Polynomial Functions

Some Terminology:• Monomial: Real number, or variable, or product of a real

number and variables with exponents which are whole numbers. The degree of the monomial (in one variable) is the exponent on the variable.

• Polynomial: A monomial or sum of monomials. The degree of the polynomial is the highest degree among its monomial terms.

• Polynomial Function: Say, of “”, would be the a polynomial with terms involving the variable “”. For example: would be a polynomial function of of degree 3, also called a cubic function of .

• Terms are arranged in descending order of degree:

• Names of polynomials by number of terms: Monomial: polynomial with one term Binomial: polynomial with two terms Trinomial: polynomial with three terms

• Names of polynomial functions by highest degree:

Standard Form for a Polynomial Function

Name Constant Linear Quadratic Cubic Quartic Quintic

Degree 0 1 2 3 4 5

Leading Term

Behavior of Polynomial Functions

Important characteristics of functions typically include: End behavior: How function behaves as

( means approaches) Turning Points: Places where the function changes

direction (that is, where the slope changes sign) An n-degree polynomial has, at most, (n-1) turning points. A polynomial of odd degree has an even number of turning

points. A polynomial of even degree has an odd number of turning

points.

Some Exercises

• Write each polynomial in standard form, classify it by degree and by number of terms:3x + 9x2 + 54x – 6x2 + x4 + 10x2 – 12

• For each function, find the end behavior, then using a graphing calculator, find the turning points, and the intervals on which the function is increasing and decreasing.

How Can We Find the Degree of a Polynomial from Data?

• Given a set of outputs for consecutive and evenly spaced (differing by a constant) inputs for a polynomial, find its degree.For degree 1, the first differences of the data will be the same.For degree 2, the second differences of the data will be the

same….For degree n, the nth differences of the data will be the same.

Find the order of the following polynomials given the data.x -3 -2 -1 0 1 2 3

y -1 -7 -3 5 11 9 -7

x -3 -2 -1 0 1 2 3

y 23 -16 -15 -10 -13 -12 29

Finding Degree of Polynomial from Data

Find the order of the following polynomials given the data.

x -3 -2 -1 0 1 2 3

y -1 -7 -3 5 11 9 -7

x -3 -2 -1 0 1 2 3

y 23 -16 -15 -10 -13 -12 29

We can solve some polynomial equations: = 0by finding linear factors: = 0Some terminology and properties:For a polynomial function: If is a linear factor of then b is a zero of P(x). b is a root of the equation: P(x) = 0. b is an x-intercept of the graph of y = P(x). Theorem: The expression is a factor of a polynomial if and only

if the value b is a zero of the related polynomial function.

Linear Factors and Zeros of a Polynomial

Some Exercises

• For y = (x+2)(x-1)(x-3), Find the zeros, graph the function, and put it in standard form.

1) First find the zeros2) Find y for values of x between zeros3) Identify the end behavior4) Sketch the function5) Multiply out the factors, combine like terms, and

rearrange terms to put in standard form.

1) A cubic polynomial function has zeros: -2, 2, 3. Write it in standard form.

2) A quartic polynomial function has zeros -2, -2, 2, 3. Write it in standard form.

More Exercises

The quartic polynomial function of the last example can be written:f(x) = (x+2)2 (x-2)(x-3)We say it has a “multiple zero” and that -2 is a zero with a multiplicity of 2.A function with a zero of multiplicity n behaves like a function of degree n near that zero. So: f(x) behaves like: a quadratic near x =-2 (zero has multiplicity of 2) a line near x =2 and x = 3 (zeros each with multiplicity of 1)

Multiplicity of Zeros

Exercise

• For f(x) = x4 - 2x3 - 8x2

Find the zeros and their multiplicities. How does the graph behave near these zeros?

• Repeat this exercise for: f(x) = x3 - 4x2 + 4x

• At a Turning Point, the function achieves an extreme value relative to neighboring points. We call this a Relative (or local ) Maximum or Minimum.

• More generally these are “relative extrema”.• If a function, at a point, achieves its lowest or highest value

over its entire range, we call this a GLOBAL maximum or minimum.

• A whole field of optimization is devoted to trying to find the global extrema of functions without getting “trapped” or deceived by local extrema.

Relative Maxima and Minima

Modeling with a Polynomial• A digital box camera maximizes the volume while keeping the

sum of dimensions at 6 inches. Also, the length must be 1.5 times the height. So what should each dimension be?

1) Write the Volume as a function of length, width, and height.2) Write equations for the two constraints in the problem.3) Using the constraint equations, rewrite the volume as a

function of one variable (say, the height).4) Plot the function on the TI-nspire. Then choose :

Menu->Analyze Graph(6)->Maximum(3). Use the touch pad to choose lower and upper bounds bracketing the relative maximum.

5) Step 4 will give the height and maximum volume. Calculate the length and width using the constraint equations.

Try this problem• Page 294: Problem 39.

Always start with all polynomials in standard form. Include “zero” terms (terms where zero multiplies a lower power of x) for clarity.Divide:1) 6x2 + 7x +2 by 2x+1

2) 4x2 + 23x – 16 by x+5(Note: This example will leave a remainder)

3) 8x3 – 1 by (x - ½ ) How can this example be used to find the zeros of 8x3 – 1 ?

How can we divide Polynomials?

Synthetic Division• Streamlines the division process. • Valid for dividing polynomials by linear factors (x-a) only.Write the coefficients of all the terms in descending order from the left,

including zero terms.In a bracket to the left, write the “a” from the (x-a) factor.Bring down the first (leftmost) coefficient.Multiply the coefficient by the divisor (a) and add it to the next

coefficient bringing down the result.Continue the process, working your way to the right, until you reach the

last coefficient.The result brought down from the last coefficient will be the remainder.

Try these:

1) x3 – 14x2 + 51x -54 by x-2

2) x3 + 7x2 - 38x – 240 by x+5

Synthetic Division Exercises

Hwk 22 (due Tues 1/20)

• Page 285-286: 12, 24, 34, 40• Page 293-295: 25, 30, 44• Page 308-310: 12, 26

If you divide a Polynomial P(x) of degree n ≥ 1 by (x-a), then the remainder is P(a).

Proving it:Let P(x) = the polynomialLet D(x) = (x-a) = the divisorLet Q(x) = the quotient.Let R = Remainder = a real number since it must be of lower degree than (x-a) which is degree 1.

So: P(x) = D(x) Q(x) + Ror: P(x) = (x-a) Q(x) + RSubstituting “a” for “x” in the above equation: P(a) = (a-a) Q(a) + R = R

Remainder Theorem

Find the remainder when f(x) = 3x2 + 7x – 20 is divided by a) x-2b) x+1c) x+4

Check, using synthetic division.

Using the Remainder Theorem

Solving Polynomial Equations• Methods we’ll explore:

• Graphing: Versatile.

• Factoring: Relies on recognizing familiar patterns in polynomial expressions; won’t always work.

• Dividing Polynomials: If you can find one root; say x = r1, then dividing the polynomial by the linear factor (x – r1) will reduce its degree by 1. Once you get to a quadratic, you can use the quadratic formula.

See Page 297 of your text.

A Bag of Tricks

a) x4 – 3x2 = 4 (Hint: try a “change of variables”)

b) x3 = 1

c) x4 = 16

d) x3 = 8x – 2x2

e) x(x2 + 8) = 8(x+1)

Solving Polynomial Equations

Solve: x3 + 5 = 4x2 + xThe equation is satisfied when the Right Hand Side (RHS) equals the Left Hand Side (LHS).

1. So: Plot f1(x) = x3 + 5 and f2(x) = 4x2 + x simultaneously and see where they intersect. You’ll have to zoom out to get all three roots. Hit menu-> analyze graph-> intersect.

2. Or: conventionally: plot: f(x) = x3 – 4x2 –x + 5 then Hit menu-> analyze graph-> zero

Use trace function to locate minima and maxima too.

An Interesting Graphing Method

1. Stacy, Una, and Amir were all born on July 4th . Stacy is one year younger than Una. Una is two years younger than Amir. On July 4th, 2010 the product of their ages was 2300 more than the sum of their ages. How old was each person on that day?(Note: How come we have to leave units out here?)2. Find three consecutive integers whose product is 480 more than their sum.

Problems that come up Everyday

Math Hwk 23Algebra2TrigPage 300-302: 16, 20, 37 (set up an equation in one variable for solving on a graphing calculator. Solve it with a graphing calculator or at before class on Wednesday.), 64.Page 308: 49-52.Due Wednesday 1/21

Hwk 23

1. Rational Root Theorem:

Let be a polynomial with

integer coefficients. Then there are a limited number of possible roots to the equation: a) Integer roots must be factors of .b) Rational roots must have a reduced form: where p is an integer factor of and q is an integer factor of . Ex: 21x2 + 29x + 10 = 0Simply form all possible candidates for roots and then test them using synthetic division until you find a root.

Theorems about Roots of Polynomials

Strategy: Use synthetic division with candidate roots until you find a root (no

remainder). A success leaves you with a polynomial with lower degree.

Repeat until you get a quadratic. Then use the quadratic formula. Examples:Find the rational roots of: a) 3x3 + 4x2 - 5x - 2 = 0 b) x2 - 2 = 0 (hmm....the theorem does NOT guarantee an answer.

Using the Rational Root Theorem

What are the roots of: 2x3 + x2 - 7x – 6 = 0?

Rational Root Theorem: Another Example

There are Two Parts to this theorem:1. Irrational Root Theorem: If P(x) is a polynomial with rational

coefficients, then irrational roots of P(x) = 0 occur in conjugate pairs: and , where and are rational.

2. Imaginary Root Theorem: If P(x) is a polynomial with real coefficients, then complex roots of P(x) = 0 occur in complex conjugate pairs: and where and are real.

Where does it come from? If you get enough factors to reduce a polynomial down to a quadratic, the last two solutions will come in a conjugate pair from: .

The Conjugate Root Theorem

What are the Conjugates of the Following Terms?

1) A quartic Polynomial function P(x) has rational coefficients with roots and

a) What are the other roots?b) What is the polynomial function?c) Given the polynomial we found in (b), can we use the

rational root theorem to go the other way and get a root?2) A cubic polynomial P(x) has real coefficients with roots: and .

a) What are the other roots?b) What is the polynomial function?

Using the Conjugate Root Theorem

1) A third degree polynomial function P(x) has roots for P(x) = 0 of -4 and 2i. Find P(x).

2) A quartic polynomial function P(x) has roots for P(x) = 0 of 2-3i , 8, and 2. Find P(X).

More Using the Conjugate Root Theorem

Let P(x) be a polynomial in standard form with real coefficients.

The number of positive real roots equals the number of sign changes in P(x), or is less than that by an even number.

The number of negative real roots equals the number of sign changes in P(-x), or is less than that by an even number.

Descartes’ Rule of Signs

What does the rule tell you about:a) x3 – x2 + 1 = 0

b) 2x4 – x3 + 3x2 - 1 = 0

c) 2x5 -3x2 – 3x + 6

Using Descartes’ Rule of Signs

Math Hwk 24 Due WedPage 316-317: 14, 18, 25, 30, 32, 38, 40

Math Hwk 24 Due WedPage 316-317: 14, 18, 25, 30, 32, 38, 40

Math Hwk 24 Due WedPage 316-317: 14, 18, 25, 30, 32, 38, 40

Hwk 24 Due Wednesday

If P(x) is a polynomial function of degree n ≥ 1, then P(x) = 0 has exactly n roots, including multiple and complex roots.Steps to find zeros of a polynomial:1. Use a graphing calculator to find real roots2. Factor out linear factors of the form (x-a) using synthetic

division.3. Use the Quadratic Formula to find complex roots.

Example: Find all zeros of:

Fundamental Theorem of Algebra

If you raise a binomial to the nth power, you’ll get an expression of this form: (a + b)n = P0 an + P1 an-1b + P2 an-2b2 + … + Pn-2 a2bn-2 +Pn-1 abn-1 + Pn bn

Where the “P”s are constant coefficients given by the nth row of “Pascal’s Triangle”:

Examples: Pascal’s Triangle Row

(a+b)0 = 1 1 0

(a+b)1 = 1a+1b 1 1 1

(a+b)2 = 1a + 2ab + 1b 1 2 1 2

(a+b)3 = 1a3+3a2b+3ab2+1b3 1 3 3 1 3

(a+b)4 = 1a4+4a3b+6a2b2+4ab3+1b4 1 4 6 4 1 4

The Binomial Theorem

Expand the following using the Binomial Theorem:1) (x + a)3

2) (x - 2)5

3) (2x + 4)2

4) (3a - 2)3

5) The side length of a cube is (x2 - ). Express the volume of the cube in standard form.

Applying the Binomial Theorem

We’ve seen how to transform absolute value functions: Parent Function: Into the Transformed Function: And how to transform quadratic functions:Parent Function: Into the Transformed Function:

In the same way we can transform general parent functions: into transformed functions:

Transformations Revisited

For the parent function: y=x3, what function represents :1) A compression by ½, a reflection across the x-axis, a shift

to the right by 3, and a vertical shift upwards by 2?

2) A vertical stretch by 2, a shift to the left by 3, and a vertical shift down by 4?

Examples

Page 322: 18, 39 (state each value of x at which a bridge is needed.)Pages: 328-330: 10, 36, 44, 48, 52 (You must use the binomial theorem for all of these problems; don’t multiply the binomials long hand.)Page: 343: 12Due Friday

Hwk 25