# Polynomials in Several Variables

**§ 6.4 Polynomials in Several
Variables – Objectives (p352)**

1. Evaluate polynomials in several

variables.

2. Understand the vocabulary of

polynomials in two variables.

3. Add and subtract polynomials in

several variables.

4. Multiply polynomials in several

variables.

**§ 6.4 Obj # 1 Evaluate polynomials
in several variables: p352-353**

P358#82 The storage shed shown in the

figure has a volume given by the

polynomial

P358#82 a. A small businesss is

• This is the given polynomial

• Replace x by 13 and y by 27

• Evaluate the exponential

expressions first!

x^{2} = 13^{2} = (13)(13) = 169

(avoid the temptation to do

2(13) = 26 first!)

• Perform indicated

multiplication.

• This is the EXACT answer,

but it’s not practical!

• π is approx. 3.14

• Need to answer in a sentence!

**The volume of the storage shed is
about 16290 cubic feet.**

• Note: Use the π button on your

calculator. It carries π out to more

decimal places-how does it effect

your answer?

• 9126 + 2281.5π Why is this NOT

11407.5 π?

b) The business requires at least

18,000 cubic feet. Should they

construct the storage shed

described in part a?

Evaluating a polynomial in several

variables.

1. Substitute the given value for each

variable

2. Perform the resulting computation

using the order of operations.

(Especially remember to do exponents

before multiplication! Then

multiplication before addition)

**§ 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 ax^{n}y^{m}. The constant

a is the coefficient. (KNOW THIS!)

The exponents n and m represent whole

numbers. The degree of the monomial

ax^{n}y^{m} is n+m.

(Nice to know-will not be tested on this)

**§ 6.4 Obj # 3 Add and Subtract
polynomials in several variablesp353-
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.

**§ 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: (7x^{2}y)(5x^{3}y^{2})

**§ 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:

**§ 6.4 Obj # 4 Multiplying Polynomials in
Two Variables-Same as one variable! p355**

The Square of a Binomial Sum

(6x + 4y)^{2} = 36x^{2} + 2(24)xy + 16y ^{2}

= 36x ^{2} + 48xy + 16y ^{2}

The Square of a Binomial Difference

(9x - 3y)^{2} = 81x ^{2} – 2(27)xy + 9y ^{2}

= 81x ^{2} – 54xy + 9y ^{2}

Product of a Sum & Difference of Two Terms

(5x + 3y) (5x - 3y) = 25x ^{2} - 9y ^{2}

**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} + v_{o} t + s_{o} 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/sec^2

v_{o} is original speed in ft/sec of object

s_{o} 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

s = -½ g t^{2} + v_{o} t + s_{o}

g = 32, v_{o} = 80 s_{o} = 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

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

83. 2 seconds after being thrown?

84. 4 seconds after being thrown

85. 6 seconds after being thrown? Describe

what this means in practical terms.

**§ 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 ball’s 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?

88. Identify your answer

from Exercise 84 as a

point oh the graph.

89. Identify your answer

from Exercise 83 as a

point oh the graph.

90. After how many seconds

doew the ball strike the

ground?

91. After how many seconds

does the ball reach its

maximum height above

the ground? What is a

reasonable estimate of

this maximum height?

**On a sheet of paper for one Quiz point
Print your name and:**

From TODAY’S 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.