chapter 6 polynomials and polynomial functions. in this chapter, you will … learn to write and...

Chapter 6

Polynomials and Polynomial Functions

In this chapter, you will …

Learn to write and graph polynomial functions and to solve polynomial equations.Learn to use important theorems about the number of solutions to polynomial equations.Learn to solve problems involving permutations, combinations and binomial probability.

6-1 Polynomial Functions

What you’ll learn …To classify polynomialsTo model data with polynomial functions

2.04 Create and use best-fit mathematical models of linear, exponential, and quadratic functions to solve problems involving sets of data.

a. Interpret the constants, coefficients, and bases in the context of the data.

b. Check the model for goodness-of-fit and use the model, where appropriate, to draw conclusions or make predictions.

2.06 Use cubic equations to model and solve problems.a. Solve using tables and graphs. b. Interpret constants and coefficients in the context of the

problem.

A monomial is an expression that is a real number, a variable or a product of real numbers and variables.13,    3x,    -57,   x²,   4y²,  -2xy,  or  520x²y²                   A binomial is the sum of two monomials.  It has two unlike terms.  3x + 1,    x² - 4x,     2x + y,    or    y - y²

  A trinomial is the sum of three monomials.  It has three unlike terms. x2 + 2x + 1,     3x² - 4x + 10,      2x + 3y + 2

  A polynomial is the sum of one or more terms.  x2 + 2x,  3x3 + x² + 5x + 6, 4x - 6y + 8

• The exponent of the variable in a term determines the degree of that term.

• The terms in the polynomial are in descending order by degree.

• This order demonstrates the standard form of a polynomial.

3 2Leading Coefficient

Cubic Term

QuadraticTerm

LinearTerm

ConstantTerm

Degree Name Using Degree

Polynomial Example

Number of

Terms

Name Using Number of

Terms

0 Constant 6 1 monomial

1 Linear x + 3 2 binomial

2 Quadratic

3x2 1 monomial

3 Cubic 2x3 -5x2 -2x 3 trinomial

4 Quartic x4 + 3x2 2 binomial

5 Quintic -2x5+3x2-x+4

4 polynomial

Example 1 Classifying Polynomials

Write each in standard form and classify it by degree and number of terms.-7x + 5x4

x2 – 4x + 3x3 +2x

4x – 6x + 5

Linear Model

Quadratic Model

Cubic Model

Example 2 Comparing Models

Using a graphing calculator, determine whether a linear model, a quadratic model, or a cubic model best fits the values in the table.

x 0 5 10 15 20

y 10.1 2.8 8.1 16.0 17.8

Example p.303 #17

The data at the right indicate that the life expectancy for residents of the US has been increasing.

a. Find a quadratic model for the data set.

b. Find a cubic model for the data set.

c. Graph each model. Compare the quadratic and cubic models to determine which one is a better fit.

Year of Birth

Males Females

1970 67.1 74.7

1980 70.0 77.4

1990 71.8 78.8

2000 73.2 80.2

2010 74.5 81.3

Example p.303 #23Find a cubic model for each function. Then use your model to estimate the value of y when x=17.

X 0 2 4 6 8 10 12 14 16 18 20

Y 4.1 6 15.7 21.1 23.6 23.1 24.7 24.9 23.9 25.2 29.5

Example p.304 #31The diagram shows a cologne bottle that consists of a cylindrical base and a hemispherical top.

a. Write an expression for the cylinder’s volume.

b. Write an expression for the volume of the hemispherical top.

c. Write a polynomial to represent the total volume

r

h=10

6-2 Polynomials and Linear Factors

What you’ll learn …To analyze the factored form of a polynomialTo write a polynomial function from its zeros

1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems.

Example 1 Writing a Polynomial in Standard Form

Write the expression (x+1)(x+2)(x+3) as a polynomial in standard form.

Write the expression (x+1)(x+1)(x+2) as a polynomial in standard form.

Example 2 Writing a Polynomial in Factored Form

Write 2x3 +10x2 + 12x in factored form.

Write 3x3 - 3x2 - 36x in factored form.

Example 3 Real World Connection

Several popular models of carry-on luggage have a length 10 in. greater than their depth. To comply with airline regulations, the sum of the length, width and depth may not exceed 40 in.

a. Assume that the sum of the length, width and depth is 40 in. Graph the function relating volume V to depth x.

b. Describe a realistic domain.

c. What is the maximum possible volume of a piece of luggage? What are the corresponding dimensions of the luggage?

The maximum value in Example 3 is the greatest value of the points in a region of the graph.It is called a relative maximum.Similarly, a relative minimum is the least y-value among nearby points on a graph. Relative maximum

Value of y

x interceptsRelative minimumValue of y

Example 4 Finding Zeros of a Polynomial Function

Find the zeros of y= (x-2)(x+1)(x+3).Then graph the function.

Find the zeros of y= (x-7)(x-5)(x-3).Then graph the function.

You can reverse the process and write linear factors when you know the zeros. The relationship between the linear factors of a polynomial and the zeros of a polynomial is described by the Factor Theorem.

Factor Theorem The expression x-a is a linear factor of a

polynomial if and only if the value a is a zero of the related polynomial function.

Example 5 Writing a Polynomial Function From Its Zeros

Write a polynomial function in standard form with zeros at -2, 3, and 3.

Write a polynomial function in standard form with zeros at -4, -2, and 1.

While the polynomial function in Example 5 has three zeros, it has only two distinct zeros: -2 and 3. If a linear factor of a polynomial is repeated, then the zero is repeated. A repeated zero is called a multiple zero. A multiple zero has a multiplicity equal to the number of times the zero occurs. In example 5 the zero 3 has a multiplicity of 2.

Example 6 Finding the Multiplicity of a Zero

Find any multiple zeros of f(x)=x4 +6x3+8x2 and state the multiplicity.

Find any multiple zeros of f(x)=x3 - 4x2+4x and state the multiplicity.

Equivalent Statements about Polynomials

1. -4 is a solution of x2 +3x -4 =0.2. -4 is an x-intercept of the graph

of y= x2 +3x -4.3. -4 is a zero of y= x2 +3x -4.4. x+4 is a factor of x2 +3x -4.

6-3 Dividing Polynomials

What you’ll learn …To divide polynomials using long divisionTo divide polynomials using synthetic division

1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems.

You can use polynomial division to help find all the zeros of a polynomial function. Division of polynomials is similar to numerical long division.

8 56 217 65465 x 2x2

Example 1a Polynomial Long Division

Divide x2 +3x – 12 by x - 3

Example 1b Polynomial Long Division

Divide x2 -3x + 1 by x - 4

Example 2 Checking Factors

x2 + 6x + 8

Determine whether x+4 is a factor of each polynomial

x3 + 3x2 -6x - 7

To check (divisor) (quotient) + r = dividend

To divide by a linear factor, you can use a simplified process that is known as synthetic division. In synthetic division, you omit all variables and exponents. By reversing the sign in the divisor, you can add throughout the process instead of subtracting.

Example 3a Using Synthetic Division

Use synthetic division to divide 3x3 – 4x2 +2x – 1 by x +1

Example 3b Using Synthetic Division

Use synthetic division to divide x3 + 4x2 + x – 6 by x +1

Example 4 Real World Connection

The volume in cubic feet of a sarcophagus (excluding the cover) can be expressed as the product of its three dimensions: V(x) = x3 – 13x + 12. The length is x + 4.

a. Find linear expressions with integer coefficients for the other dimensions. Assume that the width is greater than the height.

Check for Understanding

Use synthetic division to divide x3 - 2x2 - 5x + 6 by x + 2

Remainder TheoremIf a polynomial P(x) of degree n>1 is

divided by (x-a). Where a is a constant, then the remainder is P(a).

Example 5a Evaluating a Polynomial by Synthetic Division

Use synthetic division to find P(-4) for P(x) = x4 - 5x2 + 4x + 12

Example 5b Evaluating a Polynomial by Synthetic Division

Use synthetic division to find P(-1) for P(x) = 2x4 + 6x3 – x2 - 60

6-4 Solving Polynomial Equations

What you’ll learn …To solve polynomials equations by graphingTo solve polynomials equations by factoring

1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems.

Example 1 Solving by Graphing

Solve x3 + 3x2 = x + 3

Solve x3 - 19x = -2x2 + 20

Steps Graph y1= x3 + 3x2

Graph y2= x + 3 Use the

intersection feature to find the x values at the points of intersection.

Example 3 Real World Connection

The dimensions in inches of a portable kennel can be expressed as width x, length x+7 and height x-1. The volume is 5.9 ft3. Find the portable kennel’s dimensions.

Sometimes you can solve polynomial equations by factoring the polynomial and using the factor Theorem. Recall that a quadratic expression that is the difference of squares has a special factoring pattern. Similarly, a cubic expression may be the sum of cubes or the difference of cubes.

Sum and Differences of Cubesa3 + b3 = (a + b)(a2 – ab + b2)a3 - b3 = (a - b)(a2 + ab + b2)

Example 3 Factoring a Sum of Cubes

Factor 8x3 + 1 Factor 64x3 + 27

Example 3 Factoring a Difference of Cubes

Factor 8x3 - 27 Factor 125x3 - 64

Example 4 Solving a Polynomial Equation

Factor x3 + 8 = 0 Factor 27x3 + 1

Example 5 Factoring by Using a Quadratic Form

Factor x4 - 2x2 – 8 = 0

Factor x4 + 7x2 + 6 = 0

6-5 Theorems About Roots of Polynomial Equations

What you’ll learn …To solve equations using the Rational Root TheoremTo use the Irrational Root Theorem and the Imaginary Root Theorem

1.02 Define and compute with complex numbers. 1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems.

Consider the equivalent equations….

x3 – 5x2 -2x +24 = 0 and (x+2)(x-3)(x-4) =0

-2, 3 and 4 are the roots of the equation.

• The product of -2,3 and 4 is 24.• Notice that all the roots are factors of the

constant term 24.• In general, if the coefficients in a polynomial

equation are integers, then any integer root of the equation is a factor of the constant term.

Both the constant and the leading coefficient of a polynomial can play a key role in identifying the rational roots of the related polynomial equation.

The role is expressed in the Rational Root Theorem.

If is in simplest form and is a rational root of the polynomial equation with integer coefficients, then p must be a factor of the constant term and q must be a factor of the leading coefficient.

pq

Example 1a Finding Rational Roots

x3 - 4x2 - 2x + 8 = 0Steps

1. List the possible rational roots of the leading coefficient and the constant.

2. Test each possible root.

Example 1b Finding Rational Roots

2x3 - x2 +2x - 1 = 0Steps

1. List the possible rational roots of the leading coefficient and the constant.

2. Test each possible root.

Example 1c Finding Rational Roots

x3 - 2x2 - 5x + 10 = 0

Steps1. List the possible

rational roots of the leading coefficient and the constant.

2. Test each possible root.

Example 1d Finding Rational Roots

3x3 + x2 - x + 1 = 0Steps

1. List the possible rational roots of the leading coefficient and the constant.

2. Test each possible root.

In Chapter 5 you learned to find irrational solutions to quadratic equations. For example, by the Quadratic Formula, the solutions of x2 – 4x -1 =0 are 2+√5 and 2 - √5.

Number pairs of the form a+√b and a-√b are called conjugates.

You can often use conjugates to find the irrational roots of a polynomial

equation.

Irrational Root Theorem Let a and b be rational numbers

and let √b be an irrational number. If a+ √b is a root of a polynomial equation with rational coefficients, then the conjugate a- √b also is a root.

Example 3 Finding Irrational Roots

1 + √3 and -√11

A polynomial equation with integer coefficients has the following roots. Find two additional roots..

2 - √7 and √5

Number pairs of the form a+bi and a-bi are complex conjugates. You can use complex conjugates to find an equation’s imaginary

roots.

Imaginary Root Theorem If the imaginary number a+bi is a

root of a polynomial equation with real coefficients then the conjugate a-bi also is a root.

Example 4 Finding Imaginary Roots

3i and -2 + i

A polynomial equation with integer coefficients has the following roots. Find two additional roots..

3 - i and 2i

Example 5a Writing a Polynomial Equation from its Roots

Find a third degree polynomial equation with rational coefficients that has roots -1 and 2-i.

Steps1. Find the other root

using the Imaginary Root Theorem.

2. Write the factored form of the polynomial using the Factor Theorem.

3. Multiply the factors.

Example 5b Writing a Polynomial Equation from its Roots

Find a third degree polynomial equation with rational coefficients that has roots 3 and 1+i.

Steps1. Find the other root

using the Imaginary Root Theorem.

2. Write the factored form of the polynomial using the Factor Theorem.

3. Multiply the factors.

Example 5c Writing a Polynomial Equation from its Roots

Find a fourth degree polynomial equation with rational coefficients that has roots i and 2i.

Steps1. Find the other root

using the Imaginary Root Theorem.

2. Write the factored form of the polynomial using the Factor Theorem.

3. Multiply the factors.

6-6 The Fundamental Theorem of Algebra

What you’ll learn …To use the Fundamental Theorem of Algebra in solving polynomial equations with complex roots

1.02 Define and compute with complex numbers. 1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems.

You have solved polynomial equations and found that their roots are included in the set of complex numbers. That is, the roots have been integers, rational numbers, irrational numbers and imaginary numbers.

But can all polynomial equations be But can all polynomial equations be solved using complex numbers?solved using complex numbers?

In 1799, the German mathematician Carl Friedrich Gauss proved that the answer to this question is yesyes. The roots of every polynomial equation, even those with imaginary coefficients, are complex numbers.

The answer is so important that his theorem is called the Fundamental Theorem of Algebra.

Carl Friedrich Gauss

Fundamental Theorem of Algebra If P(x) is a polynomial of degree n>1

with complex coefficients, then P(x) = 0 has at least one complex root.

Corollary Including imaginary roots and multiple

roots, an nth degree polynomial equation has exactly n roots; the related polynomial function has exactly n zeros.

Example 1a Using the Fundamental Theorem of Algebra

Find the number of complex roots, the possible number of real roots and possible number of rational roots.

x4 - 3x3 + x2 – x +3 = 0

Example 1b Using the Fundamental Theorem of Algebra

Find the number of complex roots, the possible number of real roots and possible number of rational roots.

x3 - 2x2 + 4x -8 = 0

Example 1c Using the Fundamental Theorem of Algebra

Find the number of complex roots, the possible number of real roots and possible number of rational roots.

x5 + 3x4 - x - 3 = 0

6-8 The Binomial Theorem

What you’ll learn …To use Pascal’s TriangleTo use the Binomial Theorem

1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems.

You have learned to multiply binomials using the FOIL method and the Distributive Property. If you are raising a single binomial to a power, you have another option for finding the product.

Consider the expansion of several binomials. To expand a binomial being raised to a power, first multiply; then write the result as a polynomial in standard form.

Pascal’s Triangle

(a + b)2 = (a + b) (a + b)

a2 + 2ab + b2

(a + b)3 = (a + b) (a + b) (a + b)

a3 + 3a2b + 3ab2 + b3

The coefficients of the product are 1,2 1.

The coefficients of the product are 1,3,3,1.

Pascal’s Triangle

Example 1a Using Pascal’s Triangle

Expand (a+b)8

Example 1b Using Pascal’s Triangle

Expand (x - 2)4

Example 1c Using Pascal’s Triangle

Expand (m + 3)5

Example 1d Using Pascal’s Triangle

Expand (3 – 2x)6

Quadratic Inequalities

Quadratics

Before we get started let’s review. A quadratic equation is an equation that canbe written in the form , where a, b and c are real numbers and a cannot

equalzero.

In this lesson we are going to discuss quadraticinequalities.

02 cbxax

Quadratic Inequalities

What do they look like? Here are some examples:

0732 xx

0443 2 xx

162 x

Quadratic Inequalities

When solving inequalities we are trying to find all possible values of the variablewhich will make the inequality true.

Consider the inequality

We are trying to find all the values of x for which the

quadratic is greater than zero or positive.

062 xx

Solving a quadratic inequality

We can find the values where the quadratic equals zero

by solving the equation, 062 xx

023 xx

02or03 xx

2or3 xx

Solving a quadratic inequality

For the quadratic inequality,we found zeros 3 and –2 by solving the equation

. Put these values on a number line and we can see three intervals that we will test in the inequality. We will test one value from each interval.

062 xx

062 xx

-2 3

Solving a quadratic inequality

Interval Test Point

Evaluate in the inequality True/False

2,

3,2

,3

06639633 2

06600600 2

066416644 2

3x

0x

4x

True

True

False

062 xx

062 xx

062 xx

Example 2:

Solve First find the zeros by solving the equation,

0132 2 xx0132 2 xx

0132 2 xx

0112 xx

01or012 xx

1or2

1 xx

Example 2:

Now consider the intervals around the zeros and test a value from each interval in the inequality.

The intervals can be seen by putting the zeros on a number line.

1/2 1

Example 2:Interval Test Point Evaluate in Inequality True/False

2

1,

1,2

1

,1

0x

4

3x

2x

0110010302 2

08

11

4

9

8

91

4

33

4

32

2

0316812322 2

False

True

False

0132 2 xx

0132 2 xx

0132 2 xx

Summary

In general, when solving quadratic inequalities 1. Find the zeros by solving the equation you

get when you replace the inequality symbol with an equals.

2. Find the intervals around the zeros using a number line and test a value from each interval in the number line.

3. The solution is the interval or intervals which make the inequality true.

02452 xx

012 2 xx

0116 2 x

0452 xx

In this chapter, you should have …

Learned to write and graph polynomial functions and to solve polynomial equations.Learned to use important theorems about the number of solutions to polynomial equations.Learned to solve problems involving permutations, combinations and binomial probability.