For Exercises 105–110, factor the expressions over the set of complex numbers. For assistance, consider these examples. In Chapter R we saw that some expressions factor over the set of integers. For example: x 2 − 4 = ( x + 2 ) ( x − 2 ) . Some expressions factor over the set of irrational numbers. For example: x 2 − 5 = ( x + 5 ) ( x − 5 ) . To factor an expression such as x2 1 4, we need to factor over the set of complex numbers. For example, verify that x 2 + 4 = ( x + 2 i ) ( x − 2 i ) . a. x 2 − 64 b. x 2 + 64
For Exercises 105–110, factor the expressions over the set of complex numbers. For assistance, consider these examples. In Chapter R we saw that some expressions factor over the set of integers. For example: x 2 − 4 = ( x + 2 ) ( x − 2 ) . Some expressions factor over the set of irrational numbers. For example: x 2 − 5 = ( x + 5 ) ( x − 5 ) . To factor an expression such as x2 1 4, we need to factor over the set of complex numbers. For example, verify that x 2 + 4 = ( x + 2 i ) ( x − 2 i ) . a. x 2 − 64 b. x 2 + 64
Solution Summary: The author explains how to calculate the factor of the expression x2-64.
For Exercises 105–110, factor the expressions over the set of complex numbers. For assistance, consider these examples.
In Chapter R we saw that some expressions factor over the set of integers. For example:
x
2
−
4
=
(
x
+
2
)
(
x
−
2
)
.
Some expressions factor over the set of irrational numbers. For example:
x
2
−
5
=
(
x
+
5
)
(
x
−
5
)
.
To factor an expression such as x2 1 4, we need to factor over the set of complex numbers. For example, verify that
x
2
+
4
=
(
x
+
2
i
)
(
x
−
2
i
)
.
a.
x
2
−
64
b.
x
2
+
64
Combination of a real number and an imaginary number. They are numbers of the form a + b , where a and b are real numbers and i is an imaginary unit. Complex numbers are an extended idea of one-dimensional number line to two-dimensional complex plane.
Chapter 4 Quiz 2 As always, show your work. 1) FindΘgivencscΘ=1.045.
2) Find Θ given sec Θ = 4.213.
3) Find Θ given cot Θ = 0.579. Solve the following three right triangles.
B
21.0
34.6° ca
52.5
4)c
26°
5)
A
b
6) B 84.0 a
42°
b
Q1: A: Let M and N be two subspace of finite dimension linear space X, show that if M = N
then dim M = dim N but the converse need not to be true.
B: Let A and B two balanced subsets of a linear space X, show that whether An B and
AUB are balanced sets or nor.
Q2: Answer only two
A:Let M be a subset of a linear space X, show that M is a hyperplane of X iff there exists
ƒ€ X'/{0} and a € F such that M = (x = x/f&x) = x}.
fe
B:Show that every two norms on finite dimension linear space are equivalent
C: Let f be a linear function from a normed space X in to a normed space Y, show that
continuous at x, E X iff for any sequence (x) in X converge to Xo then the sequence
(f(x)) converge to (f(x)) in Y.
Q3: A:Let M be a closed subspace of a normed space X, constract a linear space X/M as
normed space
B: Let A be a finite dimension subspace of a Banach space X, show that A is closed.
C: Show that every finite dimension normed space is Banach space.
• Plane II is spanned by the vectors:
P12
P2 = 1
• Subspace W is spanned by the vectors:
W₁ =
-- () ·
2
1
W2 =
0
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