UNIT 4: MODELING WITH POLYNOMIAL & RATIONAL

FUNCTIONS & EQUATIONS

Ms. C. TaylorCommon Core Math 3

Warm-Up Factor the following

Polynomials Degree-highest exponent Leading Coefficient-coefficient of

the first term. Left Behavior- what the graph does

on the left side as Right Behavior- what the graph

does on the right side as

Handout Fill in the handout on Investigation

of Polynomial Functions

Relative Maxima or Minima

Relative Maxima is when the graph of changes from increasing to decreasing.

Relative Minima is when the graph of changes from decreasing to increasing.

Higher Degree Polynomial Functions

and Graphs

an is called the leading coefficient n is the degree of the polynomial a0 is called the constant term

Polynomial Function

A polynomial function of degree n in the variable x is a function defined by

where each ai is real, an 0, and n is a whole number.01

11)( axaxaxaxP n

nn

n

Polynomial Functions

f(x) = 3

ConstantFunction

Degree = 0

Maximum Number of

Zeros: 0

f(x) = x + 2LinearFunction

Degree = 1

Maximum Number of

Zeros: 1

Polynomial Functions

f(x) = x2 + 3x + 2QuadraticFunction

Degree = 2Maximum Number of

Zeros: 2

Polynomial Functions

f(x) = x3 + 4x2 + 2

Cubic Function

Degree = 3

Maximum Number of

Zeros: 3

Polynomial Functions

Quartic Function

Degree = 4

Maximum Number of

Zeros: 4

Polynomial Functions

The Leading Coefficient Test

As x increases or decreases without bound, the graph of the polynomial function

f (x) = anxn + an-1x

n-1 + an-2xn-2 +…+ a1x + a0 (an ¹ 0)

eventually rises or falls. In particular,

For n odd: an > 0 an < 0

As x increases or decreases without bound, the graph of the polynomial function

f (x) = anxn + an-1x

n-1 + an-2xn-2 +…+ a1x + a0 (an ¹ 0)

eventually rises or falls. In particular,

For n odd: an > 0 an < 0

If the leading coefficient is positive, the graph falls to the left and rises to the right.

If the leading coefficient is negative, the graph rises to the left and falls to the right.

Rises right

Falls left

Falls right

Rises left

As x increases or decreases without bound, the graph of the polynomial function

f (x) = anxn + an-1x

n-1 + an-2xn-2 +…+ a1x + a0 (an ¹ 0)

eventually rises or falls. In particular,

For n even: an > 0 an < 0

As x increases or decreases without bound, the graph of the polynomial function

f (x) = anxn + an-1x

n-1 + an-2xn-2 +…+ a1x + a0 (an ¹ 0)

eventually rises or falls. In particular,

For n even: an > 0 an < 0

If the leading coefficient is positive, the graph rises to the left and to the right.

If the leading coefficient is negative, the graph falls to the left and to the right.

Rises right

Rises left

Falls left

Falls right

The Leading Coefficient Test

ExampleUse the Leading Coefficient Test to determine the end behavior of the graph of f (x) = x3 + 3x2 - x - 3.

Falls left

yRises right

x

Determining End BehaviorMatch each function with its graph.

4 2

3 2

( ) 5 4

( ) 3 2 4

f x x x x

h x x x x

47)(

43)(7

26

xxxkxxxxg

A. B.

C. D.

x-Intercepts (Real Zeros)

Number Of x-Intercepts of a Polynomial Function

A polynomial function of degree n will have a maximum of n x- intercepts (real zeros).

Find all zeros of f (x) = -x4 + 4x3 - 4x2. -x4 + 4x3 - 4x2 = 0 We now have a polynomial

equation. x4 - 4x3 + 4x2 = 0 Multiply both sides by -1. (optional step)

x2(x2 - 4x + 4) = 0 Factor out x2.

x2(x - 2)2 = 0 Factor completely.

x2 = 0 or (x - 2)2 = 0 Set each factor equal to zero.

x = 0 x = 2 Solve for x.

(0,0) (2,0)

Multiplicity and x-Intercepts

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. Regardless of whether a zero is even or odd, graphs tend to flatten out at zeros with multiplicity greater than one.

Example Find the x-intercepts and

multiplicity of f(x) =2(x+2)2(x-3) Zeros are at

(-2,0)(3,0)

Extrema Turning points – where the graph of a function changes from

increasing to decreasing or vice versa. The number of turning points of the graph of a polynomial function of degree n 1 is at most n – 1.

Local maximum point – highest point or “peak” in an interval function values at these points are called local maxima

Local minimum point – lowest point or “valley” in an interval function values at these points are called local minima

Extrema – plural of extremum, includes all local maxima and local minima

Extrema

Warm-Up Given the following

polynomial , state how many zeros are possible, left behavior, right behavior, and name the polynomial.

End BehaviorDegree Leading Coefficient End Behavior

Even Positive

Even Negative

Odd Positive

Odd Negative

Domain & Range It depends on the polynomial

graph that you are looking at. Most likely the domain will be Most likely the range will be

Effects of The “h” will shift the graph

left or right depending on sign

The “k” will shift the graph up or down depending on the sign.

Remainder Theorem If I divide the polynomial by the

factor given then I will have a remainder other than zero.

If you plug in the constant of the factor into the equation then you should come up with the remainder when doing synthetic division.

Examples-Remainder Theorem

Warm-Up Determine the

remainder of the following:

Factor Theorem In order to prove that (x-a)

is a factor of the polynomial f(a)=0.

In other words, the remainder from synthetic division has to be zero.

Examples

Fundamental Theorem of Algebra

Every polynomial equation with degree greater than zero has at least one root in the set of complex numbers.

Rational Zero Theorem

If is a rational number in simplest form and is a zero of the polynomial then p is a factor of and q is a factor of .

Examples

Warm-Up Determine if the factor is a

factor of State all of the possible

rational zeros for the following: .

Rational Equations Step 1: Set all denominators equal

to zero and solve. Step 2: Find the CD (Factor Denom.) Step 3: Look for “what is missing” &

multiply each fraction by the appropriate expression. DUMP the denominators.

Step 4: Solve for x. Step 5: Check to be sure that x

doesn’t equal any number from step 1.

Rational Equations Examples:

Warm-Up Solve the following:

Literal Equations Solve the following for the

specified variable, solve for h, solve for c, solve for r

Warm-Up Solve the following for the indicated variable:

Practice An open box is to be made

from a rectangular piece of material 20 inches by 12 inches by cutting equal squares from the corners and turning up the sides.

What is the maximum volume that the box can hold?

Practice An open box is to be made

from a rectangular piece of material 15 inches by 10 inches by cutting equal squares from the corners and turning up the sides.

What is the maximum volume that the box can hold?

Practice An open box is to be made

from a rectangular piece of material 6 inches by 1 inch by cutting equal squares from the corners and turning up the sides.

What is the maximum volume that the box can hold?

Practice An open box is to be made

from a rectangular piece of material 25 inches by 14 inches by cutting equal squares from the corners and turning up the sides.

What is the maximum volume that the box can hold?

Practice An open box is to be made

from a rectangular piece of material 35 inches by 22 inches by cutting equal squares from the corners and turning up the sides.

What is the maximum volume that the box can hold?

Practice An open box is to be made

from a rectangular piece of material 40 inches by 24 inches by cutting equal squares from the corners and turning up the sides.

What is the maximum volume that the box can hold?

Warm-Up Write the following

polynomial in standard form and give the degree, name of the polynomial, leading coefficient.

Finding k in a Polynomial

Substitute the value from the factor in for x and then set the equation equal to zero.

Check by using synthetic division to make sure that you get a remainder of 0.

Examples given (x-2) is a factor given (x-2) is a factor

Practice given is a factor given is a factor given is a factor given is a factor

Warm-Up If is a factor of , then find the value of k.

STEP 1: REWRITE THE ZEROS AS FACTORS (X – ( )) (X – ( )) (X – ( ))

To write the polynomial equation

given the zeros:

Step 2: Multiply the factors that have i’s or first

Remember: Imaginary numbers and irrational numbers come in conjugate pairs (ex: 2+3i, 2-3i)

Step 3: Multiply the trinomial and binomial by the box

When you are given the zeros/roots:

write them as factors:

(x - )(x - )(x - )

x = { , , }2-1

-2

Change the signs if you can:(x + 2)(x +1)(x-2)

Now write your equation by multiplying out!

)2)(23( 2 xxx

44)( 23 xxxxf2

x

(x + 2)(x +1)(x-2)

Box!

Box! AGAIN!

2x x3 23x

4x6

23x22x

x2

2. When you are given the zeros/roots:

034

344

3

x

x

x

write them as factors:

(x - )( x - )

x = { , }3/4 -2

Change the signs if you can:(x + 2)(4x-3)

4x - 3

Now write your equation by multiplying out!

(x + 2)(4x -3)

Box!

654)( 2 xxxf

3. When you are given the zeros/roots:

write them as factors:

(x - )(x - )(x - )

x = { , , }3-2

5

Change the signs if you have to:

(x -5)(x +2)(x-3)

Now write your equation by multiplying out!

)3)(103( 2 xxx

306)( 23 xxxxf3

x

(x -5)(x +2)(x-3)

Box!

Box! AGAIN!

2x x3 103x

30x9

23x23x

x10

Warm-Up Write the equation given the roots