a
letter or symbol representing such a quantity.
Resources cited from http://www.mathwords.com
nth Root
The number that must be multiplied times itself n times to equal a given value. The nth
root of x is written
or
. For example,
since 25 = 32.
Notes: When n = 2 an nth root is called a square root. Also, if n is even and x is
negative, then is nonreal.
Nonreal Numbers
The complex numbers that are not real. That is, the complex numbers with a nontrivial
imaginary part.
For example, 3 + 2i is nonreal, 2i is nonreal, but 3 is real.
Real Numbers
All numbers on the number line. This includes (but is not limited to) positives and
negatives, integers and rational numbers, square roots, cube roots , π (pi), etc. Real
numbers are indicated by either or .
Integers
All positive and negative whole numbers (including zero). That is, the set {... , –3, –2, –
1, 0, 1, 2, 3, ...}. Integers are indicated by either or J.
Resources cited from http://www.mathwords.com
Rational Numbers
All positive and negative fractions, including integers and so-called improper fractions.
Formally, rational numbers are the set of all real numbers that can be written as a ratio
of integers with nonzero denominator. Rational numbers are indicated by the symbol
.
Note: Real numbers that aren't rational are called irrational.
Resources cited from http://www.mathwords.com
Irrational Numbers
Real numbers that are not rational. Irrational numbers include numbers such as
,
,
, π, e, etc.
Algebraic Numbers
Real numbers that can occur as roots of polynomial equations that have integer
coefficients. For example, all rational numbers are algebraic. So are all surds such as
, as well as numbers built from surds such as
.
Note: Real numbers which are not algebraic are known as transcendental numbers.
Resources cited from http://www.mathwords.com
Transcendental Numbers
Real numbers that are not algebraic. That is, real numbers that cannot be a root of a
polynomial equation with integer coefficients. e and π are transcendental.
Resources cited from http://www.mathwords.com
Imaginary Numbers
Pure Imaginary Numbers
Complex numbers with no real part, such as 5i.
Complex Numbers
Numbers like 3 – 2i or
that can be written as the sum or difference of a real
number and an imaginary number. Complex numbers are indicated by the symbol
.
Note: All real numbers and all pure imaginary numbers are complex. Sometimes,
however, mathematicians use the phrase complex numbers to refer strictly to numbers
which have both nonzero real parts and nonzero imaginary parts.
Resources cited from http://www.mathwords.com
Difference
The result of subtracting two numbers or expressions. For example, the difference
between 7 and 12 is 12 – 7, which equals 5.
Expression
Any mathematical calculation or formula combining numbers and/or variables using
sums, differences, products, quotients (including fractions), exponents, roots,
logarithms, trig functions, parentheses, brackets, functions, or other mathematical
operations. Expressions may not contain the equal sign (=) or any type of inequality.
Examples:
Quotient
The result of dividing two numbers or expressions. For example, the 40 divided by 5
has a quotient of 8.
Note: 43 divided by 5 has a quotient of 8 and a remainder of 3.
Remainder
The part left over after long division.
Equation
A mathematical sentence built from expressions using one or more equal signs (=).
Examples:
Resources cited from http://www.mathwords.com
Properties of Equality
Equation Rules
Equivalence properties and algebra rules for manipulating equations are listed below.
Definitions
1. a = b means a is equal to b.
2. a ≠ b means a does not equal b.
Operations
1. Addition: If a = b then a + c = b + c.
2. Subtraction: If a = b then a – c = b– c.
3. Multiplication: If a = b then ac = bc.
4. Division: If a = b and c ≠ 0 then a/c = b/c.
Reflexive
Property
a=a
Symmetric Property
If a = b then b = a.
Transitive Property
If a = b and b = c then a = c.
Compound Fraction
Complex Fraction
A fraction which has, as part of its numerator and/or denominator, at least one other
fraction.
Examples:
1.
2.
Resources cited from http://www.mathwords.com
Solution
Solution Set
Any and all value(s) of the variable(s) that satisfies an equation, inequality, system of
equations, or system of inequalities.
With a system of equations or system of inequalities, the solution set is the set
containing value(s) of the variable(s) that satisfy all equations and/or inequalities in the
system.
Exponent
x in the expression ax. For example, 3 is the exponent in 23.
Base in an Exponential Expression
a in the expression ax. For example, 2 is the base in 23. Similar to the base of a
logarithm.
Exponent Rules
Algebra rules and formulas for exponents are listed below.
Definitions
1. an = a·a·a···a (n times)
2. a0 = 1 (a ≠ 0)
3.
4.
(a ≠ 0)
(a ≥ 0, m ≥ 0, n > 0)
Resources cited from http://www.mathwords.com
Combining
1. multiplication: axay = ax + y
2. division:
(a ≠ 0)
3. powers: (ax)y = axy
Distributing (a ≥ 0, b ≥ 0)
1. (ab)x = axbx
2.
(b ≠ 0)
Careful!!
1. (a + b)n ≠ an + bn
2. (a – b)n ≠ an – bn
Compute
To figure out or evaluate. For example, "compute 2 + 3" means to figure out that the
answer is 5.
Resources cited from http://www.mathwords.com