Question

. Rewrite each of the following polynomials, ordering
the terms using the lex order, the

grlex order, and the grevlex order, giving LM(f), LT(f), and
multideg(f) in each case.

a. f(x, y,z) = 2x + 3y + z + x2 − z

2 + x3

.

b. f(x, y,z) = 2x2

y8 − 3x5

yz4 + xyz3 − xy4

.

Answer #1

Find the GCD d(x) of the following pairs of polynomials
a(x),b(x) in Z3[x]. In each case,
express d(x) as a linear combination of a(x) and b(x).
a) a(x)= x3 + x2 +1 , b(x)=
x2+1
b) a(x)= x3 + x2+ 1 , b(x)= x2
+ x + 1
c) a(x)= x3 +x2 +1 ,b(x)= x2 +
x

1)Solve the system of linear equations, using the Gauss-Jordan
elimination method. (If there is no solution, enter NO SOLUTION. If
there are infinitely many solutions, express your answer in terms
of the parameters t and/or s.)
x1
+
2x2
+
8x3
=
6
x1
+
x2
+
4x3
=
3
(x1,
x2, x3)
=
2)Solve the system of linear equations, using the Gauss-Jordan
elimination method. (If there is no solution, enter NO SOLUTION. If
there are infinitely many solutions, express...

For each of the following statements:
• Rewrite the symbolic sentence in words,
• Determine if the statement is true or false and justify your
answer,• Negate the statement (you may write the negation
symbolically).
√
(a) ∀x∈R, x2 =x.
(b) ∃y∈R,∀x∈R,xy=0.
(c) ∃x ∈ R, ∀y ∈ R, x2 + y2 > 9.

1) Determine whether x3 is O(g(x)) for the following:
a. g(x) = x2 + x3
b. g(x) = x2 + x4
c. g(x) = x3 / 2 2)
Show that each of these pairs of functions are of the same
order:
a. 3x + 7, x
b. 2x2 + x - 7, x2

Find the derivatives of each of the following functions. DO NOT
simplify your answers.
(a) f(x) = 103x (3x5+ x − 1)4
(b) g(x) = ln(x3 + x) /
x2 − 4
(c) h(x) = tan-1(xex)
(d) k(x) = sin(x)cos(x)

6) Cramer’s Rule
Solve each using Cramer’s rule
a) 3x + .5y =10
10x - -y =
-3
b) x + z +y = 12
-x + 3y - z = 2
-2x + z = -2
7) Proofs
Prove the following by induction
1. n3 – n is divisible by 3 for n
>=2
2. . Prove that n! <
nn forn ≥ 2.
Prove directly that:
The sum of an integer and its cube...

For each of the following pairs of polynomials f(x) and g(x),
write f(x) in the form
f(x) = k(x)g(x) + r(x)
with deg(r(x)) < deg(g(x)).
a) f(x) = x^4 + x^3 + x^2 + x + 1 and g(x) = x^2 −
2x + 1.
b) f(x) = x^3 + x^2 + 1 and g(x) = x^2 − 5x + 6.
c) f(x) = x^22 − 1 and g(x) = x^5 − 1.

For each of the
following vector fields F , decide whether it is
conservative or not by computing curl F . Type in
a potential function f (that is, ∇f=F∇f=F
). If it is not conservative, type N.
A.
F(x,y)=(−10x+3y)i+(3x+10y)jF(x,y)=(−10x+3y)i+(3x+10y)j
f(x,y)=f(x,y)=
B.
F(x,y)=−5yi−4xjF(x,y)=−5yi−4xj
f(x,y)=f(x,y)=
C.
F(x,y,z)=−5xi−4yj+kF(x,y,z)=−5xi−4yj+k
f(x,y,z)=f(x,y,z)=
D.
F(x,y)=(−5siny)i+(6y−5xcosy)jF(x,y)=(−5siny)i+(6y−5xcosy)j
f(x,y)=f(x,y)=
E.
F(x,y,z)=−5x2i+3y2j+5z2kF(x,y,z)=−5x2i+3y2j+5z2k
f(x,y,z)=f(x,y,z)=
Note: Your answers should be either expressions of x, y and z
(e.g. "3xy + 2yz"), or the letter "N"

For each of the following vector fields F , decide whether it is
conservative or not by computing curl F . Type in a potential
function f (that is, ∇f=F). If it is not conservative, type N. A.
F(x,y)=(−4x+3y)i+(3x+16y)j f(x,y)= B. F(x,y)=−2yi−1xj f(x,y)= C.
F(x,y,z)=−2xi−1yj+k f(x,y,z)= D. F(x,y)=(−2siny)i+(6y−2xcosy)j
f(x,y)= E. F(x,y,z)=−2x2i+3y2j+8z2k

1. a) Find the solution to the system of linear equations using
matrix row operations. Show all your work.
x + y + z = 13
x - z = -2
-2x + y = 3
b) How many solutions does the following system have? How do you
know?
6x + 4y + 2z = 32
3x - 3y - z = 19
3x + 2y + z = 32

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 24 minutes ago

asked 32 minutes ago

asked 42 minutes ago

asked 42 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago