IntroductionPolynomial 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.

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2.2.2: Complex Polynomial Identities

Key Concepts• An imaginary number is any number of the form bi,

where b is a real number, , and b ≠ 0. The

imaginary unit i is used to represent the non-real

value .

• 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.

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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.

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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

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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.

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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

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2.2.2: Complex Polynomial Identities

Guided Practice

Example 1Find the result of (10 + 7i)(10 – 7i).

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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.

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2.2.2: Complex Polynomial Identities

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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.

The rewritten identity is (10 + 7i)(10 – 7i) = 102 + 72.

(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.

Guided Practice: Example 1, continued

3. Simplify the equation as needed.

The result of (10 + 7i)(10 – 7i) is 149.

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2.2.2: Complex Polynomial Identities

(10 + 7i)(10 – 7i) = 102 + 72 Equation from the previous step

= 100 + 49 Evaluate the exponents.

= 149 Sum the terms.

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Guided Practice: Example 1, continued

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2.2.2: Complex Polynomial Identities

Guided Practice

Example 2Factor the expression 9x2 + 169.

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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.

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2.2.2: Complex Polynomial Identities

Guided Practice: Example 2, continued

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2.2.2: Complex Polynomial Identities

9x2 + 169 Original expression

= (3x)2 + (13)2 Rewrite each term using exponents.

= 32x2 + 132 Rewrite (3x)2 as the product of two squares.

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.

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2.2.2: Complex Polynomial Identities

Guided Practice: Example 2, continued

3. Factor the sum of squares.

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2.2.2: Complex Polynomial Identities

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.

Guided Practice: Example 2, continuedWhen factored, the expression 9x2 + 169 is written as (3x + 13i)(3x – 13i).

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2.2.2: Complex Polynomial Identities

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Guided Practice: Example 2, continued

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2.2.2: Complex Polynomial Identities