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Intro to Polynomials

Terminology Review: p244-248 College Algebra book

polynomial function

leading coefficient

leading term

degree

constant

linear

quadratic

cubic

quartic

quadratic graphs

cubic graphs

domain

range

intercepts

continuous

Leading Term Test

Even degree: Odd degree:

Example 1

(a) f(x) = 3x4 - 2x3 + 3 (b) f(x) = -5x3 - 2x2 + 4x + 2

(c) f(x) = x5 + ¼x + 1 (d) f(x) = -x6 + x5 - 4x3

Beware of problems like this!

f(x) = 15x2 - 10 + 0.11x4 - 7x3 f(x) = -6

Zeros/Roots/x-intercepts/solutions

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

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

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

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

♣ -3, -1, 2 are _____________________ 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 ______________ 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 __________________of h(x) = x3 + 2x2 - 5x - 6.

To find all of these:

Every polynomial function of degree n, with n > 1, has at least _______________ zero and at most _______________ zeros.

Ex: Are 2 and -5 zeros of P(x) = x3 + x2 -17x + 15?

Ex: Find the zeros of f(x) = 5(x - 2)(x - 2)(x - 2)(x + 1)

Ex: Find the zeros of f(x) = -(x - 1)2(x + 2)2

Multiplicity – ______________ of a zero.

What was multiplicity of the zeros from the last two examples?

f(x) = 5(x - 2)(x - 2)(x - 2)(x + 1) f(x) = -(x - 1)2(x + 2)2

We know that zeros are x-intercepts.

How do different multiplicities affect the graph?

Factor by Grouping

Ex: Find the zeros of f(x) = x3 - 2x2 - 9x + 18

Z-substitution

Ex: Find the zeros of f(x) = x4 + 4x2 - 45

Graphing Polynomials

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

x-intercepts

y-intercepts

end behavior

Shape we expect

Change window?

[pic]

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

x-intercepts

y-intercepts

end behavior

Shape we expect

Change window?

[pic]

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

x-intercepts

y-intercepts

end behavior

Shape we expect

Change window?

[pic]

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

x-intercepts

y-intercepts

end behavior

Shape we expect

Change window?

[pic]

Polynomial Models

The polynomial function M(t) = 0.5t4 + 3.45t3 - 96.65t2 + 347.7t can be used to estimate the number of milligrams of the pain relief medication ibuprofen in the bloodstream t hours after 400 mg of the medication has been taken. Find the number of milligrams in the bloodstream at t = 0, 0.5, 1, 1.5, and so on, up to 6 hr. Round the function values to the nearest 10th.

Calculator Regression

Example from p261 College Algebra

by Beecher et. al.

Today U.S. Farm acreage is about the same as it was in the early part of the twentieth century, but the number of farms has decreased. Find a quartic equation to model the trend, where x is the number of years since 1900. Use your function to estimate the number of farms in 1975, 2005, and 2010.

Example

The wiggle-level in whahoozals of a Whatsit t seconds after a wobble can be modeled by [pic]

(a) Find the wiggle-level of a Whatzit 12 seconds after a wobble.

(b) A wiggle-level of 8000 whahoozals will occur how long after a wobble?

Remainder and Factor Theorems

Polynomial Long Division

Long Division Practice

Ex: 568 ÷ 8 Ex: 42 ÷ 8 Ex: 2x2 ÷ x

Example 1 Example 2

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

Is it a factor? Is it a factor?

Example 3 Example 4

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

Is it a factor? Is it a factor? PLACE HOLDERS!

Synthetic Division

Example 1 Example 2 Example 3

Divide x2 + 3x - 12 by x – 3 Divide x3 + 2x2 - 5x - 6 by x – 3 Divide x3 + 1 by x + 1

Remainder Theorem

If a number c is substituted for x in the polynomial f(x), then the result __________ is the ____________________________ that would be obtained by dividing f(x) by x - c.

Example:

Given f(x) = 2x5 - 3x4 + x3 - 2x2 + x - 8, find f(10).

OLD WAY:

NEW WAY: Synthetic Division:

Factor Theorem

For a polynomial f(x), if f(c) = 0, then ___________________ is a factor of f(x).

Example:

Let f(x) = x3 + 2x2 - 5x - 6. Factor f(x) and solve the equation f(x) = 0.

Example:

Let f(x) = x4 - 6x3 + 13x2 - 12x +4. Factor f(x).

Suppose the following are true for some function: f(-5)=-2, f(-2)=0, f(0)=5, f(5)=-5

1. Which is a "zero" of the function?

A. -5 B. -2 C. 0 D. 5

2. Where does f(x) cross the x-axis?

3. What do we know is a factor of f ?

Polynomial Root Theorems

Fundamental Theorem of Algebra & Corollary

If P(x) is a polynomial of degree n ≥ 1 with complex coefficients, then P(x)=0 has at least ___________ complex root.

Including imaginary roots and multiple roots, an nth degree polynomial equation has exactly ___________ roots; the related polynomial function has exactly _______ zeros.

Imaginary and Irrational Root Theorems

For polynomials with rational coefficients…

Irrational Root Theorem - Irrational Roots of the form [pic]always come in pairs, called _____________.

If [pic] is a root, __________________ is too.

If (2 - √5) is a root, ___________________ is too.

Imaginary Root Theorem - Imaginary Roots of the form a+bi always come in pairs, called ___________.

If a+bi is a root, _____________________ is too.

If (3 - i) is a root, _____________________ is too.

Example

Write a 4th degree polynomial that has roots (2 - √7) and √5.

Example

Write a 4th degree polynomial that has roots (14 - i) and 2i.

Rational Root Theorem

For a polynomial with integer coefficients where the constant ≠ 0

The only possible rational roots of a polynomial =

Example: Find the roots of x3 + x2 - 3x - 3 = 0.

What are possible rational roots?

How can we test?

Find the rest

Example: Find the roots of 2x3 - x2 + 2x – 1 = 0.

What are possible rational roots?

Test

Find the rest

Example: Find the roots of 2x5 – x4 – 4x3 + 2x2 – 30x + 15 = 0 then factor.

What are possible rational roots?

Test

Find the rest

Write as factors

Polynomial Inequalities

Solving Polynomial Inequalities

1. Get 0 on one side.

2. Solve related equation.

3. Sketch graph.

4. Compare sketch with inequality symbol.

Example: Solve x3 - 1 > 0

Example: Solve 3x4 + 10x < 11x3 + 4

Rational Functions

Rational Function Definition

A rational function is a quotient function of two polynomials, that is, f(x) = p(x) where p(x) and q(x) are polynomials and q(x) is not zero polynomial. q(x)

Domain of Rational Functions

Cannot have ____________ in denominator.

Every x value that does not give 0 in denominator is in the ______________.

Ex: Find the domain of f(x) = [pic] Ex: Find the domain of f(x) = [pic].

Holes in the graph occur when the p(x) and q(x) have _________________________.

Ex: f(x) = [pic]

Vertical Asymptotes occur at zeros of denominator (not including common factors with numberator.)

Ex: f(x) = [pic]

What about (x-3)? How do you find the y-intercept?

Summary:

Holes V.A. x-int.

Horizontal Asymptotes - what happens as f(x) approaches ∞ and -∞

If degree of top ______ degree of bottom, no horiz. asympt

If degree of top ______ degree of bottom, fraction of leading coeff's is horiz asymp

If degree of top ______ degree of bottom, y = 0 is horiz. asymp.

Ex: f(x) = [pic]

Ex: f(x) = [pic]

Ex: Find both types of asymptotes of f(x) = [pic]

Slant/Oblique Asymptotes

Only happens when degree of numerator is ________________________ more than degree of denom.

To find it: ________________________ numerator by denominator. The quotient (NOT including remainder) is slant asymptote.

Note: you will never have both horizontal and slant asymptote. Why?

Ex: Find asymptotes of f(x) = [pic]

Graphing Rational Functions

Find asymptotes and holes

Find x-intercepts: where numerator = 0 (except holes)

Find y-intercept: when x = 0

Find other values to help get shape

Note: The graph of a rational function never crosses a vertical

asymptote. The graph of a rational function might or might not

cross a horizontal or slant asymptote.

Ex: f(x) = [pic]

Ex: f(x) = [pic]

Ex: f(x) = [pic]

[pic]

When there is a hole:

What is up with the asymptote?

Application (Example 11 from p299)

The temperature T, in degrees Fahrenheit, of a person

during an illness is given by the function T(t) = [pic]

where time t is given in hours since the onset of the illness.

The graph of the function is shown at right.

a ) Find the temperature at t=0, 1, 2, 5, 12, 24.

b) Find the horizontal asymptote of the graph of T(t). Complete: T(t) → _____ as t→∞.

c) Give the meaning of the answer to part b in terms of the application.

Variation and Application

Direct Variation

f(x) = kx or __________________ where k is a positive constant: we say y varies directly as x or y is directly proportional to x. The number k is called the variation constant or constant of proportionality.

As x gets bigger, y gets _____________________.

Ex: The number of centimeters W of water produced from melting snow varies directly as S, the number of centimeters of snow. Meteorologists have found that 150 cm of snow will melt to 16.8 cm of water. To how many centimeters of water will 200 cm of snow melt?

Inverse Variation

f(x) = k/x or ______________________ where k is a positive constant: we say y varies inversely as x or y is inversely proportional to x. The number k is called the variation constant or constant of proportionality.

as x gets bigger, y gets __________________________.

Ex: The time t required to do a job varies inversely as the number of people P who work on the job (assuming that all work at the same rate). If it takes 72 hr for 9 people to frame a house, how long will it take 12 people to complete the same job?

Combined Variation

Other kinds of variation:

y varies directly as the nth power of x: y =

y varies inversely as the nth power of x: y =

y varies jointly as x and z: y =

Ex: Find the equation of variation in which y varies directly as the square of x,

and y = 12 when x = 2.

Ex: Find an equation of variation in which y varies jointly as x and z, and y = 42 when

x = 2 and z = 3.

Ex: Find an equation of variation in which y varies jointly as x and z and inversely as the square of w, and y = 105 when x = 3, z = 20, and w = 2.

Ex: The volume of wood V in a tree varies jointly as the height h and the square of the girth g (girth is distance around). If the volume of a redwood tree is 216 m3 when the height is 30 m and the girth is 1.5 m, what is the height of a tree whose volume is 960 m3 and girth is 2m?

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Factor of constant (last) .

Factor of leading coefficient (first)

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