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 − 11 b. x 2 + 11
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 − 11 b. x 2 + 11
Solution Summary: The author explains how to calculate the factor of the expression x2-11.
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
−
11
b.
x
2
+
11
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.
For Exercises 8–10,
a. Simplify the expression. Do not rationalize the denominator.
b. Find the values of x for which the expression equals zero.
c. Find the values of x for which the denominator is zero.
4x(4x – 5) – 2x² (4)
8.
-6x(6x + 1) – (–3x²)(6)
(6x + 1)2
9.
(4x – 5)?
-
10. V4 – x² - -() 2)
For questions 10 – 11, use the table to answer the questions. It is set up to multiply
two polynomials. (show your work)
Exercises 143–145 will help you prepare for the material covered
in the next section. In each exercise, factor completely.
143. 2r + 8x? + 8x
144. 5x3 – 40x?y + 35xy2
145. 96?x + 9b²y – 16x – 16y
-
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