Roots of Polynomials

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Roots of Polynomials

a quadratic polynomial: [pic] a cubic polynomial: [pic]

a quintic polynomial: [pic]

Polynomial equation in factored form: y = (x + 3)(x + 1)(x - 2)

Example: Write it in standard form: y = (x + 3)(x + 1)(x - 2)

Factoring Review

Factoring Out a Monomial: Undistributing Special Pattern: [pic]

Ex: [pic] Ex: [pic]

Polynomial equation in standard form: y = 2x3 + 10x2 +12x

Example: Write it in factored form:

Check it by multiplying:

Example: Write it in factored form: y = 3x3 - 3x2 -36x

Example: Write it in factored form: y = 75x3 - 20x

Zeros/Roots/x-intercepts/solutions

y = x3 + 2x2 - 5x - 6 = (x + 3)(x + 1)(x - 2)

• (x + 3), (x + 1), & (x - 2) are factors of x3 + 2x2 - 5x - 6.

• -3, -1, 2 are solutions of x3 + 2x2 - 5x - 6 = 0.

• Plugging each of these numbers in for x makes equation true.

• -3, -1, 2 are x-intercepts of the graph of y = x3 + 2x2 - 5x - 6.

• When x = -3, x = -1, or x = 2, the graph is on the x-axis.

• -3, -1, 2 are zeros of h(x) = x3 + 2x2 - 5x - 6.

• When the graph is on the x-axis, the y-value - or function value - equals zero.

• -3, -1, 2 are roots of h(x) = x3 + 2x2 - 5x - 6.

To find all of these: set y = ______ and ________________.

Examples: Find the zeros of the functions.

f(x)=(x-3)(x+4)(x-1)

g(x) = 2x3 + 10x2 +12x

h(x) = 75x3 - 20x

Intro to Graphing Polynomial Functions

Some graphing calculator instructions:

To enter an equation:

Press the [pic] button to get to the screen to the top right

Use the [pic] button for the variable x.

Use the parenthesis buttons for parenthesis.

Use the ^ button for exponents.

To see the graph:

Press the GRAPH button to get to the third screen shown.

To adjust the window:

If the window on your graph does not count 10 in each direction, you can reset the window by pressing the ZOOM button and choosing option 6:ZStandard.

To see the table:

Press the 2ND button then GRAPH.

(See TABLE written above the graph button.)

You should be able to scroll up and down with the arrow buttons to see different x-values and the y-values that go with them. It should look something like the third screen shot. If it does not, you will have to reset the table.

To reset the table:

Press the 2ND button then WINDOW. (See TBLSET above the window button.)

Change the settings to match the bottom screen shot. To highlight a word, use the arrow and ENTER buttons.

Some ANY calculator instructions:

To evaluate an equation for a given x-value:

You can “plug in” an x-value to the standard OR factored form. Both should give you the same result.

Ex: To plug in x=-6 to the last example using the standard form. To avoid making errors with a negative, use parenthesis.

Ex: To plug in x=-6 to the last example using the factored form.

Classifying Polynomials

[pic]

Degree (biggest exponent on variable):

Leading term (term with highest degree):

Leading coefficient (coefficient of leading term):

More vocab: see chart p301 in text.

Examples:

Write the polynomial in standard form.

Then classify it by degree and by number of terms.

[pic] [pic]

The greatest value (y-value) of the points in a region of a graph is called a __________________.

(Think of the top of a hill.)

The least value (y-value) of the points in a region of a graph is called a __________________.

(Think of the bottom of a valley.)

Multiplicity of a Zero

A repeated zero is called a __________________. A multiple zero has a ____________________ equal to the number of times the zero occurs.

Example: Find the zeros of the function. State the multiplicity of multiple zeros.

[pic] [pic]

Examples:

Write a polynomial function in standard form with zeros at -2 and 3 (multiplicity 2).

Write a polynomial function in standard form with zeros at -4, -2, and 0 (multiplicity 3).

Graphing Polynomials on the TI-84

To find minimum and maximum:

2ND TRACE

3: minimum

4: maximum

[pic] [pic] [pic] [pic]

Example: Graph g(x) = x4 - 7x3 + 12x2 + 4x - 16 = (x + 1)(x - 2)2(x - 4)

Before we graph what x-intercepts and y-intercepts do we expect?

Graph using calculator. Do we need to change window?

ZOOM 6:ZStandard WINDOW

[pic] [pic]

Example: Graph h(x) = -2x4 + 3x3

x-intercepts and y-intercepts: end behavior: Shape we expect:

Change window?

[pic] Relative minimums and maximums:

Example: Graph f(x) = 10x3 + 5x2 - 40x - 20

x-intercepts and y-intercepts: end behavior: Shape we expect:

Change window?

[pic] Relative minimums and maximums:

Example: Graph g(x) = x3 + 10x2 - 25x - 250

x-intercepts and y-intercepts: end behavior: Shape we expect:

Change window?

[pic] Relative minimums and maximums:

Dividing Polynomials

Divide x2 + 3x - 12 by x - 3 Divide x3 + 2x2 - 5x - 6 by x+3

Is x - 3 a factor? Is x+3 a factor?

Divide x3 + 2x2 - 5x - 6 by x2 - x - 2 Divide x3 + 1 by x + 1

Is it a factor? Is it a factor?

Synthetic Division

Review Long Division Synthetic Division

3x3 - 4x2 + 2x - 1 3x3 - 4x2 + 2x - 1

x – 1 x - 1

(5x3 - 6x2 + 4x - 1) ÷ (x - 3) (x4 - 5x2 + 4x + 12) ÷ (x + 2)

Factor (x3 - 13x + 12).

Hint: Here is what the graph of y = x3 - 13x + 12 looks like.

Factor (x3 - 6x2 + 3x +10).

Hint: One factor is (x - 5).

Use Synthetic Division to Factor (x3 - 4x2 - 3x +18).

Use Synthetic Division to Factor (x4 + 6x3 + 8x2).

3x2 + 5x + 2

3x +2

Use Synthetic Division to Factor (x3 + 12x2 + 47x + 60).

Using Synthetic Division to Evaluate a Polynomial Equation

f(x) = x4 + 3x3 – x2 – 3x + 5

x y

-6

-5

-4

-3

-2

-1

-0.5

0

0.5

1

2

3

4

5

6

We can find f (a) using Synthetic Division.

Put ____ in the box.

The ______________ = f (a)

Use Synthetic Division to find the following for the above function.

a = -6

a = -2

a = 0

a = 1

Special Factoring Patterns

Pattern we already know: [pic]

Example: [pic]

By the way: [pic] is NOT factorable.

Example: [pic]

New patterns: [pic]

[pic]

Examples: [pic]

[pic]

More Factoring and Solving by Factoring

Find the zeros...

x4+3x2-10 = y x5+3x3-10x = y

x3+ 27= y 375x5+ x2= y

Solving Polynomials by Graphing

Check out the Resources page of my website: purtle. for links to free graphing software.

[pic]

[pic] [pic]

What if one side has a zero?

Example: Solve [pic]

Sometimes the solution isn't always an integer.

Example: Solve [pic]. Round to the nearest hundredth.

Notes to help me graph:

-----------------------

Hint:

Multiply

things at a time.

[pic]

Hint:

Sometimes you will have to use more than one factoring technique.

[pic]

[pic]

-222.8

-76.8

-17.01

0

1.05

0

1.35

3.84

4.05

0

[pic]

[pic]

IF (x - 1) DOES end up being a factor,

what would the associated zero be?

That is the number that goes in the box.

IF (3x + 2) DOES end up being a factor,

what would the associated zero be?

That is the number that goes in the box.

Graph each of these and find the ______________ where they ________________.

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