Download Complex Polynomial Identities

Introduction
Polynomial identities can be used to find the product of
complex numbers. A complex number is a number of
the form a + bi, where a and b are real numbers and i is
the imaginary unit. An expression that cannot be written
using an identity with real numbers can be factored
using the imaginary unit i.
1
2.2.2: Complex Polynomial Identities
Key Concepts
• An imaginary number is any number of the form bi,
where b is a real number, i = -1 , and b ≠ 0. The
imaginary unit i is used to represent the non-real
value i = -1.
• Recall that i 2 = –1.
• Polynomial identities and properties of the imaginary
unit i can be used to expand or factor expressions
with complex numbers.
2
2.2.2: Complex Polynomial Identities
Key Concepts, continued
• Complex conjugates are two complex numbers of the
form a + bi and a – bi. Both numbers contain an
imaginary part, but multiplying them produces a value
that is wholly real. Therefore, the complex conjugate of
a + bi is a – bi, and vice versa.
• The sum of two squares can be rewritten as the product
of complex conjugates: a2 + b2 = (a + bi)(a – bi), where
a and b are real numbers and i is the imaginary unit.
3
2.2.2: Complex Polynomial Identities
Key Concepts, continued
• Rewriting the sum of two squares in this way can allow
you to either factor the sum of two squares or to find
the product of complex conjugates.
• To prove this, find the product of the conjugates and
simplify the expression.
(a + bi)(a – bi) =
a • a + a • bi + a(–bi) + bi(–bi) =
a2 + abi – abi – b2i 2 =
a2 – b2 (–1) =
a2 + b2
4
2.2.2: Complex Polynomial Identities
Key Concepts, continued
• This factored form of the sum of two squares can also
include variables, such as a2x2 + b2 = (ax + bi)(ax – bi),
where x is a variable, a and b are real numbers, and i
is the imaginary unit.
5
2.2.2: Complex Polynomial Identities
Common Errors/Misconceptions
• confusing the sum of squares with the Square of
Sums Identity
• incorrectly calculating a and b in the expression
a2 + b2 = (a + bi)(a – bi), when a2 and b2 are given
• incorrectly squaring quantities
6
2.2.2: Complex Polynomial Identities
Guided Practice
Example 1
Find the result of (10 + 7i)(10 – 7i).
7
2.2.2: Complex Polynomial Identities
Guided Practice: Example 1, continued
1. Determine whether an identity can be
used to rewrite the expression.
Since (10 + 7i) and (10 – 7i) are complex conjugates,
the expression (10 + 7i)(10 – 7i) can be rewritten as
the sum of squares: (a + bi)(a – bi) = a2 + b2.
8
2.2.2: Complex Polynomial Identities
Guided Practice: Example 1, continued
2. Identify a and b in the sum of squares.
Let 10 = a and 7 = b.
(10 + 7i)(10 – 7i)
Given expression
(a + bi)(a – bi) = a2 + b2
The product of
two complex
conjugates is the
sum of squares.
[(10) + (7)i ][(10) – (7)i ] = (10)2 + (7)2
Substitute 10 for
a and 7 for b.
The rewritten identity is (10 + 7i)(10 – 7i) = 102 + 72.
2.2.2: Complex Polynomial Identities
9
Guided Practice: Example 1, continued
3. Simplify the equation as needed.
(10 + 7i)(10 – 7i) = 102 + 72
Equation from the
previous step
= 100 + 49
Evaluate the
exponents.
= 149
Sum the terms.
The result of (10 + 7i)(10 – 7i) is 149.
✔
10
2.2.2: Complex Polynomial Identities
Guided Practice: Example 1, continued
11
2.2.2: Complex Polynomial Identities
Guided Practice
Example 2
Factor the expression 9x2 + 169.
12
2.2.2: Complex Polynomial Identities
Guided Practice: Example 2, continued
1. Determine whether an identity can be
used to rewrite the expression.
Both 9 and 169 are perfect squares; therefore,
9x2 + 169 is a sum of squares.
The original expression can be rewritten using
exponents.
13
2.2.2: Complex Polynomial Identities
Guided Practice: Example 2, continued
9x2 + 169
Original expression
= (3x)2 + (13)2
Rewrite each term using exponents.
= 32x2 + 132
Rewrite (3x)2 as the product of two
squares.
14
2.2.2: Complex Polynomial Identities
Guided Practice: Example 2, continued
2. Identify a and b in the sum of squares.
The expression 32x2 + 132 is in the form a2x2 + b2,
which can be rewritten using the factored form of the
sum of squares: a2x2 + b2 = (ax + bi)(ax – bi).
In the rewritten expression 32x2 + 132, let 3 = a and
13 = b.
15
2.2.2: Complex Polynomial Identities
Guided Practice: Example 2, continued
3. Factor the sum of squares.
a2x2 + b2 = (ax + bi)(ax – bi)
The sum of
squares is the
product of
complex
conjugates.
(3)2x2 + (13)2 = [(3)x + (13)i][(3)x – (13)i ] Substitute 3 for a
and 13 for b.
9x2 + 169 = (3x + 13i)(3x – 13i)
Evaluate the
exponents.
16
2.2.2: Complex Polynomial Identities
Guided Practice: Example 2, continued
When factored, the expression 9x2 + 169 is written
as (3x + 13i)(3x – 13i).
✔
17
2.2.2: Complex Polynomial Identities
Guided Practice: Example 2, continued
18
2.2.2: Complex Polynomial Identities