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A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power. SeeExample 4. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient. SeeExample 5. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function. SeeExample 6andExample 7. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
A polynomial of degreennwill have at mostnnx-intercepts and at mostnâ1nâ1turning points. SeeExample 8,Example 9,Example 10,Example 11, andExample 12. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
Polynomial functions of degree 2 or more are smooth, continuous functions. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero. SeeExample 2,Example 3,andExample 4. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
Another way to find thex-x-intercepts of a polynomial function is to graph the function and identify the points at which the graph crosses thex-x-axis. SeeExample 5. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
The multiplicity of a zero determines how the graph behaves at thex-x-intercepts. SeeExample 6. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
The graph of a polynomial will cross the horizontal axis at a zero with odd multiplicity. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
The graph of a polynomial will touch the horizontal axis at a zero with even multiplicity. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
The end behavior of a polynomial function depends on the leading term. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
The graph of a polynomial function changes direction at its turning points. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
A polynomial function of degreennhas at mostnâ1nâ1turning points. SeeExample 7. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at mostnâ1nâ1turning points. SeeExample 8andExample 10. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
Graphing a polynomial function helps to estimate local and global extremas. SeeExample 11. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
The Intermediate Value Theorem tells us that iff(a)andf(b)f(a)andf(b)have opposite signs, then there exists at least one valueccbetweenaaandbbfor whichf(c)=0.f(c)=0.SeeExample 9. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree. SeeExample 1andExample 2. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
The Division Algorithm tells us that a polynomial dividend can be written as the product of the divisor and the quotient added to the remainder. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
Synthetic division is a shortcut that can be used to divide a polynomial by a binomial in the formxâk.xâk.SeeExample 3,Example 4,andExample 5. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
Polynomial division can be used to solve application problems, including area and volume. SeeExample 6. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
To findf(k),f(k),determine the remainder of the polynomialf(x)f(x)when it is divided byxâk.xâk.This is known as the Remainder Theorem. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
According to the Factor Theorem,kkis a zero off(x)f(x)if and only if(xâk)(xâk)is a factor off(x).f(x).SeeExample 2. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
According to the Rational Zero Theorem, each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient. SeeExample 3andExample 4. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
When the leading coefficient is 1, the possible rational zeros are the factors of the constant term. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
Synthetic division can be used to find the zeros of a polynomial function. SeeExample 5. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
According to the Fundamental Theorem, every polynomial function has at least one complex zero. SeeExample 6. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
Every polynomial function with degree greater than 0 has at least one complex zero. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Each factor will be in the form(xâc),(xâc),whereccis a complex number. SeeExample 7. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
The number of positive real zeros of a polynomial function is either the number of sign changes of the function or less than the number of sign changes by an even integer. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
The number of negative real zeros of a polynomial function is either the number of sign changes off(âx)f(âx)or less than the number of sign changes by an even integer. SeeExample 8. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
Polynomial equations model many real-world scenarios. Solving the equations is easiest done by synthetic division. SeeExample 9. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
We can use arrow notation to describe local behavior and end behavior of the toolkit functionsf(x)=1xf(x)=1xandf(x)=1x2.f(x)=1x2.SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
A function that levels off at a horizontal value has a horizontal asymptote. A function can have more than one vertical asymptote. SeeExample 2. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
Application problems involving rates and concentrations often involve rational functions. SeeExample 3. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
The domain of a rational function includes all real numbers except those that cause the denominator to equal zero. SeeExample 4. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
The vertical asymptotes of a rational function will occur where the denominator of the function is equal to zero and the numerator is not zero. SeeExample 5. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
A removable discontinuity might occur in the graph of a rational function if an input causes both numerator and denominator to be zero. SeeExample 6. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
A rational functionâs end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions. SeeExample 7,Example 8,Example 9, andExample 10. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
Graph rational functions by finding the intercepts, behavior at the intercepts and asymptotes, and end behavior. SeeExample 11. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
If a rational function hasx-intercepts atx=x1,x2,â¦,xn,x=x1,x2,â¦,xn,vertical asymptotes atx=v1,v2,â¦,vm,x=v1,v2,â¦,vm,and noxi=anyvj,xi=anyvj,then the function can be written in the form | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
SeeExample 12. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
The inverse of a quadratic function is a square root function. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
Iffâ1fâ1is the inverse of a functionf,f,thenffis the inverse of the functionfâ1.fâ1.SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
While it is not possible to find an inverse of most polynomial functions, some basic polynomials are invertible. SeeExample 2. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
To find the inverse of certain functions, we must restrict the function to a domain on which it will be one-to-one. SeeExample 3andExample 4. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
When finding the inverse of a radical function, we need a restriction on the domain of the answer. SeeExample 5andExample 7. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
Inverse and radical and functions can be used to solve application problems. SeeExample 6andExample 8. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
A relationship where one quantity is a constant multiplied by another quantity is called direct variation. SeeExample 1. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
Two variables that are directly proportional to one another will have a constant ratio. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
A relationship where one quantity is a constant divided by another quantity is called inverse variation. SeeExample 2. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
Two variables that are inversely proportional to one another will have a constant multiple. SeeExample 3. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
In many problems, a variable varies directly or inversely with multiple variables. We call this type of relationship joint variation. SeeExample 4. | https://openstax.org/books/college-algebra-2e/pages/5-key-concepts |
f ( x ) = a x 2 + b x + c f ( x ) = a x 2 + b x + c | https://openstax.org/books/college-algebra-2e/pages/5-key-equations |
f ( x ) = a ( x â h ) 2 + k f ( x ) = a ( x â h ) 2 + k | https://openstax.org/books/college-algebra-2e/pages/5-key-equations |
f ( x ) = a n x n + ... + a 2 x 2 + a 1 x + a 0 f ( x ) = a n x n + ... + a 2 x 2 + a 1 x + a 0 | https://openstax.org/books/college-algebra-2e/pages/5-key-equations |
f ( x ) = d ( x ) q ( x ) + r ( x ) where q ( x ) â 0 f ( x ) = d ( x ) q ( x ) + r ( x ) where q ( x ) â 0 | https://openstax.org/books/college-algebra-2e/pages/5-key-equations |
f ( x ) = P ( x ) Q ( x ) = a p x p + a p â 1 x p â 1 + ... + a 1 x + a 0 b q x q + b q â 1 x q â 1 + ... + b 1 x + b 0 , Q ( x ) â 0 f ( x ) = P ( x ) Q ( x ) = a p x p + a p â 1 x p â 1 + ... + a 1 x + a 0 b q x q + b q â 1 x q â 1 + ... + b 1 x + b 0 , Q ( x ) â 0 | https://openstax.org/books/college-algebra-2e/pages/5-key-equations |
y = k x n , k y = k x n , k is a nonzero constant. | https://openstax.org/books/college-algebra-2e/pages/5-key-equations |
y = k x n , k y = k x n , k is a nonzero constant. | https://openstax.org/books/college-algebra-2e/pages/5-key-equations |
arrow notation : a way to represent symbolically the local and end behavior of a function by using arrows to indicate that an input or output approaches a value | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
axis of symmetry : a vertical line drawn through the vertex of a parabola, that opens up or down, around which the parabola is symmetric; it is defined byx=âb2a.x=âb2a. | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
coefficient : a nonzero real number multiplied by a variable raised to an exponent | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
constant of variation : the non-zero valuekkthat helps define the relationship between variables in direct or inverse variation | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
continuous function : a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
degree : the highest power of the variable that occurs in a polynomial | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
Descartesâ Rule of Signs : a rule that determines the maximum possible numbers of positive and negative real zeros based on the number of sign changes off(x)f(x)andf(âx)f(âx) | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
direct variation : the relationship between two variables that are a constant multiple of each other; as one quantity increases, so does the other | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
Division Algorithm : given a polynomial dividendf(x)f(x)and a non-zero polynomial divisord(x)d(x)where the degree ofd(x)d(x)is less than or equal to the degree off(x)f(x), there exist unique polynomialsq(x)q(x)andr(x)r(x)such thatf(x)=d(x)q(x)+r(x)f(x)=d(x)q(x)+r(x)whereq(x)q(x)is the quotient andr(x)r(x)is the remaind... | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
end behavior : the behavior of the graph of a function as the input decreases without bound and increases without bound | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
Factor Theorem : kkis a zero of polynomial functionf(x)f(x)if and only if(xâk)(xâk)is a factor off(x)f(x) | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
Fundamental Theorem of Algebra : a polynomial function with degree greater than 0 has at least one complex zero | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
general form of a quadratic function : the function that describes a parabola, written in the formf(x)=ax2+bx+cf(x)=ax2+bx+c, wherea,b,a,b,andccare real numbers andaâ0.aâ0. | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
global maximum : highest turning point on a graph;f(a)f(a)wheref(a)â¥f(x)f(a)â¥f(x)for allx.x. | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
global minimum : lowest turning point on a graph;f(a)f(a)wheref(a)â¤f(x)f(a)â¤f(x)for allx.x. | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
horizontal asymptote : a horizontal liney=by=bwhere the graph approaches the line as the inputs increase or decrease without bound. | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
Intermediate Value Theorem : for two numbersaaandbbin the domain off,f,ifa<ba<bandf(a)âf(b),f(a)âf(b),then the functionfftakes on every value betweenf(a)f(a)andf(b)f(b); specifically, when a polynomial function changes from a negative value to a positive value, the function must cross thex-x-axis | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
inverse variation : the relationship between two variables in which the product of the variables is a constant | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
inversely proportional : a relationship where one quantity is a constant divided by the other quantity; as one quantity increases, the other decreases | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
invertible function : any function that has an inverse function | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
joint variation : a relationship where a variable varies directly or inversely with multiple variables | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
leading coefficient : the coefficient of the leading term | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
leading term : the term containing the highest power of the variable | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
Linear Factorization Theorem : allowing for multiplicities, a polynomial function will have the same number of factors as its degree, and each factor will be in the form(xâc)(xâc), whereccis a complex number | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
multiplicity : the number of times a given factor appears in the factored form of the equation of a polynomial; if a polynomial contains a factor of the form(xâh)p(xâh)p,x=hx=his a zero of multiplicityp.p. | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
polynomial function : a function that consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power. | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
power function : a function that can be represented in the formf(x)=kxpf(x)=kxpwherekkis a constant, the base is a variable, and the exponent,pp, is a constant | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
rational function : a function that can be written as the ratio of two polynomials | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
Rational Zero Theorem : the possible rational zeros of a polynomial function have the formpqpqwhereppis a factor of the constant term andqqis a factor of the leading coefficient. | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
Remainder Theorem : if a polynomialf(x)f(x)is divided byxâkxâk, then the remainder is equal to the valuef(k)f(k) | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
removable discontinuity : a single point at which a function is undefined that, if filled in, would make the function continuous; it appears as a hole on the graph of a function | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
roots : in a given function, the values ofxxat whichy=0y=0, also called zeros | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
smooth curve : a graph with no sharp corners | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
standard form of a quadratic function : the function that describes a parabola, written in the formf(x)=a(xâh)2+kf(x)=a(xâh)2+k, where(h,k)(h,k)is the vertex | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
synthetic division : a shortcut method that can be used to divide a polynomial by a binomial of the formxâkxâk | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
term of a polynomial function : anyaixiaixiof a polynomial function in the formf(x)=anxn+...+a2x2+a1x+a0f(x)=anxn+...+a2x2+a1x+a0 | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
turning point : the location at which the graph of a function changes direction | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
varies directly : a relationship where one quantity is a constant multiplied by the other quantity | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
varies inversely : a relationship where one quantity is a constant divided by the other quantity | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
vertex : the point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
vertex form of a quadratic function : another name for the standard form of a quadratic function | https://openstax.org/books/college-algebra-2e/pages/5-key-terms |
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