CHAPTER 2Polynomial and Rational Functions

SECTION 1Quadratic Functions

Quadratic Functions

Let a, b, and c be real number with a ≠ 0. The function f(x) = ax2 + bx = c is called a quadratic function.

The graph of a quadratic function is a special type of U-shaped curve that is called a parabola.

All parabolas are symmetric with respect to a line called the axis of symmetry, or simply the axis of the parabola.

The point where the axis intersects the parabola is called the vertex.

Quadratic Functions

If a >0, then the graph opens upward.

If a < 0, then the graph opens downward.

The Standard Form of a Quadratic Function

The standard form of a quadratic functions

f(x) = a(x-h)2+ k, a ≠ 0 Vertex is (h, k) |a| produces a vertical stretch or

shrink (x – h)2 represents a horizontal shift

of h units k represents a vertical shift of k

units Graph by finding the vertex and the

x-intercepts

Vertex of a Parabola

The vertex of the graphf(x) = a(x)2+ bx + c is( -b/2a, f(-b/2a))

EXAMPLEFind the vertex and x-intercepts-4x2 +x + 3

SECTION 2Polynomial Functions of Higher Degree

Polynomial Functions

Let n be a nonnegative integer and let an, an-1, ….. …a2, a1, a0 be real numbers with an ≠ 0.

The function f(x) = anxn + an-1xn-1 +…… a2x2 + a1x + a0 is called a polynomial function of x with degree n.

EXAMPLE

f(x) = x3

Characteristics of Polynomial Functions

1. The graph is continuous.

2. The graph has only smooth rounded turns.

Sketching Power Functions

Polynomials with the simplest graphs are monomial of the form f(x) = xn and are referred to as power functions.

REMEMBER ODD and EVEN FUNCTIONS1. Even : f(-x) = f(x) and symmetric to y-axis

2. Odd: f(-x) = - f(x) and symmetric to origin

Leading Coefficient Test

1. When n is odd: If the leading coefficient is positive (an >0), the

graph falls to the left and rises to the right

2. When n is odd: If the leading coefficient is negative (an <0), the graph rises to the left and falls to the right

Leading Coefficient Test

1. When n is even: If the leading coefficient is positive (an >0), the graph rises to the left and right.

2. When n is even: If the leading coefficient is negative (an <0), the graph falls to the left and right

EXAMPLE

1. Identify the characteristics of the graphs

2. f(x) = -x3 + 4x

3. f(x) = -x4 - 5x2 + 4

4. f(x) = x5 - x

Real Zeros of Polynomial Functions

If f is a polynomial function and a is a real number, the following statements are equivalent.

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

Repeated Zeros

A factor (x-a)k, k >0, yields a repeated zero x = a of multiplicity k.

1. If k is odd, the graph crosses the x-axis at x = a

2. If k is even, the graph touches the x-axis at x = a (it does not cross the x-axis)

EXAMPLE

1. Graph using leading coefficient test, finding the zeros and using test intervals

2. f(x) = 3x4 -4x3

3. f(x) = -2x3 + 6x2 – 4.5x

4. f(x) = x5 - x

SECTION 3Long Division of Polynomials

Long Division Algorithm

If f(x) and d(x) are polynomials such that d(x) ≠ 0, and the degree of d(x) is less than or equal to the degree of f(x), there exist unique polynomials q(x) and r(x) such that:

f(x) = d(x) q(x) + r(x)

EXAMPLE

Divide the following using long division.1. x3 -1 by x – 12. 2x4 + 4x3 – 5x2 + 3x -2 by x2 +2x – 3

Remember to use zero coefficients for missing terms

Synthetic Division

Synthetic Division is simply a shortcut for long division, but you still need to use 0 for the coefficient of any missing terms.

EXAMPLE Divide x4 – 10x2 – 2x +4 by x + 3 -3 1 0 -10 -2 4 -3 9 3 -3 1 -3 -1 1 1 = x3 – 3x2 -

x + 1 R 1

The Remainder and Factor Theorems

Remainder Theorem:If a polynomial f(x) is divided by x-k,

then the remainder is r = f(k) EXAMPLEEvaluate f(x) = 3x3 + 8x2 + 5x – 7 at

x = -2 Using synthetic division you get r = -

9, therefore,f(-2) = -9

The Remainder and Factor Theorems

Factor Theorem:A polynomial f(x) has a factor (x-k) if and only if

f(k) =0 EXAMPLEShow that (x-2) and ( x+3) are factors of

f(x) = 2x4 + 7x3 -4x2 -27x – 18Using synthetic division with x-2 and then again

with x+3 you get f(x) = (x-2)(x+3)(2x+3)(x+1) implying 4 real

zeros

Uses of the Remainder in Synthetic Division

The remainder r, obtained in the synthetic division of f(x) by x-k, provides the following information:

1. The remainder r gives the value of f at x=k. That is, r= f(k)

2. If r=0, (x-k) is a factor of f(x) 3. If r=0, (k,0) is an x-intercept of the

graph of f

SECTION 4Complex Numbers

The Imaginary Unit i

Because some quadratic equations have no real solutions, mathematicians created an expanded system of numbers using the imaginary unit i, defined as i = -1

i2 = -1 i3 = -i

i4 = 1

Complex Numbers

The set of complex numbers is obtained by adding real numbers to real multiples of the imaginary unit. Each complex number can be written in the standard form a + bi . If b = 0, then a + bi = a is a real number. If b ≠ 0, the number a + bi is called an imaginary number. A number of the form bi, where b ≠ 0, is called a pure imaginary number.

Some Properties of Complex Numbers

1. a + bi = c+ di if and only if a=c and b=d.

2. (a + bi) + (c+ di) = (a +c) + (b + d)i

3. (a + bi) – (c+ di) = (a – c) + (b – d)i

4. – (a + bi) = – a – bi5. (a + bi ) + (– a – bi) = 0 + 0i = 0

Complex Conjugates

a + bi and a –bi are complex conjugates

(a + bi) (a –bi ) = a2 + b2

EXAMPLE(4 – 3i) (4 + 3i) = 16 + 9 = 25

Complex Solutions of Quadratic Equations

Principal Square Root of a Negative Number

If a is a positive number, the principal square root of the negative number –a is defined as

– a = a i

EXAMPLE – 13 = 13i

SECTION 5TheFundamental Theorem of Algebra

The Fundamental Theorem of Algebra

If f(x) is a polynomial of degree n, where n > 0, then f has at least one zero in the complex number system.

If f(x) is a polynomial of degree n, where n > 0, then f has precisely n linear factors.

f(x) = an(x – c1)(x-c2)…(x–cn)

Where c1,c2…cn are complex numbers

Linear Factorization Theorem

EXAMPLE

Find the zeros of the following:1. f(x) = x – 22. f(x) = x2 – 6x + 93. f(x) = x3 + 4x4. f(x) = x4 – 1

Rational Zero Test

If the polynomial f(x)= anxn + an-1xn-1 +…a2x2+a1x1 +a0 has integer coefficients, every rational zero of f has the form

Rational zero = p/q or constant term/leading coefficent

Where p and q have no common factors other than 1, and

p = a factor of the constant term a0

q = a factor of the leading coefficient an

EXAMPLE

Find the rational zeros of f(x) = 2x3+3x2 – 8x + 3

Rational zeros p/q = ± 1, ± 3 / ± 1, ± 2

Possible rational zeros are ± 1, ± 3, ± ½, ± 3/2

Use synthetic division by trial and error to find a zero

Conjugate Pairs

Let 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 zero of the function.

Rational zero = p/qWhere p and q have no common factors

other than 1, andp = a factor of the constant term a0

q = a factor of the leading coefficient an

EXAMPLE

Find a 4th degree polynomial function with real coefficients that has – 1, – 1, and 3i as zeros

Thenf(x) = a(x+1)(x+1)(x – 3i)(x+3i)For simplicity let a = 1Multiply the factors to find the

answer.

EXAMPLE

Find all the zeros of f(x) = x4 – 3x3 + 6x2 + 2x – 60 where 1 + 3i is

a zeroKnowing complex zeros occur in pairs,

then 1 – 3i is a zeroMultiply (1+3i)(1 – 3i) = x2 – 2x +10 and use

long division to find the other zeros of -2 and 3

x4 – 3x3 + 6x2 + 2x – 60/(x2 – 2x +10)

EXAMPLE

Find all the zeros of f(x) = x5 + x3 + 2x2 – 12x

+8Find possible rational roots

and use synthetic division

EXAMPLE

You are designing candle-making kits. Each kit will contain 25 cubic inches of candle wax and a mold for making a pyramid-shaped candle. You want the height of the candle to be 2 inches less than the length of each side of the candle’s square base. What should the dimension of your candle mold be? Remember V = 1/3Bh

SECTION 6Rational Functions

Rational Function

A rational function can be written in the form

f(x) = N(x)/D(x) where N(x) and D(x) are polynomials and D(x) is not the zero polynomial. Also, this sections assumes N(x) and D(x) have no common factors.

In general, the domain of a rational function of x includes all real numbers except x-values that make the denominator zero.

EXAMPLE

Find the domain of the following and explore the behavior of f near any excluded x-values (graph)

1. f(x) = 1/x

2. f(x) = 2/(x2 – 1) 2

ASYMPTOTE

Is essentially a line that a graph approaches but does not intersect.

Horizontal and Vertical Asymptotes

1. The line x= a is a vertical asymptote of the graph of f if f(x) → ∞ or f(x) → – ∞

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 f(x) → b

as x → ∞ or x → – ∞

Asymptotes of a Rational Function

Let f be the rational function given byf(x) = N(x)/D(x) where N(x) and D(x) have no common

factors then: anxn + an-1xn-1…./(bmxm +bm-1xm-1…)

1. The graph of f has vertical asymptotes at the zeros of D(x).

2. The graph of f has one or no horizontal asymptote determined by comparing the degrees of N(x) and D(x)

a. If n < m, the graph of f has the line y = 0 as a horizontal asymptote.

b. If n= m, the graph of f has the line y = an/bm as a horizontal asymptote.

c. If n>m, the graph of f has no horizontal asymptote

EXAMPLE

Find the horizontal and vertical asymptotes of the graph of each rational function.

1. f(x) = 2x/(x4 + 2x2 + 1)

2. f(x) = 2x2 /(x2 – 1)

Graphing Rational Functions

Let f be the rational function given byf(x) = N(x)/D(x) where N(x) and D(x) have no common factors1. Find and plot the y-intercept (if any) by evaluating f(0).2. Find the zeros of the numerator (if any) by solving the

equation N(x) =0 and plot the x-intercepts3. Find the zeros of the denominator (if any) by solving the

equation D(x) = 0, then sketch the vertical asymptotes4. Find and sketch the horizontal asymptote (if any) using the

rule for finding the horizontal asymptote of a rational function5. Test for symmetry (mirror image)6. Plot at least one point between and one point beyond each

x-intercept and vertical asymptote7. Use smooth cures to complete the graph between and beyond

the vertical asymptotes

EXAMPLE

Graph1. f(x) = 3/(x – 2)

2. f(x) = (2x – 1)/x

3. f(x) = (x2 – 9)/(x2 – 4)

Slant Asymptotes

Consider a rational function whose denominator is of degree 1 or greater. If the degree of the numerator is exactly one more than the degree of the denominator, the graph of the function has a slant (or oblique) asymptote.

EXAMPLE

f(x) = (x2 – x) /( x+ 1) has a slant asymptote.

To find the equation of a slant asymptote, use long division.

You get x – 2 + 2/(x+1)y = x – 2 because the remainder term

approaches 0 as x increases or decreases without bound

EXAMPLE

f(x) = (x2 – x – 2) /( x – 1) 1. Find the x-intercepts2. Find the y-intercepts3. Vertical asymptotes4. Slant asymptote

5. Try graphing using your calculator.

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