Math 65 Blitzer 3rd Edition

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Section 6.4
Polynomials in Several Variables

• Math 65 Blitzer 3rd Edition

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6.4 Polynomials in Several Variables
Objectives (p352)

• Evaluate polynomials in several variables.
• Understand the vocabulary of polynomials in two
  variables.
• Add and subtract polynomials in several
  variables.
• Multiply polynomials in several variables.

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6.4 Obj 1 Evaluate polynomials in several
variables p352-353

• P35882 The storage shed shown in the figure has
  a volume given by the polynomial

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6.4 Obj 1 Evaluate polynomials in several
variables p352-353

• P35882 a. A small businesss is considering
  having a shed installed like the one shown in the
  figure. The sheds height, 2x, is 26 feet and
  its length, y is 27 feet. Using x 13 and y
  27, find the volume of the storage shed.

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6.4 Obj 1 Evaluate polynomials in several
variables p352-353

• This is the given polynomial
• Replace x by 13 and y by 27
• Evaluate the exponential expressions first!
• x2 132 (13)(13) 169(avoid the
  temptation to do 2(13) 26 first!)
• Perform indicated multiplication.
• This is the EXACT answer, but its not practical!
• ? is approx. 3.14

• 9126 2281.5?
• 9126 2281.5(3.14)
• 9126 7163.91
• 16289.91
• ? 16290 cubic feet

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6.4 Obj 1 Evaluate polynomials in several
variables p352-353

• Need to answer in a sentence!
• The volume of the storage shed is about 16290
  cubic feet.
• Note Use the ? button on your calculator. It
  carries ? out to more decimal places-how does it
  effect your answer?
• 9126 2281.5? Why is this NOT 11407.5 ??

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6.4 Obj 1 Evaluate polynomials in several
variables p352-353

• b) The business requires at least 18,000 cubic
  feet. Should they construct the storage shed
  described in part a?

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6.4 Obj 1 Evaluate polynomials in several
variables p352-353

• Evaluating a polynomial in several variables.
• Substitute the given value for each variable
• Perform the resulting computation using the order
  of operations.(Especially remember to do
  exponents before multiplication! Then
  multiplication before addition)

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6.4 Obj 2 Understand the vocabulary of
polynomials in two variables-p352-353

• In general, a polynomial in two variables, x and
  y, contains the sum of one or more monomials in
  the form axnym. The constant a is the
  coefficient. (KNOW THIS!)
• The exponents n and m represent whole numbers.
  The degree of the monomial axnym is nm.
• (Nice to know-will not be tested on this)

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6.4 Obj 3 Add and Subtract polynomials in
several variables- p353-354
Adding and subtracting polynomials in 2
variables, is the same as in one variable. Add
Just combine like terms. (Same variables with
same exponents!) Subtract Distribute the
subtraction sign to every term of the polynomial
being subtracted. (Change all its signs!) Then
add.
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6.4 Obj 4 Multiplying Monomials-p354-355

• Multiplying Monomials
• To multiply monomials, multiply the coefficients
  and add the exponents on variables with the same
  base.

Example 5 p355 Multiply (7x2y)(5x3y2)
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6.4 Obj 4 Multiplying a Monomial and Poly-p355

• Same as with one variable! We use the
  distributive property to multiply a monomial and
  a polynomial that is not a monomial. For
  example
• 3x2y (2x35x1) 3x2y (2x3) 3x2y (5x) 3x2y
  (1)
• 3(2) x2 x3 y 3(5) x2
  x1 y 3x2y
• 6 x5y 15 x3 y 3x2y

Both of these steps can be done mentally!
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6.4 Obj 4 Multiplying Polynomials in Two
Variables-Same as one variable! p355
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Consider the equation of motion for an object
tossed straight up into the air (projected
vertically)-or dropped (falling) p359)
s -½ g t 2 vo t so where
s is vertical position (height above ground) of
the tossed object in feet (ft) t is time in
seconds that the object has been in motion
(sec) g is acceleration due to gravity 32
ft/sec2 vo is original speed in ft/sec of
object so is original position (i.e. at t
0) of object
This figure shows that a ball is thrown straight
up from a roof top at an original (initial)
velocity of 80 ft/sec from an original (initial)
height of 96 feet above the ground. The ball
missed the rooftop on its way down and eventually
strikes the ground
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Consider the equation of motion for an object
tossed straight up into the air (projected
vertically)-or dropped (falling) p359)
This figure shows that a ball is thrown straight
up from a roof top at an original (initial)
velocity of 80 ft/sec from an original (initial)
height of 96 feet above the ground. The ball
missed the rooftop on its way down and eventually
strikes the ground
s -½ g t 2 vo t so g 32, vo 80 so
96 So the above very general formula becomes a
little more specific for an object with this
original velocity starting at this height s -
16t 2 80 t 96
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Consider the equation of motion for an object
tossed straight up into the air (projected
vertically)-or dropped (falling) p359)
For an object with an original velocity original
(initial) velocity of 80 ft/sec from an original
(initial) height of 96 feet above the ground s
- 16t 2 80 t 96 s is still the vertical
position (height above ground) of the tossed
object in feet (ft) t is time in seconds
(sec) that the object has been in motion

• How high above the ground will the ball be
• 2 seconds after being thrown?
• 4 seconds after being thrown
• 6 seconds after being thrown? Describe what this
  means in practical terms.

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6.4 Obj 2 The Power Rule for Exponents-p334
The graph visually displays the information about
the thrown ball described in Exercises 83-85.
The horizontal axis represents the balls time in
motion in seconds. The vertical axis represents
the balls height above the ground. 86. During
which time period is the ball rising? 87. During
which time period is the ball falling?
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6.4 Obj 2 The Power Rule for Exponents-p334

• Identify your answer from Exercise 84 as a point
  oh the graph.
• Identify your answer from Exercise 83 as a point
  oh the graph.
• After how many seconds doew the ball strike the
  ground?
• After how many seconds does the ball reach its
  maximum height above the ground? What is a
  reasonable estimate of this maximum height?

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On a sheet of paper for one Quiz point Print your
name and
From TODAYS lesson 1) Describe one main math
idea 2) Identify one math concept or idea that
interests you. 3) Ask at least one question
about the math we covered.