math 16020 handouts spring 2018 - purdue universityweld/spring2018/handoutsspring2018.pdf · math...

MATH 16020 HANDOUTSSPRING 2018

1

2 MATH 16020 SPRING 2018

MATH 16020 SPRING 2018 3

Lesson 5: Integration by Parts (II)

Example 1. Find the area under the curve of f(x) = x(x − 3)6 over the interval0 ≤ x ≤ 3.

Example 2. Suppose a turtle is moving at a speed of 18(t+1)3 ln(t+1)1/9 miles/hour.How far does the turtle travel in half an hour? Round your answer to the nearestthousandth.


Example 3. A factory produces pollution at a rate of14 ln(7t+ 1)

(7t+ 1)3tons/week. How

much pollution does the factory produce in a day? Round your answer to the nearesthundredth.

Example 4. Suppose the probability of a gold necklace having a gold purity of 100xpercent (so 0 ≤ x ≤ 1) is given by

P (x) =9e3

e3 − 4xe−3x.

Find the probability that a gold necklace has a purity of at least 75%. Round youranswer to the nearest percent.


Lesson 8: Differential Equations: Separation of Variables (II)

Example 1. Find the general solution to

x3y′ = y′ + x2e−y.

Example 2. Find y(2) if y is a function of x such that

xy6y′ = 2 and y = 1 when x = 1.


Example 3. Suppose during a chemical reaction, a substance is converted into adifferent substance at a rate inversely proportional to the amount of the originalsubstance at any given time t. If, there were initially 10 grams of the original substanceand after an hour only half remained, how much of the original substance is thereafter 2 hours?

Example 4. A 500-gallon tank initially contains 250 gallons of brine, a salt andwater combination. Brine containing 2 pounds of salt per gallon flows into the tankat a rate of 4 gallons per minute. Suppose the well-stirred mixture flows out of thetank at a rate of 2 gallons per minute. Set up a differential equation for the amountof salt (in pounds) in the tank at time t (minutes).


Example 5. A 700-gallon tank initially contains 400 gallons of brine containing 50pounds of dissolved salt. Brine containing 6 pounds of salt per gallon flows into thetank at a rate of 3 gallons per minute, and the well-stirred mixture flows out of thetank at a rate of 3 gallons per minute. Find the amount of salt in the tank after 10minutes. Round your answer to 3 decimal places.


Lesson 10: First Order Linear Differential Equations (II)

Example 1. Suppose a silo contains 50 tons of grain and that a farmer is movingthe grain to another silo. If the amount of grain in the second silo changes at a rateproportional to the amount of grain in the first silo, find a differential equation thatrepresents this situation.

Example 2. Find the integrating factor of

(sin 2x)y′ − 2(cot 2x)y = − cos 2x.


Example 3. A store has a storage capacity for 50 printers. If the store currentlyhas 25 printers in inventory and the management determines they sell the printersat a daily rate equal to 10% of the available capacity, when will the store sell out ofprinters?

Example 4. An 850-gallon tank initially contains 250 gallons of brine containing 50pounds of dissolved salt. Brine containing 4 pounds of salt per gallon flows into thetank at a rate of 5 gallons per minute. The well-stirred mixture then flows out of thetank at a rate of 2 gallons per minute. How much salt is in the tank when it is full?(Round your answer to the nearest hundredth.)


Lesson 17: Geometric Series and Convergence (II)

Example 1. A ball has the property that each time it falls from a height h onto theground, it will rebound to a height of rh for some 0 < r < 1. Find the total distance

traveled by the ball if r =1

3and it is dropped from a height of 9 feet.

Example 2. Suppose that in a country, 75% of all income the people receive is spentand 25% is saved. What is the total amount of spending generated in the long runby a $10 billion tax rebate which is given to the country’s citizens to stimulate theeconomy if saving habits do not change? Include the government rebate as part ofthe total spending.


Example 3. How much should you invest today at an annual interest rate of 4%compounded continuously so that in 3 years from today, you can make annual with-drawals of $2000 in perpetuity? Round your answer to the nearest cent.

Example 4. 500 people are sent to a colony on Mars and each subsequent year 500more people are added to the population of the colony. The annual death proportionis 5%. Find the eventual population of the Mars colony after many years have passed,just before a new group of 500 people arrive.


Lesson 21: Differentials of Multivariable Functions

Let z = f(x, y) be a function. All the differential problems come down to applyingthe incremental approximation formula for functions of two variables:

∆z ≈ ∂z

∂x∆x+

∂z

∂y∆y.

The ∆ simply refers to a change in the values z, x, or y. The∂z

∂xand

∂z

∂yare the

usual partial derivatives. Each problem involves determining what is known and thensolving for the unknown.

Example 1. Use increments to estimate the change in z at (1,−1) if∂z

∂x= 3x + y

and∂z

∂y= 9y given ∆x = .01 and ∆y = .02.

Example 2. Suppose that when a babysitter feeds a child x donuts and y pieces of

cake, the child needs to run√x2y + 7 laps in the backyard to be able to go to bed

before the parents get home. If one evening the babysitter gives the child 3 donutsand 2 pieces of cake and the next time babysitting, 3.5 donuts and 1.5 pieces of cake,estimate the difference in the number of laps the child will need to run.


Example 3. A company produces boxes with square bases. Suppose they initiallycreate a box that is 10 cm tall and 4 cm wide but they want to increase the box’sheight by .5 cm. Estimate how they must change the width so that the box stays thesame volume.

Example 4. Suppose the function S = W 2F + F 2W describes the number of fernspores (in millions) released into the air where F is the number of ferns in an areaand W is the speed of the wind in miles per hour. Suppose F = 56 and W = 10with maximum errors of 2 ferns and 3 miles per hour. Find the approximate relativepercentage error in calculating S. Round your answer to the nearest percent.


Lesson 24: Extrema of Functions of Two Variables (II)

Example 1. We are tasked with constructing a rectangular box with a volume of64 cubic feet. The material for the top costs 8 dollars per square foot, the materialfor the sides costs 10 dollars per square foot, and the material for the bottom costs4 dollars per square foot. To the nearest cent, what is the minimum cost for such abox? (Round your answer to 2 decimal places.)


Example 2. The post office will accept packages whose combined length and girthis at most 50 inches (girth is the total perimeter around the package perpendicularto the length and the length is the largest of the 3 dimensions). What is the largestvolume that can be sent in a rectangular box? (Round your answer to the nearestinteger.)


Example 3. A biologist must make a medium to grow a type of bacteria. Thepercentage of salt in the medium is given by S = 0.01x2y2z, where x, y, and z areamounts in liters of 3 different nutrients mixed together to create the medium. Theideal salt percentage for this type of bacteria is 48%. The costs of x, y, and z nutrientsolutions are respectively, 6, 3, and 8 dollars per liter. Determine the minimum costthat can be achieved. (Round your answer to the nearest 2 decimal places.)


Example 4. A manufacturer is planning to sell a new product at the price of 310dollars per unit and estimates that if x thousand dollars is spent on developmentand y thousand dollars is spent on promotion, consumers will buy approximately

270y

y + 4+

300x

x+ 9units of the product. If manufacturing costs for the product are 220

dollars per unit, how much should the manufacturer spend on development and howmuch on promotion to generate the largest possible profit? Round your answer tothe nearest cent.


Lesson 26: Lagrange Multipliers (II)

Example 1. There is an ant on a circular heated plate which has a radius of 10meters. Let x and y be the meters from the center of the plate measured horizontallyand vertically respectively. Suppose the temperature of the plate is given by f(x, y) =x2− y2 + 150 F◦ and that the ant is walking along the edge of the plate. What is thewarmest spot the ant can find?

Example 2. A rectangular box with a square base is to be constructed from materialthat costs $5/ft2 for the bottom, $4/ft2 for the top, and $10/ft2 for the sides. Findthe box of the greatest volume that can be constructed for $216. Round your answerto 4 decimal places.


Example 3. A rectangular building with a square front is to be constructed frommaterials that cost $10 per ft2 for the flat roof, $20 per ft2 for the sides and back,and $15 per ft2 for the glass front. We will ignore the bottom of the building. Ifthe volume of the building is 10,000 ft3, what dimensions will minimize the cost ofmaterials?

Example 4. On a certain island, at any given time, there are R hundred rats and Shundred snakes. Their populations are related by the equation

(R− 16)2 + 20(S − 16)2 = 81.

What is the maximum combined number of rats and snakes that could ever be on theisland at the same time? (Round your answer to the nearest integer).


Lesson 30: Systems of Equations, Matrices, Gaussian Elimination

A system of equations is just a list of equations. The goal is to find the inputswhich make the list true; we call these inputs solutions.

Types of Solutions

InconsistentThere are no solutions

Ex

{x+ y = 1x+ y = −1

No (x, y) work

Consistent IndependentThere is 1 solution

Ex

{x+ y = 2−x+ y = 0

(x, y) = (1, 1)

Consistent DependentThere are many solutions

Ex

{x+ y = 1

2x+ 2y = 2(x, y) = (1− t, t) for any t

The following 3 operations on a system of equations will not change the set ofsolutions:

(1)

(2)

(3)

Solving a system of equations by algebraic manipulation the equations is calledthe elimination method.

We can write a system of equations as an augmented matrix.

A matrix looks like

[1 03 12

]and an augmented matrix looks like

[6 −1 32 0 −7

].

Ex 1.

x y const[13

012

79

]l

x + y =

x + y =

x y z const

l{6x + (−1)y + 3z = 02x + (0)y + (−7)z = 6

We can do similar operations to matrices as we can to systems of equations. Wecall them row operations. We may

(1)

(2)

(3)


We can also solve systems of equations when they are in matrix form. To solvethese systems of equations using matrices, we put matrices into row-echelon form:[

1 # #0 1 #

],

[1 # #0 0 0

],

1 # # #0 1 # #0 0 1 #

, 1 # # #

0 1 # #0 0 0 0

Solving a system of equations by putting a matrix in row-echelon form is called

Gaussian elimination.

Examples.

1. Solve the following system of equations using matrices:{3x + 2y = 76x + 3y = 12

Translate

−−→

−2R1+R2→R2

−−→

−R2→R2

−−→

1

3R1→R1

−−→

Translate

−−→

{x + y =

y =

Solution: (x, y) = ( , )

2. Solve {2x + 6y = 103x + 5y = 11

Translate

−−→

1

2R1→R1

−−→

−3R1+R2→R2

−−→

− 1

4R2→R2

−−→

Translate

−−→

{x + y =

y =

Solution: (x, y) = ( , )


3. Put the following matrix into row-echelon form: 1 −2 3 90 1 0 52 5 5 29

−−−−−→

−−−−−→

−−−−−→

4. A goldsmith has two alloys of gold with the first having a purity of 90% andthe second having a purity of 70%. If x grams of the first are mixed with ygrams of the second such that we get 100 grams of an alloy containing 80%gold, find x to the nearest gram.

5. Solve and classify the following system of equations: 3x + 2y + z = 1x + y + 2z = 0

4x + 3y + 3z = 1

−−−−−→

−−−−−→


−−−−−→

−−−−−→

−−−−−→

−−−−−→


Lesson 31: Gauss-Jordan Elimination

A matrix is in reduced row-echelon form if it looks like

[1 0 #0 1 #

],

1 0 0 #0 1 0 #0 0 1 #

, 1 0 0 #

0 1 0 #0 0 0 #

.

The method of putting a matrix into reduced row-echelon form is called Gauss-Jordan elimination.

Examples.

1. Use Gauss-Jordan elimination to solve

{2x+ 3y = −5−x+ 2y = −8

Translate

−−−−−→

−−−−−→

−−−−−→

−−−−−→

−−−−−→

−−−−−→

Solution: (x, y) =( , )

2. Put the following matrix into reduced row-echelon form:

−2 3 3 −41 −1 2 5−1 2 −1 −5

−−−−−→

−−−−−→

−−−−−→

−−−−−→


−−−−−→

−−−−−→

−−−−−→

−−−−−→

3. Solve the following using Gauss-Jordan elimination: 3x − 2y − 6z = 1x + 2y + z = 0−x + 2y − z = 4

−−−−−→

−−−−−→

−−−−−→

−−−−−→

−−−−−→

−−−−−→

−−−−−→

−−−−−→


−−−−−→

−−−−−→

Solution: (x, y, z) =( , , )

4. Use Gauss-Jordan elimination to solve the system of equations:

x + y + z = 145x + 2y + 5z = 52

y − 2z = 2

−−−−−→

−−−−−→

−−−−−→

−−−−−→

−−−−−→

−−−−−→

−−−−−→

−−−−−→

Solution: (x, y, z) =( , , )


5. Use Gauss-Jordan elimination to solve the system of equations 5x + 5y + 3z = 55−5x + 5y + 4z = 115

2x + 4y + z = 40

−−−−−→

−−−−−→

−−−−−→

−−−−−→

−−−−−→

−−−−−→

−−−−−→

−−−−−→

−−−−−→

−−−−−→

−−−−−→

Solution: (x, y, z) =( , , )


Lesson 32: Matrix Operations

The dimensions of a matrix are always given by row×column.

Ex 1.[1 0 −10 7 2

]2× 3 matrix

−201

3× 1 matrix

[1 1−1 1

]2× 2 matrix

Elements in a matrix are specified by the ordered pair (row, column).

Ex 2. 6 is the (2, 3)-entry of the matrix

[1 2 34 5 6

]Matrix Addition : We add two matrices component-wise, that is, by adding

each entry that has the same (row, column). We can only add matrices that have thesame dimensions.

Ex 3. [2 1−1 3

]+

[3 −10 −5

]=

[2 + (3) 1 + (−1)−1 + (0) 3 + (−5)

]=

[5 0−1 −2

]

−1 3 20 1 −11 2 5

+

1 1 −52 3 4−1 −2 −1

=

=

Scalar Multiplication : A scalar is a number that isn’t in a matrix. We use

the term scalar to differentiate it from the entries of a matrix. We can multiplymatrices by scalars, which amounts to multiplying each entry in the matrix by thescalar.

Ex 4.

3

[1 2−1 3

]=

[3(1) 3(2)

3(−1) 3(3)

]=

[3 6−3 9

]

2

2 1 −1−1 3 0

0 7 5

=

=

Matrix Multiplication : We can multiply matrices together. This is not done

component-wise. There is an excellent reason why we do matrix multiplication thisway but the reason is beyond the scope of this class.


Ex 5. If

A =

[1 0 02 1 3

]and B =

3−1

2

,find AB.

[1 0 02 1 3

] 3−1

2

=

[1(3) + 0(−1) + 0(2)2(3) + 1(−1) + 3(2)

]=

[3

11

]

Notice that in terms of the dimensions of the matrix, we have (2×3)(3×1) = 2×1.This is not an accident. In a similar way, (5× 2)(2× 3) = 5× 3.

Sometimes matrix multiplication doesn’t make sense. For example, BA doesn’tmake sense because the number of columns on the left has to equal the number ofrows on the right.

Examples.

1. Let A =

[2 1−1 0

]and B =

[3 04 −1

]. Find 3A, 3A−B, AB, and BA.

3A = 3

[2 1−1 0

]=

=

3A−B = 3

[2 1−1 0

]−[

3 04 −1

]=

−

=

=

AB =

[2 1−1 0

] [3 04 −1

]=

=

BA =

[3 04 −1

] [2 1−1 0

]=

=

In general, AB 6= BA. So order matters for matrix multiplication.


2. If A =

1 0 11 2 −1−1 −1 3

, find A2. What is the (3, 2)-entry of A2?

A2 = A · A =

1 0 11 2 −1−1 −1 3

1 0 11 2 −1−1 −1 3

=

=

The (3, 2)-entry is .

3. If A =

[1 −1 20 3 −2

]and B =

1 1−1 0

2 3

, find AB.

AB =

[1 −1 20 3 −2

] 1 1−1 0

2 3

=

=

4. Let M =

[1 −11 0

]. Find M2 − 3M .

M2 − 3M =

[1 −11 0

] [1 −11 0

]− 3

[1 −11 0

]


=

−

=

−

=


Lesson 33: Inverses and Determinants of Matrices (I)

Definition 1. The square matrix with 1s along the diagonal and 0s elsewhere iscalled the identity matrix.

Ex 1.

I2 =

[1 00 1

]2× 2 identity matrix

I3 =

1 0 00 1 00 0 1

3× 3 identity matrix

If In is the n× n identity matrix, then for any n× n matrix A,

AIn = A = InA.

For some square matrices A, there exists a inverse matrix A−1, i.e.,

AA−1 = In = A−1A.

Method for Finding Matrix Inverses

Let A be an n× n matrix. Create a new matrix

B =[A In

].

Use row-operations to put B into reduced-row echelon form. If A has aninverse, A−1, then the resulting matrix, B′, will be of the form

B′ =[In A−1

].

Ex 2. Let A =

[2 1−1 0

]. Find A−1.

[2 1 1 0−1 0 0 1

]R1↔R2

−−−−−→[−1 0 0 1

2 1 1 0

]−R1→R1

−−−−−→[

1 0 0 −12 1 1 0

]−2R1+R2→R2

−−−−−→[

1 0 0 −10 1 1 2

]Thus,

A−1 =

[0 −11 2

].

Quick Check: Show that

AA−1 =

[2 1−1 0

] [0 −11 2

]=

[1 00 1

]and A−1A =

[0 −11 2

] [2 1−1 0

]=

[1 00 1

].


Examples.

1. Given A =

1 1 −10 1 3−1 0 −1

, find A−1 if it exists.

From our method above, our matrix B is

1 1 −1 1 0 00 1 3 0 1 0−1 0 −1 0 0 1

. We

put this in reduced row-echelon form.

−−−−−→

−−−−−→

−−−−−→

−−−−−→

−−−−−→

−−−−−→

So

A−1 =

.

Finding Solutions using Inverse Matrices

A column vector is a matrix of the form[10

]or

−182

.Let AX = Y be a system of equations where A is the n×n coefficient matrixand X and Y are column vectors, then

X = A−1Y.


Examples.

2. Find the solution of

{2x+ 3y = −5−x+ 2y = −8

using inverse matrices.

Here A =

[2 3−1 2

], X =

[xy

], Y =

[−5−8

]. By the method described

above, X = A−1Y . So we compute A−1.

[2 3 1 0−1 2 0 1

]−−−−−→

−−−−−→

−−−−−→

−−−−−→

−−−−−→

So, A−1 =

. Then, since X = A−1Y ,

[xy

]=

[ −5−8

]=

=

=

Solution: (x, y) =( , )

3. Find a solution to

−x+ 5y + 2z = 393y + 5z = 39

2x+ y + 2z = 28given that the inverse of the

coefficient matrix is

1

37

1 −8 1910 −6 5−6 11 −3

.Here,

A =

, X =

, Y =


We are also told that

A−1 =

By our method we know that X = A−1Y , so we can write =

=

=

=

Solution: (x, y, z) =( , , )


Lesson 34: Inverses and Determinants of Matrices (II)

The determinant of a matrix A (detA or |A|) is a function on square matricesthat returns a number, not a matrix.

Fact 2.

• If detA 6= 0, then A−1 exists.

• If A−1 exists, then detA 6= 0.

Definition 3. A matrix is called singular if detA = 0. A matrix is called non-singular if detA 6= 0.

Determinant of 2× 2 Matrices:

Important 2× 2 Formulas

Let A =

[a bc d

], then

(i) detA =

∣∣∣∣ a bc d

∣∣∣∣ = ad− bc

(ii) A−1 =1

detA

[d −b−c a

]if detA 6= 0

Ex 3. Let A =

[2 0−1 1

]. Find detA and, if it exists, find A−1.

We write

detA = |A| =∣∣∣∣ 2 0−1 1

∣∣∣∣ =

Because detA 6= 0, we know that A−1 exists. Thus, by (ii),

A−1 =1

detA

[d −b−c a

]=

1

.

Determinant of 3× 3 Matrices:

The determinant of 3 × 3 matrices is defined using 2 × 2 matrices. We computethe minors and cofactors of the matrix. Let

A =

0 2 11 −2 −11 0 1

.The minor of the (3, 2)-entry, M32, is the determinant of the matrix A after

deleting the 3rd row and 2nd column, that is,


0 2 11 −2 −11 0 1

(3, 2)-entry

−→

0 2 11 −2 −11 0 1

3rd Row,

2nd Column

−→

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

M32

= =

The cofactor of the (3, 2)-entry, C32, is (−1) M32 = (−1) =

Ex 4. Consider the matrix A =

1 2 −20 1 0−1 3 2

.

Minors and Cofactors of A

(2, 1)-entry 1 2 −20 1 0−1 3 2

M21 =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

=

C21 =

=

(2, 2)-entry 1 2 −20 1 0−1 3 2

M22 =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

=

C22 =

=

(2, 3)-entry 1 2 −20 1 0−1 3 2

M23 =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

=

C23 =

=

detA = [(2,1)-entry]C21 + [(2,2)-entry]C22 + [(2,3)-entry]C23

=

=

Determinant of a 3× 3 Matrix:

If A is a 3× 3 matrix,then for any row r,

detA = [(r, 1)-entry]Cr1 + [(r, 2)-entry]Cr2 + [(r, 3)-entry]Cr3.

Examples.

1. Is A =

[2 10 −1

]singular?


2. Find the minor and cofactor of the (1, 3)-entry of A =

1 −1 10 2 13 −1 0

.

3. Find the determinant of A from # 2.

4. Given

∣∣∣∣ x− 3 30 x+ 1

∣∣∣∣ = 0, find x.


5. Given

∣∣∣∣∣∣x− 6 0 −2

33 x+ 4 1−3 2 x− 6

∣∣∣∣∣∣ = 0, find x.


Lesson 35: Eigenvalues and Eigenvectors (I)

Recall, a vector is a column matrix. Given a matrix A, there exists a number λand a collection of vectors ~vλ such that

A~vλ = λ~vλ.

λ is called a eigenvalue and ~vλ is an eigenvector associated to λ.

Ex 5. Let A =

[3 −12 0

], λ = 2, and ~v2 =

[11

].

[3 −12 0

] [11

]=

[3(1) + (−1)(1)

2(1) + 0(1)

]=

[22

]= 2 ·

[11

].

λ = 2 is an eigenvalue and the vector ~v2 =

[11

]is an eigenvector associated

to λ = 2.

Note 4. Recall that the following are identity matrices:

I2 =

[1 00 1

]and I3 =

1 0 00 1 00 0 1

.Fact 5. If A is a matrix, the expression det(λI − A) is a polynomial in λ. If A is a2× 2 matrix, then det(λI −A) is a degree 2 polynomial. If A is a 3× 3 matrix, thendet(λI − A) is a degree 3 polynomial.

Finding Eigenvalues and Eigenvectors

Let A be a square matrix.

(1) Solve det(λI − A) = 0 for λ. These are the eigenvalues.

(2) For each eigenvalue λ, solve (λI − A)~vλ = 0 for ~vλ. These are theeigenvectors.

Ex 6. Let A =

[1 20 2

]. Find its eigenvalues and eigenvectors.

Eigenvalues: We have to solve

det(λI − A) = 0

for λ.

det(λI − A) =

∣∣∣∣λ [ 1 00 1

]−[

1 20 2

]∣∣∣∣ =

∣∣∣∣∣∣∣

−

∣∣∣∣∣∣∣ =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣


Thus, the solution to det(λI − A) = 0 is λ =

Eigenvectors: We want to solve (λI − A)~vλ = 0 for some ~vλ. Here, we need tobreak this into cases.

Case 1:

Case 2:

Examples.

1. Let A =

[3 −12 0

]. Find the eigenvalues of A and then find an eigenvector

for each eigenvalue of A.


2. Determine if ~v =

[−1

1

]is an eigenvector of A =

[3 21 −1

].

3. Let A =

[1 1−1 3

]. Find the eigenvalues of A and, for each eigenvalue, find

an associated eigenvector.


4. Find the eigenvalues and eigenvectors of A =

[2 51 −2

].


Lesson 36: Eigenvalues and Eigenvectors (II)

Definition 6. A root is a number that makes a polynomial equal to 0.

If A is a 3 × 3 matrix, then det(λI − A) is a polynomial of degree 3 and hencehas 3 roots (although they may not all be distinct). The roots of det(λI −A) are theeigenvalues of A so finding eigenvalues comes down to factoring.

Ex 7. Suppose we know that x = 2 is a root of the polynomial f(x) = x3−4x2+x+6.How do we find its other roots?

Since we know that x = 2 is a root, we use synthetic division to factor f(x) intoa linear term and a quadratic term:

Hence, f(x) =

Definition 7. If f(x) is a polynomial, we say f(x) is monic if the coefficient of thehighest degree term is 1.

Ex 8.

• x4 + 2x+ 1 is a monic polynomial

• 2x2 + 1 is not a monic polynomial

Rational Root Test. If f(x) is a monic polynomial, then all roots of the f(x) dividethe constant term.

Ex 9. Let f(x) = x3 − 6x2 + 11x − 6. We assume for this homework that all ourpolynomials have integer roots. Thus, by the rational root test, the only possibleroots of f(x) divide −6, that is, ±1, ±2, ±3, or ±6.

Method: Let f(x) be a monic polynomial of degree 3. To factor f(x), applythe following:

(1) Write out all the divisors of the constant term

(2) Plug those values into f(x) until you find a root

(3) Use polynomial long division or synthetic division to factor f(x) intoa linear term and quadratic term

(4) Factor the quadratic term

Ex 10. Suppose we want to factor x3 − 3x2 − 4x + 12. We go through the methodto factor:

(1) Divisors:


(2) Finding 1 root:

(3) We use synthetic division:

(4)

Examples.

1. Find the eigenvalues and eigenvectors of A =

−1 −1 4−12 0 12−12 4 −1

.

Eigenvalues: We solve det(λI − A) = 0 for λ.


Eigenvectors:

2. An eigenvalue of A =

−8 6 3−6 4 3−12 6 7

is λ = −2. Are any of

112

, 1

32

, 2

12

, 1

11

an eigenvector for λ = −2?


3. Find the eigenvalues and eigenvectors for A =

6 −4 00 −6 10 −20 3

.

math 16020 handouts spring 2018 - purdue universityweld/spring2018/handoutsspring2018.pdf · math...

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