Flashcards in Ch.5 - 8 Vocabulary Deck (80)

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## Polynomial in one variable

### An expression of the form an x^n + an-1 x^n-1 + ... + a2 x^2 + a1 x + a0, where an is not 0, all the coefficients are real numbers, and n is a nonnegative integer.

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## Leading coefficient

### The coefficient of the first term of a polynomial in standard form.

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## Polynomial function

### A continuous function that can be described by a polynomial in one variable.

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## Power function

### The simplest form of a polynomia function in the form of f(x)= ax^b, where a and b are nonzero real numbers.

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## Quartic function

### A polynomial function with the degree of four.

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## Quintic function

### A polynomial function with the degree of five.

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## End behavior

### The behavior of the graph of f(x) as x approaches positive or negative infinity, determined by the degree and leading coefficient of the function.

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## Location principle

### If the value of f(x) changes signs from one value of x to the next, then there is a zero between those two values.

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## Relative maximum

### A point with no other nearby points that have a greater y-coordinante.

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## Relative minimum

### A point with no other nearby points that have a lesser y-coordinate.

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## Extrema

### The maximum and minimum values of a function.

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## Turning point

### A point that is a relative maximum or minimum of the graph in which the graph changes direction.

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## Prime polynomial

### A polynomial that cannot be factored.

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## Quadratic form

### au^2+bu+c, which a polynomial in x could be rewritten as.

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## Sum of two cubes (formula)

### a^3 + b^3 = (a+b)(a^2-ab+b^2)

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## Difference of two tubes (formula)

### a^2-b^3 = (a-b) (a^2+ab+b^2)

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## Synthetic substitution

### Applying the Remainder Theorem using synthetic division to evaluate a function.

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## Depressed polynomial

### The quotient after dividing a polynomial by a binomial, which would have a degree one less than the original polynomial.

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## Remainder theorem

### If a polynomial is divided by x-r, the remainder isa constant P(r).

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## Factor theorem

### If the binomial x-r is a factor of the polynomial P(x) if and only if P(r)=0.

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## The fundamental theorem of algebra

### Every polynomial equation with degree greater than zero has at least one root in the set of complex numbers.

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## Corollary to the fundamental theorem of algebra

### A polynomial equation of degree n has exactly n roots in the set of complex numbers, including repeated roots.

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## Descartes' rule of signs

###
The number of positive real zeros of P(x) is the same as the number of changes in sign of the coefficient of the terms, or is less than this by an even number.

The number of negative real zeros of P(x) is that in terms of P(-x).

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## Complex conjugates theorem

### Let a and b be real numbers and b is not 0. If a+bi is a zero of a polynomial function with real coefficients, then a-bi is also a zero of the function.

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## Rational zero theorem

### If P(x) is a polynomial function with integral coefficients, then every rational zero of P(x) = 0 is of the form p/q, a rational number in simplest form, where p is a factor of the constant term and q is a factor of the leading coefficient.

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## Composition of functions

### The results of one function are used to evaluate a second function.

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## Inverse relation

### The set of ordered pair obtained by exchanging the coordinates of each ordered pair.

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## Inverse function

### The inverse function of f(x) is written as f-1(x), f(a) = b if and only if f-1(b) = a.

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## Square root function

### A function that contains the square root of a variable.

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## Radical function

### A function in which the variable is under the radical sign.sign.

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## Square root inequality

### An inequality involving square roots of a variable.

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## Radicand

### The number under the radical sign

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## Index

### For nth root (the inverse of raising a number to the nth power), the index is n.

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## Primary root

### The nonnegative root when there is more than one real root and the index is even.

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## Rationalizing the denominator

### Multiplying the numerator and denominator by a quantity so that the radicand of the denominator has an exact root to eliminate radicals from the denominator.

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## Like radical expressions

### Radicals with the same index and radicand.

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## Conjugates

### a√b + c √d and a√b - c √d are conjugates if a,b,c,d are all rational numbers.

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## Radical equation

### An equation that include radical expressions, can be solved by raising each side of the equation to a power.

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## Extraneous solution

### The result of an equation that does not satisfy the original equation.

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## Radical inequality

### An inequality that has a variable in the radicand.

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## Exponentail function

### A function where the base is a constant and the exponent is the independent variable.

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## Exponential growth

### A function of the form f(x)=b^x, where b > 1.

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## Asymptote

### A line that the graph of the function approaches.

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## Growth factor

### The base of the exponential function, 1+r.

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## Exponential decay

### A function of the form f(x) = b^x, where 0

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## Decay factor

### The base of the exponential function, 1-r.

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## Exponential equation

### An equation in which variables occur as exponents.

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## Compound interest

### Interest paid on the principal of an investment and any previously earned interest.

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## Exponential inquality

### An inequality involving exponential functions.

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## Logarithm

### X=b^y, the variable y is the logarithm of x.

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## Logarithmic function

### A function in the form y=logbx, where b is not 1.

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## Logarithmic equation

### Contains one or more logarithms.

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## Logarithmic inequality

### An inequality that involves logarithms.

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## Product property of logarithms

### Logx(ab) = logx(a)+logx(b)

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## Quotient property of logarithms

### Logx(a/b) = logx(a)-logx(b)

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## Power property of logarithms

### Logb(m^p) = p logb(m)

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## Common logarithms

### Base 10 logarithms.

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## Change of base formula

### Loga(n) = logb(n) / logb(a)

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## Natural base

### An irrational number with the value of 2.71828

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## Natural base exponential function

### An exponential function with base e.

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## Natural logarithm

### The inverse of a natural base exponential function, often abbreviated as ln.

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## Rate of continuous growth

###
Represented as k in an exponential growth function,

f(t) = ae^kt

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## Rate of continuous decay

###
Represented as k in an exponential decay function,

f(t) = ae^-kt

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## Logistic growth model

###
Represents growth that has a limiting factor,

f(t) = c / (1+ae^-bt) with a,b,c all being positive constant and b

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## Rational expression

### A ratio of two polynomial expressions.

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## Complex fraction

### A rational expression with a numerator and/or denominator that is also a rational expression.

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## Reciprocal function

### A function with an equation of the form f(x) = 1/a(x), where a(x) is a linear function and does not equal to 0.

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## Hyperbola

### The shape of the graph of a reciprocal function.

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## Rational function

### A function with an equation of the form f(x) = a(x)/b(x)m where a(x) and b(x) are both polynomial functions and b(x) does not equal to 0.

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## Vertical asymptote

### b(x) = 0.

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## Horizontal asymptote

###
If the degree of a(x) is greater, no horizontal asymptote.

If the degree of a(x) is less, f(x) = 0.

If the degrees are equal, f(x) = leading coefficient of a(x) / leading coefficient of b(x).

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## Oblique asymptote

### if the degree of a(x) minus that of b(x) is 1, f(x) = a(x) / b(x) with no remainder is the oblique asymptote.

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## Direct variation

### Can be expressed in the form y=kx.

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## Constant of variation

### The value k in the equation y=kx.

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## Joint variation

### When one quantity varies directly as the product of the two or more other quantities, y=kxz.

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## Inverse variation

### When the product of two quantities is equal to a constant k, xy=k, y=k/x.

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## Combined variation

### When one quantity varies directly and/or inversely as two or more other quantities.

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## Rational equations

### Equations that contain one or more rational expressions.

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## Weighted average

### The method for finding the mean of a set of numbers in which some elements carry more importance.

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