A course in mathematics,: For students of engineering and applied science

Frederick S Woods

Chapter 4

THE POLYNOMIAL OF THE $N$ TH DEGREE - all with Video Answers

Educators

Chapter Questions

Problem 1

Plot the graphs of the following qualratic expressions, in each case locating the vertex of the graph and determining the nature of the roots of the corresponding equation :
$2 x^2+3 x-2$.

John Nicolle

John Nicolle

Numerade Educator

Problem 2

Plot the graphs of the following qualratic expressions, in each case locating the vertex of the graph and determining the nature of the roots of the corresponding equation :

$9 x^2-3 x-2$.

James Kiss

Numerade Educator

Problem 3

Plot the graphs of the following qualratic expressions, in each case locating the vertex of the graph and determining the nature of the roots of the corresponding equation :

$4 x^2+4 x+3$.

Jessica Kluck

Numerade Educator

Problem 4

Plot the graphs of the following qualratic expressions, in each case locating the vertex of the graph and determining the nature of the roots of the corresponding equation :
$-3 x^2+5 x$.

Naveed Naemi

Numerade Educator

Problem 5

Plot the graphs of the following qualratic expressions, in each case locating the vertex of the graph and determining the nature of the roots of the corresponding equation :
$-9 x^2+12 x-7$.

Teresa Fuston

Numerade Educator

Problem 6

Plot the graphs of the following qualratic expressions, in each case locating the vertex of the graph and determining the nature of the roots of the corresponding equation :
$4 x^2-4 x-1$.

AG

Ankit Gupta

Numerade Educator

Problem 7

For what values of $a$ are the roots of $a x^2+3 x+7=0$ equal? What are the roots?

SA

Souad Al Nabulsi

Numerade Educator

Problem 8

Prove that the roots of $\left(c x+\frac{2 a}{c}\right)^2-8 a x=0$ are equal for all values of $a$ and $c$, and find them.

Mir Afzal

Numerade Educator

Problem 9

Prove that there is no real value of $m$ for which the roots of $x^2+(m x+3)^2-16=0$ are equal.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 10

For what values of $k$ are the roots of the following qualratic equations (1) equal? (2) real and unequal? (3) imaginary?
$2 x^2+3 x+2=k$.

Cory Kuzinski

Numerade Educator

Problem 11

For what values of $k$ are the roots of the following qualratic equations (1) equal? (2) real and unequal? (3) imaginary?

$x^2+(2-k) x+1=0$.

Cory Kuzinski

Numerade Educator

Problem 12

For what values of $k$ are the roots of the following qualratic equations (1) equal? (2) real and unequal? (3) imaginary?

$(k+1) x^2+(k-1) x+(k+1)=0$.

Mir Afzal

Numerade Educator

Problem 13

Plot the graphs of the following polynomials:
$x^8-a x . \quad(a>0$.

Lucas Finney

Numerade Educator

Problem 14

Plot the graphs of the following polynomials:
$x^3-4 x^2+x+1$.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 15

Plot the graphs of the following polynomials:
$x^8-8 x^2+1$.

Check back soon!

Problem 16

Plot the graphs of the following polynomials:
$x^8+x^2+2 x+5$.

John Irizar

Numerade Educator

Problem 17

Plot the graphs of the following polynomials:
$x^3-x^2+x-4$.

AG

Ankit Gupta

Numerade Educator

Problem 18

Plot the graphs of the following polynomials:
$x^8+6 x-6$.

James Kiss

Numerade Educator

Problem 19

Plot the graphs of the following polynomials:
. $x^2-12 x+8$.

James Kiss

Numerade Educator

Problem 20

Plot the graphs of the following polynomials:
$2 x^4+x^3-4 x^2-10 x-4$.

AG

Ankit Gupta

Numerade Educator

Problem 21

Plot the graphs of the following polynomials:
$4 x^4+12 x^3+7 x^2-28 x-6$.

AG

Ankit Gupta

Numerade Educator

Problem 22

Plot the graphs of the following polynomials:
$8 x^4-10 x^3-5 x^2+2 x$.

AG

Ankit Gupta

Numerade Educator

Problem 23

Plot the graphs of the following polynomials:
$x^4+6 x^3+10 x^2$.

James Kiss

Numerade Educator

Problem 24

Plot the graphs of the following polynomials:
$2 x^5+2 x^4-7 x^3-8 x^2-4 x$.

AG

Ankit Gupta

Numerade Educator

Problem 25

Find all the roots of the following equations:
$8 x^3=27$.

Harry Evans

Numerade Educator

Problem 26

Find all the roots of the following equations:
$8 x^6-63 x^8-8=0$.

AG

Ankit Gupta

Numerade Educator

Problem 27

Find all the roots of the following equations:
$x^5-5 x^8+12 x=2 x^8+3 x$.

Luke P

Numerade Educator

Problem 28

Find all the roots of the following equations:
$5 x^6+27 x^2=2 x^6-54 x^4$.

Joseph Lentino

Numerade Educator

Problem 29

Find all the roots of the following equations:
$(2 x-a)^4-(3 x+a)^4=0$.

Shima Shaw

Numerade Educator

Problem 30

Find all the roots of the following equations:
$x^4-2\left(a^2+1\right) x^2+\left(a^2-1\right)^2=0$.

Shima Shaw

Numerade Educator

Problem 31

Form the equations having the following valnes for their roots:
$0, \frac{2}{3}, \frac{3}{2}$.

Nick Johnson

Numerade Educator

Problem 32

Form the equations having the following valnes for their roots:
$a+\sqrt{b}, a-\sqrt{b},-a$.

Mir Afzal

Numerade Educator

Problem 33

Form the equations having the following valnes for their roots:
$0,0,2 a \pm b, \pm \sqrt{2 b}$.

Mir Afzal

Numerade Educator

Problem 34

Form the equations having the following valnes for their roots:
Form a quadratic equation with real coefficients having $2+3 i$ for one of its roots.

James Kiss

Numerade Educator

Problem 35

Factor the following quadratic expressions:
$4 x^2+8 x-7$.

Susan Cooper

Numerade Educator

Problem 36

Factor the following quadratic expressions:
$4 x^2+12 x+11$.

Sherrie Fenner

Numerade Educator

Problem 37

Factor the following quadratic expressions:
$4 a^2 x^2+2 a x+1$.

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 38

Factor the following quadratic expressions:
. $x^2+2 a x-a+a^2$.

Julie Silva

Numerade Educator

Problem 39

Factor the following quadratic expressions:
$a^2 x^2+2 a b x-a$.

Brandon Fox

Numerade Educator

Problem 40

Factor the following quadratic expressions:
. $a^2 x^2+2 a b x+b+b^2$.

Naresh Bagrecha

Naresh Bagrecha

Numerade Educator

Problem 41

If $r_1$ and $r_2$ are the roots of the equation $x^2+p x+q=0$, find the values of the following expressions in terms of $p$ and $q$ without solving the equation:
$r_1^2+r_2^2$.

Mir Afzal

Numerade Educator

Problem 42

If $r_1$ and $r_2$ are the roots of the equation $x^2+p x+q=0$, find the values of the following expressions in terms of $p$ and $q$ without solving the equation:
$r_1^8+r_2^3$.

James Kiss

Numerade Educator

Problem 43

If $r_1$ and $r_2$ are the roots of the equation $x^2+p x+q=0$, find the values of the following expressions in terms of $p$ and $q$ without solving the equation:
$\frac{1}{r_1}+\frac{1}{r_2}$.

Mir Afzal

Numerade Educator

Problem 44

If $r_1$ and $r_2$ are the roots of the equation $x^2+p x+q=0$, find the values of the following expressions in terms of $p$ and $q$ without solving the equation:
$\frac{1}{r_1^2}+\frac{1}{r_2^2}$.

Mir Afzal

Numerade Educator

Problem 45

If $r_1$ and $r_2$ are the roots of the equation $x^2+p x+q=0$, find the values of the following expressions in terms of $p$ and $q$ without solving the equation:
$\frac{r_1}{r_2}+\frac{r_2}{r_1}$.

Talha Khalid

Numerade Educator

Problem 46

If $r_1, r_2, r_8$ are the roots of the equation $x^2+p x^2+q x+r=0$, find the values of the following expressions in terms of the coefficients without solving the equation:
$\left(r_1^2+r_2^9+r_8^2\right)+2\left(r_1 r_2+r_2 r_3+r_3 r_1\right)+3 r_1 r_2 r_8$.

Anas Venkitta

Numerade Educator

Problem 47

If $r_1, r_2, r_8$ are the roots of the equation $x^2+p x^2+q x+r=0$, find the values of the following expressions in terms of the coefficients without solving the equation:
$r_1^2 r_2 r_8+r_2^2 r_8 r_1+r_3^2 r_1 r_2$.

Dr Harish Viswanathan

Dr Harish Viswanathan

Numerade Educator

Problem 48

If $r_1, r_2, r_8$ are the roots of the equation $x^2+p x^2+q x+r=0$, find the values of the following expressions in terms of the coefficients without solving the equation:
$\frac{1}{r_1 r_2}+\frac{1}{r_2 r_3}+\frac{1}{r_8 r_1}$.

Dr Harish Viswanathan

Dr Harish Viswanathan

Numerade Educator

Problem 49

Show that if $a+\sqrt{b}$ is a root of an equation with rational coefficients, then $a-\sqrt{b}$ is also a root.

Adam Dehollander

Adam Dehollander

Numerade Educator

Problem 50

Plot the graphs of the following expressions, and find all the roots of the corresponding equations:
$(x+1)(x-2)(x-4)$.

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 51

Plot the graphs of the following expressions, and find all the roots of the corresponding equations:
$(x-2)(x-4)(2 x+3)$.

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Problem 52

Plot the graphs of the following expressions, and find all the roots of the corresponding equations:
$(x-4)(2 x+1)(3 x+5)$.

AG

Ankit Gupta

Numerade Educator

Problem 53

Plot the graphs of the following expressions, and find all the roots of the corresponding equations:
$(x+3)(x-1)^2$.

Jessica Bisagno

Jessica Bisagno

Numerade Educator

Problem 54

Plot the graphs of the following expressions, and find all the roots of the corresponding equations:
$(2 x-1)(x-3)^2$.

Jessica Bisagno

Jessica Bisagno

Numerade Educator

Problem 55

Plot the graphs of the following expressions, and find all the roots of the corresponding equations:
$(x-2)(2 x+3)^2$.

John Nicolle

Numerade Educator

Problem 56

Plot the graphs of the following expressions, and find all the roots of the corresponding equations:
$(2 x+5)\left(x^2+2 x+3\right)$.

Matt Gibson

Numerade Educator

Problem 57

Plot the graphs of the following expressions, and find all the roots of the corresponding equations:
$(x-5)\left(2 x^2+3 x+2\right)$.

Matt Gibson

Numerade Educator

Problem 58

Plot the graphs of the following expressions, and find all the roots of the corresponding equations:
$(x+2)(x-3)(x-2)^2$.

Jessica Bisagno

Jessica Bisagno

Numerade Educator

Problem 59

Plot the graphs of the following expressions, and find all the roots of the corresponding equations:
$(x-2)(x+2)\left(x^2+2\right)$.

Check back soon!

Problem 60

Plot the graphs of the following expressions, and find all the roots of the corresponding equations:
$(x-2)^2\left(2 x^2+2 x+1\right)$.

Amy Jiang

Numerade Educator

Problem 61

Plot the graphs of the following expressions, and find all the roots of the corresponding equations:
$(x+1)(2 x-1)\left(3 x^2+2 x+3\right)$.

AG

Ankit Gupta

Numerade Educator

Problem 62

Find all the roots of the following equations:
$x^8-4 x^2-2 x+5=0$.

Check back soon!

Problem 63

Find all the roots of the following equations:
$x^3-3 x^2+4=0$.

Sneha Ravi

Numerade Educator

Problem 64

Find all the roots of the following equations:
$3 x^3-7 x^2-8 x+20=0$.

Tyler Moulton

Numerade Educator

Problem 65

Find all the roots of the following equations:
$4 x^3-8 x^2-35 x+75=0$.

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Problem 66

Find all the roots of the following equations:
$x^3+4 x^2+4 x+3=0$.

Julie Silva

Numerade Educator

Problem 67

Find all the roots of the following equations:
$8 x^3-28 x^2+80 x-9=0$.

Ashly Sunny

Numerade Educator

Problem 68

Find all the roots of the following equations:
$12 x^8-44 x^2+5 x+7=0$.

AG

Ankit Gupta

Numerade Educator

Problem 69

Find all the roots of the following equations:
. $8 x^8+10 x^2+10 x-12=0$.

Lucas Finney

Numerade Educator

Problem 70

Find all the roots of the following equations:
$3 x^3+10 x^2+2 x-8=0$.

Sriram Soundarrajan

Sriram Soundarrajan

Numerade Educator

Problem 71

Find all the roots of the following equations:
$4 x^4+8 x^3+3 x^2-2 x-1=0$.

AG

Ankit Gupta

Numerade Educator

Problem 72

Find all the roots of the following equations:
. $6 x^4-11 x^3-37 x^2+36 x+30=0$.

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 73

Find all the roots of the following equations:
. $3 x^4-17 x^3+41 x^2-53 x+30=0$.

Sriram Soundarrajan

Sriram Soundarrajan

Numerade Educator

Problem 74

Find all the roots of the following equations:
$2 x^4-9 x^8-9 x^2+57 x-20=0$.

AG

Ankit Gupta

Numerade Educator

Problem 75

Find all the roots of the following equations:
$18 x^4-27 x^8+10 x^2+12 x-8=0$.

Aman Gupta

Numerade Educator

Problem 76

Find all the roots of the following equations:
$16 x^4+16 x^3-72 x^2-20 x+25=0$.

AG

Ankit Gupta

Numerade Educator

Problem 77

Find all the roots of the following equations:
$x^5-2 x^4-4 x^3-4 x^2+15 x+18=0$.

Aman Gupta

Numerade Educator

Problem 78

Find all the roots of the following equations:
$4 x^5+12 x^4+11 x^3+5 x^2-3 x-2=0$.

Sriram Soundarrajan

Sriram Soundarrajan

Numerade Educator

Problem 79

Find all the roots of the following equations:
$12 x^5+44 x^4-55 x^3-95 x^2+63 x-9=0$.

Vishal Parmar

Numerade Educator

Problem 80

Find all the roots of the following equations:
$2 x^5-5 x^4-13 x^3+13 x^2+5 x-2=0$.

AG

Ankit Gupta

Numerade Educator

Problem 81

Determine by Descartes' rule of signs the nature of the roots of the following equations:
$x^3+5 x-7=0$.

James Kiss

Numerade Educator

Problem 82

Determine by Descartes' rule of signs the nature of the roots of the following equations:
$x^3+2 x+3=0$.

James Kiss

Numerade Educator

Problem 83

Determine by Descartes' rule of signs the nature of the roots of the following equations:
$x^8+2 x^2+5=0$.

James Kiss

Numerade Educator

Problem 84

Determine by Descartes' rule of signs the nature of the roots of the following equations:
$3 x^4+4 x^8+4 x+8=0$.

James Kiss

Numerade Educator

Problem 85

Determine by Descartes' rule of signs the nature of the roots of the following equations:
$x^4+x^2-x-6=0$.

Jeyasree R T

Numerade Educator

Problem 86

Determine by Descartes' rule of signs the nature of the roots of the following equations:
$x^4-4 \dot{x}^2+1=0$.

Amy Jiang

Numerade Educator

Problem 87

Find the real roots of the following equations, accurate to two decimal places:
$x^3+3 x-7=0$.

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Problem 88

Find the real roots of the following equations, accurate to two decimal places:
$x^3+x+5=0$.

Gregory Higby

Numerade Educator

Problem 89

Find the real roots of the following equations, accurate to two decimal places:
$x^4-12 x+7=0$.

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Problem 90

Find the real roots of the following equations, accurate to two decimal places:
$x^4-8 x^3+3=0$.

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator

Problem 91

Find the real roots of the following equations, accurate to two decimal places:
$x^8-x^2-6 x+1=0$.

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator