unit 3: lesson 1: 3.1 polynomial basics (4-1) lesson 2: 3.2 remainder theorem (4-3) lesson 2 lesson...

Unit 3:

Lesson 1: 3.1 Polynomial Basics (4-1) Lesson 2: 3.2 Remainder Theorem (4-3) Lesson 3: 3.3 RRT (4-4,4-5) Lesson 4: 3.4 Solving Rational Equations (4-6) Lesson 5: 3.5 Solving Radical Equations (4-7) Lesson 6: 3.1-3.3 Modeling Real World Data (4-

8)

Unit 3: “What The Function?”

In this unit we will cover… Standard 3.1: determine the number and type of roots of a

polynomial function or to build a polynomial function from its roots (4-1)

Standard 3.2: use the remainder and factor theorems to find roots of polynomial functions (4-3)

Standard 3.3: use the rational root theorem and rule of signs to find roots of polynomial functions (4-4, 4-5)

Standard 3.4: solve rational equations and inequalities (4-6)

Standard 3.5: solve radical equations and inequalities (4-7)

Standard 3.1: determine the number and type of roots of a polynomial function or to build a polynomial function from its roots (4-1)

In this section we will… Learn to recognize polynomial equations. Find the degree and leading coefficient of a

polynomial. Determine roots of polynomial equations. Apply the Fundamental Theorem of Algebra.

Polynomial Functions in One Variable:

Degree: Equal to the highest power of the variable present.

0 1 2

1 2 21 2 2 1 0

where:

, , ,..., represent complex numbers

either real or imaginary, is not zero, and

represents a nonnegative integer.

...

n

n

n n nn n n

a a a a

a n

a x a x a x a x ax a

Recognizing a Polynomial Function: State yes or no; if yes state degree.

No Yes, degree 5

Yes, degree 4 No

2 3 4 2 5 3

2 3 4 2

3 5 5 6

73 1 8 4

x xy y w w w w

z z z z x xx

Roots:

A polynomial of degree n has exactly…

Complex roots are real and imaginary roots in the form a + bi. Ex. 2, 3i, 4+3i, 4-3i

Real roots appear on the graph of a function as _________________. Imaginary do not.

Another name for roots is _________.

Can you ever have just one imaginary root?

These pairs are called “complex conjugates”.

Roots:

Let’s look at some scenarios: How many roots are possible? How many combinations of real and imaginary

roots can you have? 1st degree 2nd degree 3rd degree 4th degree 5th degree

Finding Roots of Polynomial Functions

State the number of complex roots of the equation x3 + 2x2 – 8x = 0. Then find the roots and graph the related function.

Corollary to the Fundamental Theorem of Algebra Every polynomial P(x) of degree n (n > 0) can be

written as the product of a constant k (k ≠ 0) and n linear factors.

Thus, a polynomial equation of degree n has exactly n complex roots.

1 2 3( ) ( )( )( )...( )nP x k x r x r x r x r

Writing Polynomial Functions

Write the polynomial equation of least degree with roots -3, 2i, and -2i.

Standard 3.1: determine the number and type of roots of a polynomial function or to build a

polynomial function from its roots (4-2)

In this section we will…

Solve quadratic equations.

Use the discriminant to describe the roots of a quadratic equation.

Quadratic Equations:

A quadratic equation is simply a polynomial of __________ degree.

Their parent graph is a ________________.

To find its roots you can _______________, or you can __________________________,

or use the __________________________.

The Quadratic Formula:

2 42

b b acx

a

The Discriminant:

Discriminant:

If the discriminant equals zero:

If the discriminant is positive:

If the discriminant is negative:

Completing the square

1. Move number away from rest of quad.

2. Take ½ the middle coeffcient.

3. Square your answer.

4. Add it to quad and to the number.

5. Factor the trinomial (Use answer from step 2)

2 2 2 0x x

Completing the square…again

1. Move number to other side of equal sign.

2. Divide out the coeff. of g2.

3. Take ½ the middle coeffcient.

4. Square your answer.

5. Add it to quad and to the number.

6. Factor the trinomial. (Use answer from step 2)

23 12 4g g

Using the Quadratic Formula:

Find all roots of f(x) = 2w2 + 3w + 3 = 0. Then graph the function.

Using the Discriminant:

Determine the nature of the roots of

x2 – 3x – 7 = 0. Find the roots then sketch the graph based on them.

Homework:

HW1 3.1: P 210 #15 – 49 odd, 55HW2 3.1: P 219 #13 – 37 odd

Reassessments must be complete by Monday 3/14

Warm-Up:

Draw the graph of a 4th degree polynomial that has:

a) No real roots

b) One real root

c) Two real roots

d) Three real roots

e) Four real roots

f) Five real roots

Homework:

Standard 3.2: use the remainder and factor theorems to find roots of polynomial functions (4-3)

In this section we will… Find the factors of polynomials using the

Remainder and Factor Theorems. Review synthetic division.

The Remainder Theorem

If a polynomial P(x) is divided by x – r, the remainder is a constant, P(r), and

P(x) = (x – r)•Q(x)+P(r)

where Q(x) is a polynomial with degree one less than the degree of P(x).

Huh?

Let’s check it out with numbers:

8 76 7 84

Okay…kinda

Now with polynomials:

Let’s review the steps for synthetic division.

3 2 2 3x x x

Synthetic Division:

Step 1: Arrange the terms of the polynomial in descending order. Hold the places of any missing powers of x with a zero. Write down the coefficients of each term (including zero’s).

3 2 2 3x x x

Synthetic Division:

1 1 0 2 Step 2: Write the r (opposite sign because it is x – r) as the divisor in the box. Set up a line leaving space for numbers and another box to hold your remainder under the last coefficient.

Synthetic Division:

Step 3: Bring down 1st coefficient.

Step 4: Multiply the 1st coefficient by r. Write the product under the 2nd coefficient. Add.

Step 5: Multiply the sum by r. Write the product under the next coefficient. Repeat for all coeff.’s

Step 6: The final sum in the box is the remainder. The other numbers represent Q(x), the quotient polynomial.

3 1 1 0 2

________________

Hooray!

We now know that…

But…The Remainder Theorem says:

P(x) = (x – r)•Q(x)+P(r) What gives?

3 2 22 ( 3)( 4 12) 38x x x x x

Find P(r): In our case P(3)

Who was P(x)?

If this really works we will have 2 methods to find the remainder!

3 2 2 3x x x

Let’s check it again!

Let P(x)=2x3 + x2 – 4x + 3. Check to see if the remainder found when

P(x) is divided by (x+1) is the same as the value of P(-1).

It works! We can find the remainder with either method!

More Remainder Stuff

Find the remainder when P(x)=x3 + 3x2 – 2x – 8 is

divided by (x + 2) using both methods.

The Factor Theorem

The binomial x – r is a factor of the polynomial P(x) iff P(r) = 0.

4-3 The Factor and Remainder Theorems

Determine whether x – 5 is a factor of P(x) if P(x) = x3 – 4x2 – 7x + 10.

Using the Factor Theorem:

Find all factors of f(x) = x3 + x2 – 17x + 15.

Answer: (x – 1)(x + 5)(x – 3)

Setting up a polynomial so that you get a certain root to be a factor: Find a value for k so that (x+2) is a factor of

g(x) = x3 + 8x2 + kx + 4.

Do another:

Find the value of j so that the remainder of (x3 + 6x2 – jx – 8) ÷ (x – 2) is 0.

Try this…

Determine how many times 1 is a root of x3 – 3x + 2 = 0.

Homework:

p 226 #15 – 39 odd and 43

Warm-Up:

Quiz!!!!

Homework:

Standard 3.3: use the rational root theorem and rule of signs to find roots of polynomial functions (4-4)

In this section we will… Use Descartes’ Rule of Signs to determine

how many positive and negative roots a polynomial function will have.

Learn to identify all the possible rational roots of a polynomial function.

Test the possible roots in order to find a real root of the function.

Descartes’ Rule of Signs

Helps us find the number of possible positive real roots and the number of possible negative real roots.


If P(x) is a polynomial whose terms are arranged in descending powers of the variable, then..

The number of possible positive real roots =

The number of possible negative real roots =

The number of sign changes in the P(x)

The number of sign changes in the P(-x)


State the number of possible complex zeros, the number of possible positive real zeros, and the number of possible negative real zeros for:

h(x) = x4 – 2x3 + 7x2 + 4x – 15


Which would you look for first?

1 positive, 3 or 1 negative

2 or 0 positive, 1 negative

4 or 2 or 0 positive, 2 or 0 negative

The Rational Root Theorem

Consider a polynomial of degree n with integral coefficients:

If this polynomial has any rational roots, they will be in the form , where p is a factor of a0 and q is a factor of an.

p

q

1 2 21 2 2 1 0...n n n

n n na x a x a x a x ax a

Consider the function

How many total complex roots does it have? How many possible positives? Possible negatives? Determine the possible rational zeros.

Which of these possible rational zeros are actual zeros of the function?

3 2( ) 2 3 8 3f x x x x

The RRT and Rule of Signs x4 – 5x3 + 9x2 – 7x + 2 = 0



Try another…4 32 6 3 0x x x



Standard 3.3: use the rational root theorem and rule of signs to find roots of polynomial functions (4-5)

In this section we will… Approximate the real zero’s of a polynomial

function.

3 2Graph the f unction 12 19 6x x x

See the graph? See the approximate location of the zeros?

Look at the table. What happens near our approximate zeros?

That’s the Location Principle!

The Location Principle

Let ( ) represent a polynomial f unction

and let and be two numbers.

I f ( ) is negative and ( ) is positive,

then the f unction has at least one real zero

between and .

y f x

a b

f a f b

a b

Do we use this much?

Nope!

How do we approximate zeros? Look at the graph again.

Choose 2nd CALC.

Pick 2: ZERO

That’s how we do it.

How can it help?

Checking the calculator may give you a integral root (an integer).

Knowing one or more roots will help you to get the depressed polynomial and final factoring faster.

Homework:

P 234 #11 –27 odd

DO THEM LIKE THIS!1. State the number of roots for the function (=degree).

2. Use Descartes’ Rule of Signs to state the number of possible positive and possible negative real roots.

3. Use RRT to find all the possible rational roots using p/q.

4. Use synthetic division to determine all the actual rational roots of the function.

Warm-Up:

P A32 Lesson 4-1: #1, 5, 9Lesson 4-2: #3, 7

Homework:

Standard 3.4: solve rational equations and inequalities (4-6)

In this section we will… Solve rational equations. Solve rational inequalities.

We will NOT be finding partial fractions.

Rational Equations:

2

Let's start simple:

1 34

2m

m m

First, find the domain of each part:

Solve the proportions by cross-multiplying.

Check answers against the known domain.

2

Step it up!

4 3 164 4

aa a a a

Find Domain

Determine LCD.

Build up each fraction so that they all have LCD.

Multiply by and remove LCD.

Solve.

Now You Try…

2

2 204 1 3 4

xx x x x

• Find Domain

• Determine LCD.

• Build up each fraction so that they all have LCD.

• Multiply by and remove LCD.

• Solve.

Rational Inequalities: Find domain. Solve. Set up solution(s) and

points of discontinuity on number line.

Test a value in each region.

Final solution set.

1 165

x x

Find domain. Solve. Set up solution(s) and

points of discontinuity on number line.

Test a value in each region.

Final solution set.

3 211 18 0x x x

Warm-up:

Homework:

Standard 3.5: solve radical equations and inequalities (4-7)

In this section we will…

Solve radical equations. Solve radical inequalities.

Radical Equations:

Radical means ANY root – not just square roots.

The big thing here is to CHECK YOUR ANSWERS in the original equation!

Radical equations like to throw false answers, so be careful.

Domain. Isolate radical. Raise both sides to

power to remove radical.

Solve. CHECK YOUR

ANSWERS! Could we see it

coming?

5 4 2x

Domain. Isolate radical. Raise both sides to


Solve. CHECK YOUR

ANSWERS!

33 4 12x

Domain. Isolate radical Uh Oh! Can’t isolate

radical! So get one on each side.

Raise both sides to power to remove radical (Don’t forget to FOIL).

Isolate radical. Raise both sides to

power to remove radical Solve. CHECK YOUR

ANSWERS!

3 4 2 7 3x x

Radical Inequalities: Domain. Isolate radical. Raise each side to


Solve. Test solution(s) and

domain restriction intervals.

Final solution set (includes any domain restrictions necessary).

5 4 8x

Domain. Isolate radical. Raise each side to


Solve. Test solution(s) and

domain restriction intervals.

Final solution set (includes any domain restrictions necessary).

6 5 4x

Homework:

P 247 #13-21 odd, 29-37 odd, 41 p 255 #13 – 39 odd

UNIT 3 TEST FRIDAY!!!!

Warm-up:

Pick up a couple pieces of graph paper for the lesson then do…

p 249 #40, 49, 51, 55 and 57

Homework:

In this section we will…In this section we will… Write polynomial functions to model real-

world data Use polynomial functions to interpret real-

world data.

Modeling Real-World Data with Polynomial Functions (4-8)

Getting ready…

Go to STAT PLOT ( press 2nd then Y = )

Choose 1:Plot 1

Turn to ON, make sure the TYPE shows scatter plot. Xlist should show L1and Ylist should show L2. Mark should be on 1st one the boxed point.

Choosing a Mathematical Model

First graph your data and sketch a best fit function.

Decide which of our function parent graphs best fits the data.

Use the calculator to develop the regression equation.

Let’s Git’er Done.

You need to set the window to fit your data. Go to WINDOW change settings: Xmin=0, Xmax=20, Xscl=2; Ymin= -5, Ymax=10, Yscl=2.

Press STAT, choose 1:Edit… In the L1 column enter the x-values; in the L2

column enter the f(x)-values. Press GRAPH. Press STAT, move to the CALC column,

choose 6:CubicReg. Press ENTER twice.

The Project:

Things to Remember…

You may need to adjust your window.

Homework:

P 262 #9 – 21 odd

Unit 3 Test Tuesday!!!