Presentation is loading. Please wait.

Presentation is loading. Please wait.

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.

Published byBlake Cook Modified over 9 years ago

Similar presentations

Presentation on theme: "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."— Presentation transcript:

1 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

2 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. 2 2.2.2: Complex Polynomial Identities

3 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. 3 2.2.2: Complex Polynomial Identities

4 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 2 4 2.2.2: Complex Polynomial Identities

5 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. 5 2.2.2: Complex Polynomial Identities

6 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 6 2.2.2: Complex Polynomial Identities

7 Guided Practice Example 1 Find the result of (10 + 7i)(10 – 7i). 7 2.2.2: Complex Polynomial Identities

8 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 2. 8 2.2.2: Complex Polynomial Identities

9 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 2 + 7 2. (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.

10 Guided Practice: Example 1, continued 3.Simplify the equation as needed. The result of (10 + 7i)(10 – 7i) is 149. 10 2.2.2: Complex Polynomial Identities (10 + 7i)(10 – 7i) = 10 2 + 7 2 Equation from the previous step = 100 + 49Evaluate the exponents. = 149Sum the terms. ✔

11 Guided Practice: Example 1, continued 11 2.2.2: Complex Polynomial Identities

12 Guided Practice Example 2 Factor the expression 9x 2 + 169. 12 2.2.2: Complex Polynomial Identities

13 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 2 + 169 is a sum of squares. The original expression can be rewritten using exponents. 13 2.2.2: Complex Polynomial Identities

14 Guided Practice: Example 2, continued 14 2.2.2: Complex Polynomial Identities 9x 2 + 169Original expression = (3x) 2 + (13) 2 Rewrite each term using exponents. = 3 2 x 2 + 13 2 Rewrite (3x) 2 as the product of two squares.

15 Guided Practice: Example 2, continued 2.Identify a and b in the sum of squares. The expression 3 2 x 2 + 13 2 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 2 + 13 2, let 3 = a and 13 = b. 15 2.2.2: Complex Polynomial Identities

16 Guided Practice: Example 2, continued 3.Factor the sum of squares. 16 2.2.2: 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 2 + 169 = (3x + 13i)(3x – 13i)Evaluate the exponents.

17 Guided Practice: Example 2, continued When factored, the expression 9x 2 + 169 is written as (3x + 13i)(3x – 13i). 17 2.2.2: Complex Polynomial Identities ✔

18 Guided Practice: Example 2, continued 18 2.2.2: Complex Polynomial Identities

Download ppt "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."

Similar presentations

7.5 – Rationalizing the Denominator of Radicals Expressions

7.5 – Rationalizing the Denominator of Radicals Expressions
4.5 Complex Numbers Objectives:

4.5 Complex Numbers Objectives:
Introduction Recall that the imaginary unit i is equal to. A fraction with i in the denominator does not have a rational denominator, since is not a rational.

Introduction Recall that the imaginary unit i is equal to. A fraction with i in the denominator does not have a rational denominator, since is not a rational.
Introduction Recall that a factor is one of two or more numbers or expressions that when multiplied produce a given product. We can factor certain expressions.

Introduction Recall that a factor is one of two or more numbers or expressions that when multiplied produce a given product. We can factor certain expressions.
Introduction Finding sums and differences of complex numbers is similar to finding sums and differences of expressions involving numeric and algebraic.

Introduction Finding sums and differences of complex numbers is similar to finding sums and differences of expressions involving numeric and algebraic.
Introduction Variables change, but constants remain the same. We need to understand how each term of an expression works in order to understand how changing.

Introduction Variables change, but constants remain the same. We need to understand how each term of an expression works in order to understand how changing.
Introduction You have learned several methods for solving polynomial equations by determining the factors, but not all equations are factorable. In this.

Introduction You have learned several methods for solving polynomial equations by determining the factors, but not all equations are factorable. In this.
6.2 – Simplified Form for Radicals

6.2 – Simplified Form for Radicals
Introduction You can determine how far a ladder will extend from the base of a wall by creating a quadratic equation and then taking the square root. To.

Introduction You can determine how far a ladder will extend from the base of a wall by creating a quadratic equation and then taking the square root. To.
Review and Examples: 7.4 – Adding, Subtracting, Multiplying Radical Expressions.

Review and Examples: 7.4 – Adding, Subtracting, Multiplying Radical Expressions.
Imaginary and Complex Numbers. Complex Number System The basic algebraic property of i is the following: i² = −1 Let us begin with i 0, which is 1. Each.

Imaginary and Complex Numbers. Complex Number System The basic algebraic property of i is the following: i² = −1 Let us begin with i 0, which is 1. Each.
Introduction While it may not be efficient to write out the justification for each step when solving equations, it is important to remember that the properties.

Introduction While it may not be efficient to write out the justification for each step when solving equations, it is important to remember that the properties.
Introduction Until now, you have been told that you cannot take the square of –1 because there is no number that when squared will result in a negative.

Introduction Until now, you have been told that you cannot take the square of –1 because there is no number that when squared will result in a negative.
Introduction Expressions can be used to represent quantities when those quantities are a sum of other values. When there are unknown values in the sum,

Introduction Expressions can be used to represent quantities when those quantities are a sum of other values. When there are unknown values in the sum,
Introduction Algebraic expressions are mathematical statements that include numbers, operations, and variables to represent a number or quantity. We know.

Introduction Algebraic expressions are mathematical statements that include numbers, operations, and variables to represent a number or quantity. We know.
16 Days. Two Days  Review - Use FOIL and the Distributive Property to multiply polynomials.

16 Days. Two Days  Review - Use FOIL and the Distributive Property to multiply polynomials.
1.3 Complex Number System.

1.3 Complex Number System.
Honors Topics.  You learned how to factor the difference of two perfect squares:  Example:  But what if the quadratic is ? You learned that it was.

Honors Topics.  You learned how to factor the difference of two perfect squares:  Example:  But what if the quadratic is ? You learned that it was.
Objectives Define and use imaginary and complex numbers.

Objectives Define and use imaginary and complex numbers.
Introduction Identities are commonly used to solve many different types of mathematics problems. In fact, you have already used them to solve real-world.

Introduction Identities are commonly used to solve many different types of mathematics problems. In fact, you have already used them to solve real-world.

Similar presentations

Ads by Google