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Unit 3Polynomial and Rational Functions

Section 1Polynomial Functions of a Higher Degree

Polynomial FunctionLet n be a nonnegative integer and let an , an−1,…,a2 , a1 , a0 be real numbers with an≠0. The function

is called a polynomial function of x with degree n.

Quadratic FunctionLet a, b, and c be real numbers with a≠0. The function

is called a quadratic function.

Graphs of Polynomial Functions

Basic Graph Shapes

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Leading Coefficient TestWhether the graph of a polynomial eventually rises or falls can be determined by the function's degree (even or odd) and by its leading coefficient.Odd Degree:

1. Leading coefficient positive:

2. Leading coefficient negative:

Even Degree1. Leading coefficient positive:

2. Leading coefficient negative:

Example 1.1Use the leading coefficient test to determine the right and left behavior of the graph of each function.1. f(x)=-x3+4x

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2. h(x)=x4-5x2+4

3. g(x)=x5-x

Zeros of Polynomial FunctionsFor a polynomial function f of degree n:

1. The graph of f has at most ___________________________2. The function f has at most ___________________________

(local maximums or minimums).**Zeros are the x-values when f(x)=0

Recall that these are all the same:1. x=a is a zero of the function f2. x=a is a solution of the polynomial equation f(x)=03. (x-a) is a factor of the polynomial f(x)

4. (a,0) is an x-intercept of the graph of f

Example 1.2a. Find all real zeros of f(x)=x3-x2-2x

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b. Find all real zeros of f(x)=-2x4+2x2

Repeated ZerosA factor _______________________ yields a repeated zero x=a of

multiplicity k.

1. When k is odd, the graph crosses the x-axis ______________

2. When k is even, the graph ____________________________

(but does not cross the x-axis) at x=a.

It is important to realize that the graph of a polynomial function can only change signs at its zeros. Between zeros, the graph must be only positive or only negative. The zeros, when put in order, divide the graph into test intervals that aid in graphing.

Example 1.3Sketch the graph of f(x)=3x4-4x3.

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The Intermediate Value TheoremLet a and b be real numbers such that a<b. If f is a polynomial

function such that _________________, then, in the interval [a,b],

f takes on _____________________________________________

Example 1.4Use the Intermediate Value Theorem to approximate the real zero of f(x)=x3-x2+1

Example 1.5Find three intervals of length 1 in which the polynomial f(x)=12x3-32x2+3x+5 is guaranteed to have a zero.

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Section 2Polynomial and Synthetic Division

Example 2.1Divide f(x)=6x3-19x2+16x-4 by x-2 using long division. Use the result to factor f(x) completely.

The Division AlgorithmAssuming the degree of d(x) is less than or equal to the degree of f(x) and d(x)≠0:

Another way to write this:

Before applying the Division Algorithm, follow two steps:

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1. Write the dividend and divisor in descending powers of the variable.2. Insert placeholders with zero coefficients for missing powers of the variable.

Example 2.2Divide x3-1 by x-1

Example 2.3Divide -5x2-2+3x+2x4+4x3 by 2x-3+x2

Example 2.4

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Divide f(x)=3x4-5x3+4x-6 by x2-3x+5

Example 2.5Divide f(x)=x3+5x2-7x+2 by x-2

Example 2.6Divide 2x4+x3+x-1 by x2+2x-1

Example 2.7

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(x3-x2+4x-10)÷(x+2)

Synthetic SubstitutionEvaluate f(x)=2x4-5x3-4x+8 when x=3Step 1: Write the coefficients of f(x) in order of descending exponents. Write the value at which f(x) is being evaluated to the left.

Step 2: Bring down the leading coefficient. Multiply the leading coefficient by the x-value. Write the product under the second coefficient. Add.

Step 3: Multiply the previous sum by the x-value. Write the product under the third coefficient. Add. Repeat for all of the remaining coefficients. The final sum is the value of f(x) at the given x-value.

Example 2.8

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Evaluate using synthetic substitutionf(x)=5x3+3x2-x+7 at x=2.

Example 2.9f(x)=5x4-3x3+4x2-x+10x=2

Example 2.10f(x)=-3x5+x3-6x2+2x+4x=-3

Synthetic Division

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The Remainder TheoremIf a polynomial f(x) is divided by x-k, then the remainder is _____.

Also, when using synthetic substitution for f(k), the other values

below the line match the _________________________________.

Example 2.11Divide f(x)=2x3+x2-8x+5 by x+3 using synthetic division.

Example 2.12Divide using synthetic division:(x3+4x2-x-1)÷(x+3)

(4x3+x2-3x+7)÷(x-1)

Example 2.13

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Divide using synthetic division(x3-6x2+5x+12)÷(x-4)

(x3-x2-22x+40)÷(x-4)

The Factor TheoremA polynomial f(x) has a factor (x-k) if and only if ___________.

Example 2.14Use synthetic division to show that (x-2) and (x+3) are factors of f(x)=2x4+7x3-4x2-27x-18 and use this result to factor f(x) completely.

Section 3

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Complex Numbers

Let a and b be real numbers. The number ___________ is called a

complex number, and it is said to be written in ______________.

The real number a is called the ____________ and the real number

b is called the ______________________ of the complex number.

When b=0, the number a+bi is a _______________. When b≠0,

the number a+bi is called an ______________________. A

number of the form bi, where b≠0, is called a pure imaginary

number.

Equality of Complex NumbersTwo complex numbers a+bi and c+di are equal to each other

if and only if a=c and b=d.

Operations with Complex Numbers

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Addition and Subtraction:

(a+bi)+(c+di)=

(a+bi)-(c+di)=

Multiplication:

(a+bi)(c+di)=

Division:

(a+bi)/(c+di)=

Example 3.1a. (4+7i)+(1-6i)

b. (1+2i)+(3-2i)

c. 3i-(-2+3i)-(2+5i)

d. (3+2i)+(4-i)-(7+i)

Example 3.2

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a. (7+3i)+(5-4i)

b. (3+4i)-(5-3i)

c. 2i+(-3-4i)-(-3-3i)

d. (5-3i)+(3+5i)-(8+2i)

Example 3.3a. 4(-2+3i)

b. (2-i)(4+3i)

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c. (3+2i)(3-2i)

d. (3+2i)2

Example 3.4a. 1

(1+i)

b. (2+3 i)(4−2i)

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Complex Conjugates

Example 3.5(2+i )(2−i)

Section 4The Fundamental Theorem of Algebra

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The Fundamental Theorem of AlgebraIf f(x) is a polynomial of degree n, where n>0, f has at least _____

______________________________________________________

Linear Factorization TheoremIf f(x) is a polynomial of degree n

where n>0, f has precisely n linear factors

where c1, c2, ..., cn are complex numbers and an is the leading coefficient of f(x).

Example 4.1Determine the number of zeros of each function and find the zeros.a. f(x)=x-2

b. f(x)=x2-6x+9

c. f(x)=x3+4x

Example 4.2

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Write f(x)=x5+x3+2x2-12x+8 as the product of linear factors, and list all of its zeros.

Conjugate PairsLet f(x) be a polynomial function that has real coefficients. If a+bi,

where b≠0, is a zero of the function, the conjugate a-bi is also a

__________________________

Example 4.3Normally, we would say that x2+1 is irreducible. With complex numbers, it is reducible:

x2+1=

Example 4.4Completely factor f(x)=x4-x2-20

Example 4.5

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Find all the zeros of f(x)=x4-3x3+6x2+2x-60 given that 1+3i is a zero of f.

Example 4.6f(x)=x2+25

Example 4.7f(x)=x2-x+56

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Finding Rational ZerosFind the rational zeros of f(x)=2x3+3x2-8x+3

Example 4.8f(x)=x2-4x+1

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Example 4.9g(x)=x2+10x+23

Example 4.10f(x)=x4-81

Example 4.11h(x)=x3-3x2+4x-2

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Example 4.12f(t)=t3-3t2-15t+125

Example 4.13Write f(x)=x5+x3+2x2-12x+8 as a product of linear factors and list all the zeros of the function.

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Example 4.14Determine the possible numbers of positive and negative real zeros of f(x)=3x3-5x2+6x-4

Section 5Rational Functions and Asymptotes

Rational Functions

where p(x) and q(x) are polynomials and q(x) is not zero. We assume that p(x) and q(x) have no common factors.

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Horizontal and Vertical Asymptotes1. The line x=a is a vertical asymptote of the graph of f if

as x→a, either from the right or from the left.2. The line y=b is a horizontal asymptote of the graph of f if

as x→∞ or x→-∞.

Asymptotes of a Rational FunctionLet f be the rational function given by:

1. The graph of f has vertical asymptotes at the zeros of ________.2. The graph of f has at most one horizontal asymptote, as follows:

a. If n<m, the ___________ is the asymptoteb. If n=m, the line _____________ is the asymptote.c. If n>m, there is no horizontal asymptote.

Example 5.1Find the horizontal asymptotes of:

f ( x )= 2 x3 x2+1

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g ( x )= 2 x2

3x2+1

h ( x )= 2 x3

3x2+1

Example 5.2Find the vertical and horizontal asymptotes of the graph of each:

a. f ( x )= x2+x−2x2−x−6

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b. f ( x )= 2 x2

x2−1

Slant AsymptotesIf the degree of the numerator is exactly _____________________

the degree of the denominator, the graph has a slant asymptote.

To find the equation of this line: simply divide the denominator

into the numerator- the ___________________________________

and the remainder (divided by the divisor) is the ______________

__________________________________

Example 5.3f ( x )= x

2−xx+1

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Example 5.4Find the asymptotes of f ( x )= x

2−x−2x−1

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