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.

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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 : 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 : 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: a 2 + b 2 = (a + bi)(a – bi), where a and b are real numbers and i is the imaginary unit : 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) = a 2 + abi – abi – b 2 i 2 = a 2 – b 2 (–1) = a 2 + b : Complex Polynomial Identities

Key Concepts, continued This factored form of the sum of two squares can also include variables, such as a 2 x 2 + b 2 = (ax + bi)(ax – bi), where x is a variable, a and b are real numbers, and i is the imaginary unit : 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 a 2 + b 2 = (a + bi)(a – bi), when a 2 and b 2 are given incorrectly squaring quantities : Complex Polynomial Identities

Guided Practice Example 1 Find the result of (10 + 7i)(10 – 7i) : 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) = a 2 + b : Complex Polynomial Identities

9 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) = (10 + 7i)(10 – 7i)Given expression (a + bi)(a – bi) = a 2 + b 2 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 : Complex Polynomial Identities (10 + 7i)(10 – 7i) = Equation from the previous step = Evaluate the exponents. = 149Sum the terms. ✔

Guided Practice: Example 1, continued : Complex Polynomial Identities

Guided Practice Example 2 Factor the expression 9x : 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, 9x is a sum of squares. The original expression can be rewritten using exponents : Complex Polynomial Identities

Guided Practice: Example 2, continued : Complex Polynomial Identities 9x Original expression = (3x) 2 + (13) 2 Rewrite each term using exponents. = 3 2 x 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 3 2 x is in the form a 2 x 2 + b 2, which can be rewritten using the factored form of the sum of squares: a 2 x 2 + b 2 = (ax + bi)(ax – bi). In the rewritten expression 3 2 x , let 3 = a and 13 = b : Complex Polynomial Identities

Guided Practice: Example 2, continued 3.Factor the sum of squares : Complex Polynomial Identities a 2 x 2 + b 2 = (ax + bi)(ax – bi)The sum of squares is the product of complex conjugates. (3) 2 x 2 + (13) 2 = [(3)x + (13)i ][(3)x – (13)i ]Substitute 3 for a and 13 for b. 9x = (3x + 13i)(3x – 13i)Evaluate the exponents.

Guided Practice: Example 2, continued When factored, the expression 9x is written as (3x + 13i)(3x – 13i) : Complex Polynomial Identities ✔

Guided Practice: Example 2, continued : Complex Polynomial Identities