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 Exercises 13–20, factor each expression.
In Exercises 126–129, determine whether each statement is true
or false. If the statement is false, make the necessary change(s) to
produce a true statement.
126. Once a GCF is factored from 6y – 19y + 10y“, the
remaining trinomial factor is prime.
127. One factor of 8y² – 51y + 18 is 8y – 3.
128. We can immediately tell that 6x? – 11xy – 10y? is prime
because 11 is a prime number and the polynomial contains
two variables.
129. A factor of 12x2 – 19xy + 5y² is 4x – y.
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