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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. 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