roots of polynomials - mrs. purtle's math classes · web viewgraphing polynomials on the ti-84...

Roots of Polynomials

a quadratic polynomial: a cubic polynomial:

a quintic polynomial:

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 ReviewFactoring Out a Monomial: Undistributing Special Pattern: Ex: Ex:

Algebra 2: Polynomials 1

Hint:Multiply

things at a time.

Polynomial equation in standard form: y = 2x3 + 10x2 +12xExample: Write it in factored form:

Check it by multiplying:

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


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

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


-222.8-76.8-17.0101.0501.353.844.050

Some graphing calculator instructions:

To enter an equation: Press the button to get to the screen to the top rightUse the 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

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.

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.


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-84To find minimum and maximum:2ND TRACE3: minimum4: maximum


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

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

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

Change window?

Relative minimums and maximums:

Example: Graph f(x) = 10x3 + 5x2 - 40x - 20x-intercepts and y-intercepts: end behavior: Shape we expect:

Change window?


Example: Graph g(x) = x3 + 10x2 - 25x - 250 x-intercepts and y-intercepts: end behavior: Shape we expect:

Change window?



Dividing PolynomialsDivide x2 + 3x - 12 by x - 3 Divide x3 + 2x2 - 5x - 6 by x+3Is x - 3 a factor? Is x+3 a factor?

Divide x3 + 2x2 - 5x - 6 by x2 - x - 2 Divide x3 + 1 by x + 1Is it a factor? Is it a factor?


Synthetic Division

Review Long Division Synthetic Division3 x 3 - 4 x 2 + 2x - 1 3 x 3 - 4 x 2 + 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).


IF (x - 1) DOES end up being a factor,what would the associated zero be?

That is the number that goes in the box.

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

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

3 x 2 + 5 x + 2 3x +2

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


IF (3x + 2) DOES end up being a factor,what would the associated zero be?

That is the number that goes in the box.

Using Synthetic Division to Evaluate a Polynomial Equation

f(x) = x4 + 3x3 – x2 – 3x + 5x 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 = -6a = -2a = 0a = 1


Special Factoring PatternsPattern we already know: Example:

By the way: is NOT factorable.Example:

New patterns:

Examples:

More Factoring and Solving by FactoringFind the zeros...x4+3x2-10 = y x5+3x3-10x = y

x3+ 27= y 375x5+ x2= y


Solving Polynomials by GraphingCheck out the Resources page of my website: purtle.weebly.com for links to free graphing software.

What if one side has a zero?Example: Solve

Sometimes the solution isn't always an integer.Example: Solve . Round to the nearest hundredth.

Notes to help me graph:


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

roots of polynomials - mrs. purtle's math classes · web viewgraphing polynomials on the ti-84...

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