Home > Polynomial and Rational Functions > Study Guide

Chapter 3: Polynomial and Rational Functions Study Guide

Section 3.1
Hints:

1.   The graph of a quadratic equation () is called a parabola.  When graphing a quadratic equation, the most important point is called the vertex.  The vertex  is where the parabola "turns" or changes direction.  

2.  The standard form of a quadratic function is .  In this form, the vertex is (h, k).

3.  The standard form also helps you determine the direction the parabola opens.  If a>0, the parabola opens up, and the vertex will be a minimum point.  If a<0, the parabola opens down, and the vertex will be a maximum point.   

4.  Parabolas are symmetrical to the line x=h.  This line is called the axis of symmetry.

        Ex.:  ; vertex (1, 1); x = 1 is the axis of symmetry.

5.  The set of steps on page 282  will help you graph a parabola in standard form.

6.  If the parabola is not in the standard form, but in the form, , then the formula below is used to find the vertex.  a still indicates the direction that the parabola opens.

Formula for vertex: 

7.  When working application problems, the x-coordinate of the vertex is the location of the maximum or minimum value of the function.  If a>0, then f has a minimum value that occurs at .  If a<0, then f has a maximum value that occurs at .

Additional Exercises:
1.	Q:  Graph the parabola in standard form.       Click Here for Answer
2.	Q:  Find the vertex of the parabola.       Click Here for Answer
3.	Q:  Graph the quadratic function.      Click Here for Answer
4.	Q:  Does the function have a minimum or maximum value?      Click Here for Answer

Section 3.2
Hints:

1.  Polynomial functions are smooth (no sharp points) and continuous (no breaks).

2.  The Lead Coefficient Test is used to determine the end behavior of a function.  The chart on page 295 gives a complete description of the test.  Here are some extra ways to remember the test.  If n is odd, think of "odd" as meaning "different".  This will help you remember that the end behaviors on the left and the right must be different from each other.  If n is even, think of "even" as meaning "the same" (For example, if two players are "even", they have the same score.) This will help you remember that the end behavior on the left and the right of the graph must be the same.  Now take it a step further.  If  , then the graph will be rising on the right.  If , then the graph will be rising on the left.  The other side of the graph follows from knowing whether n is even or odd.

3.  Odd-degree polynomial functions have graphs with opposite behavior at each end.  Even-degree polynomial functions have graphs with the same behavior at each end.

4.  Use the term "zero" when referring to a polynomial function.  Use the term "root" when referring to a polynomial equation.

5.  "Multiplicity" refers to the number of times a factor appears in a polynomial function.  Multiplicity tells you if the graph crosses the x-axis at a zero or turns around at the zero.  If r is a zero of even multiplicity, then the graph touches the x-axis and turns around at r.  If r is a zero of odd multiplicity, then the graph crosses the x-axis at r.  

6.  There is a helpful strategy for graphing polynomial functions on page 299.

Additional Exercises:
1.	Q:  Use the Lead Coefficient Test to determine the end behavior of the graph of the polynomial function.       Click Here for Answer
2.	Q:  Find the zeros of the polynomial function and using the multiplicity for each zero state whether the graph crosses the x-axis or touches the x-axis and turns around at each zero.       Click Here for Answer
3.	Q:  Graph the polynomial function.      Click Here for Answer

Section 3.3
Hints:

1. The steps for long division of polynomials are very similar to long division of numbers.  When a divisor has more than one term, here are the four steps.

        1)  Divide.

        2)  Multiply.

        3)  Subtract.

        4)  Bring down the next term.

Watch your signs when subtracting.  There is a helpful box on page 306 with the steps explained in more detail.

2.  When you are dividing polynomials, if a power of x is missing in either the dividend or the divisor, you must add that power of x with a coefficient of 0 before you divide.  This will align like terms as you divide.


3.  Synthetic division can be used if the divisor is in the form x - c.  The steps for using synthetic division are on pages 309-310.  Again, watch your signs!

4.  The Remainder Theorem allows you to use synthetic division to evaluate a function at a particular value, c.  When c is divided into the polynomial using synthetic division the answer for f(c) will be the remainder.

5.  Based on the Factor Theorem, if the remainder when x - c is divided into a polynomial is zero, then x - c is a factor of the polynomial.

Additional Exercises:
1.	Q:  Use long division to divide.       Click Here for Answer
2.	Q:  Use synthetic division to divide.       Click Here for Answer
3.	Q:   Use the Remainder Theorem to find f(-4).       Click Here for Answer

Section 3.4
Hints:

1.   The Rational Zero Theorem helps you find the possible rational zeros of a function.  

The possible rational zeros =       Factors of the constant term        

                                                  Factors of the leading coefficient   

 

You should start this process by first finding all the factors of the constant term and then finding all factors of the leading coefficient.  If you will do this on two different lines on your paper, it will help you find every combination of the constant factors over the leading coefficient factors. 

2. There are some properties of polynomial equations that can help you find their roots.  If a polynomial equation is of degree n, then counting multiple roots separately, the equation has n roots.  Also, if a + bi is a root of a polynomial equation, then its complex conjugate a - bi will also be a root.

3.  Descartes's Rule of Signs will give you more specific information about the number of real zeros a polynomial can have.  The rule is based on counting variations in sign between consecutive coefficients.  The steps for using the rule are on pages 320-321.

Additional Exercises:
1.	Q:  Use the Rational Zero Theorem to list all possible rational zeros for the given function.       Click Here for Answer
2.	Q:  Find all the solutions of the polynomial equation.       Click Here for Answer
3.	Q:  Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for the given function.      Click Here for Answer

Section 3.5
Hints:

1.  The Upper and Lower Bounds Theorem helps you rule out many of a polynomial equation's possible rational roots.  The steps for using the theorem are on page 325.


2.  The Intermediate Value Theorem tells you about the existence of real zeros and how to approximate them.  The steps for using the theorem are on pages 327-328.

 

 

 

3.  When factoring polynomials always remember that although some polynomials cannot be factored over the real numbers, they may be factorable over the imaginary numbers.

        Ex.:

  

 

Additional Exercises:
1.	Q:  Factor the polynomial completely.       Click Here for Answer
2.	Q:  Use the given root to find the solution set of the polynomial equation.       Click Here for Answer
3.	Q:  Find all the zeros of the polynomial function.      Click Here for Answer

 

Section 3.6
Hints:

1.  Rational functions are quotients of polynomials.  They look like fractions.  The domain of a rational function cannot contain any value that would make the denominator equal zero.  In order to find the domain of a rational function, you should set the denominator equal to zero and solve.  (This may require some factoring.)  The number(s) in the solution should be excluded from the domain of the rational function.

 

2.  An asymptote is a line that a curve approaches but does not touch.   A rational function may have no, one, or several  vertical asymptotes.  The graph of the rational function never intersects the vertical asymptote.  You find the vertical asymptotes the same way you find the domain of a rational function.  Set the denominator equal to zero and solve.  Any real solution will represent a vertical asymptote.  The equation of a vertical asymptote is x = a.

 

3.  The graph of a rational function may also have a horizontal asymptote.  A rational function can have at most one horizontal asymptote; and, unlike a vertical asymptote, the graph  may cross its horizontal asymptote.  The procedure for locating horizontal asymptotes is on page 341.  In general, here are the steps.

        1)  Look for the highest powers in the numerator and denominator of the rational function.

        2)  If the highest power is in the numerator, then there is no horizontal asymptote.

        3)  If the highest power is in the denominator, then the x-axis (y = 0) is the horizontal asymptote.

        4)  If the highest powers in the numerator and the denominator are the same, then the horizontal asymptote will be y = the coefficient of the highest power in the numerator / the coefficient of the highest power in the denominator.

 

4.  There is a strategy for graphing rational functions given on page 342.  Symmetry, the asymptotes, and x- and y-intercepts all play an important role in this strategy.

5.  The graph of a rational function has a slant asymptote if the degree of the numerator is one more than the degree of the denominator.  The equation of the slant asymptote is found by dividing the denominator into the numerator either by long or synthetic division.  This division will take the form

           

from the form above will be the equation of the slant asymptote.

Additional Exercises:
1.	Q:  Find the domain of the rational function.       Click Here for Answer
2.	Q:  Find all the asymptotes of the rational function.       Click Here for Answer
3.	Q:  Graph the rational function.       Click Here for Answer
4.	Q:  Find the equation of the slant asymptote of the rational function.       Click Here for Answer

 

Section 3.7
Hints:

 

1.  Variation formulas show how one quantity changes in relation to other quantities.  Quantities can vary directly, inversely, or jointly.

 

2.  Direct variation is described by an equation in the form y = kx.  The number k is called the constant of variation.  In a direct variation problem it is possible for y to vary with a power of x.

 

3.  The steps for solving a variation problem are on page 356.  In general, they are

        1)  Read the English statement and write a formula (equation) that describes it.

        2)  Find the value for k by substituting the given values into the equation and solving for k.

        3)  Substitute the value that you found for k back into the formula.

        4)  Use the equation to answer the question asked in the problem.

 

4.  Inverse variation is described by an equation in the form .  Solving an inverse variation problems requires the same steps as solving a direct variation problem.

 

5.  In combined variation direct and inverse variation occur at the same time.

 

6.  Joint variation is a variation in which a variable varies directly as the product of two or more other variables.  

7.  When solving any type of variation problem, it is very important to translate the English statement correctly.  Each variable must be in the proper position in the equation.

 

Additional Exercises:
1.	Q:  V varies jointly as x and the cube of z.  Write an equation that expresses this relationship.       Click Here for Answer
2.	Q:  P varies directly as t and inversely as s, and P=6 when t=3 and s=2.  Find k.         Click Here for Answer
3.	Q:  r varies inversely as the square root of t.  If r=18 when t=9, find r when t=36.      Click Here for Answer
4.	Q:  The cost, C, of an airplane ticket varies directly as the number of miles, M, in the trip.  A 2000 miles trip costs $500.  What is the cost of a 900 mile trip?     Click Here for Answer

Copyright © 2003 by Prentice Hall, Inc. A Pearson Company Legal Notice