1. Introduction Identities are commonly used to solve many different types of mathematics problems. In fact, you have already used them to solve real-world problems. In this lesson, you will extend your understanding of polynomial identities to include complex numbers and imaginary numbers. 1 3.4.1: Extending Polynomial Identities to Include Complex Numbers

2. Key Concepts An identity is an equation that is true regardless of what values are chosen for the variables. Some identities are often used and are well known; others are less well known. The tables on the next two slides show some examples of identities. 2 3.4.1: Extending Polynomial Identities to Include Complex Numbers

3. Key Concepts, continued 3 3.4.1: Extending Polynomial Identities to Include Complex Numbers IdentityTrue for… x + 2 = 2 + xThis is true for all values of x. This identity illustrates the Commutative Property of Addition. a(b + c) = ab + acThis is true for all values of a, b, and c. This identity is a statement of the Distributive Property.

4. Key Concepts, continued 4 3.4.1: Extending Polynomial Identities to Include Complex Numbers IdentityTrue for… This is true for all values of a and b, except for b = –1. The expression is not defined for b = –1 because if b = –1, the denominator is equal to 0. To see that the equation is true provided that b ≠ –1, note that

5. Key Concepts, continued A monomial is a number, a variable, or a product of a number and one or more variables with whole number exponents. If a monomial has one or more variables, then the number multiplied by the variable(s) is called a coefficient. A polynomial is a monomial or a sum of monomials. The monomials are the terms, numbers, variables, or the product of a number and variable(s) of the polynomial. 5 3.4.1: Extending Polynomial Identities to Include Complex Numbers

6. Key Concepts, continued Examples of polynomials include: 6 3.4.1: Extending Polynomial Identities to Include Complex Numbers rThis polynomial has 1 term, so it is called a monomial. This polynomial has 2 terms, so it is called a binomial. 3x 2 – 5x + 2This polynomial has 3 terms, so it is called a trinomial. –4x 3 y + x 2 y 2 – 4xy 3 This polynomial also has 3 terms, so it is also a trinomial.

7. Key Concepts, continued In this lesson, all polynomials will have one variable. The degree of a one-variable polynomial is the greatest exponent attached to the variable in the polynomial. For example: The degree of –5x + 3 is 1. (Note that –5x + 3 = –5x 1 + 3.) The degree of 4x 2 + 8x + 6 is 2. The degree of x 3 + 4x 2 is 3. 7 3.4.1: Extending Polynomial Identities to Include Complex Numbers

8. Key Concepts, continued A quadratic polynomial in one variable is a one- variable polynomial of degree 2, and can be written in the form ax 2 + bx + c, where a ≠ 0. For example, the polynomial 4x 2 + 8x + 6 is a quadratic polynomial. A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0, where x is the variable, a, b, and c are constants, and a ≠ 0. 8 3.4.1: Extending Polynomial Identities to Include Complex Numbers

9. Key Concepts, continued The quadratic formula states that the solutions of a quadratic equation of the form ax 2 + bx + c = 0 are given by A quadratic equation in this form can have no real solutions, one real solution, or two real solutions. 9 3.4.1: Extending Polynomial Identities to Include Complex Numbers

10. Key Concepts, continued In this lesson, all polynomial coefficients are real numbers, but the variables sometimes represent complex numbers. The imaginary unit i represents the non-real value. i is the number whose square is –1. We define i so that and i 2 = –1. An imaginary number is any number of the form bi, where b is a real number,, and b ≠ 0. 10 3.4.1: Extending Polynomial Identities to Include Complex Numbers

11. Key Concepts, continued A complex number is a number with a real component and an imaginary component. Complex numbers can be written in the form a + bi, where a and b are real numbers, and i is the imaginary unit. For example, 5 + 3i is a complex number. 5 is the real component and 3i is the imaginary component. Recall that all rational and irrational numbers are real numbers. Real numbers do not contain an imaginary component. 11 3.4.1: Extending Polynomial Identities to Include Complex Numbers

12. Key Concepts, continued The set of complex numbers is formed by two distinct subsets that have no common members: the set of real numbers and the set of imaginary numbers (numbers of the form bi, where b is a real number,, and b ≠ 0). Recall that if x 2 = a, then. For example, if x 2 = 25, then x = 5 or x = –5. 12 3.4.1: Extending Polynomial Identities to Include Complex Numbers

13. Key Concepts, continued The square root of a negative number is defined such that for any positive real number a, (Note the use of the negative sign under the radical.) For example, 13 3.4.1: Extending Polynomial Identities to Include Complex Numbers

14. Key Concepts, continued Using p and q as variables, if both p and q are positive, then For example, if p = 4 and q = 9, then But if p and q are both negative, then For example, if p = –4 and q = –9, then 14 3.4.1: Extending Polynomial Identities to Include Complex Numbers

15. Key Concepts, continued So, to simplify an expression of the form when p and q are both negative, write each factor as a product using the imaginary unit i before multiplying. 15 3.4.1: Extending Polynomial Identities to Include Complex Numbers

16. Key Concepts, continued Two numbers of the form a + bi and a – bi are called complex conjugates. The product of two complex conjugates is always a real number, as shown: Note that a 2 + b 2 is the sum of two squares and it is a real number because a and b are real numbers. 16 3.4.1: Extending Polynomial Identities to Include Complex Numbers (a + bi)(a – bi) = a 2 – abi + abi – b 2 i 2 Distribute. (a + bi)(a – bi) = a 2 – b 2 i 2 Simplify. (a + bi)(a – bi) = a 2 – b 2 (–1)i 2 = –1 (a + bi)(a – bi) = a 2 + b 2 Simplify.

17. Key Concepts, continued The equation (a + bi)(a – bi) = a 2 + b 2 is an identity that shows how to factor the sum of two squares. 17 3.4.1: Extending Polynomial Identities to Include Complex Numbers

18. Common Errors/Misconceptions substituting for when p and q are both negative neglecting to include factors of i when factoring the sum of two squares 18 3.4.1: Extending Polynomial Identities to Include Complex Numbers

19. Guided Practice Example 3 Write a polynomial identity that shows how to factor x 2 + 3. 19 3.4.1: Extending Polynomial Identities to Include Complex Numbers

20. Guided Practice: Example 3, continued 1.Solve for x using the quadratic formula. x 2 + 3 is not a sum of two squares, nor is there a common monomial. Use the quadratic formula to find the solutions to x 2 + 3. The quadratic formula is 20 3.4.1: Extending Polynomial Identities to Include Complex Numbers

21. Guided Practice: Example 3, continued 21 3.4.1: Extending Polynomial Identities to Include Complex Numbers x 2 + 3 = 0 Set the quadratic polynomial equal to 0. 1x 2 + 0x + 3 = 0 Write the polynomial in the form ax 2 + bx + c = 0. Substitute values into the quadratic formula: a = 1, b = 0, and c = 3. Simplify.

22. Guided Practice: Example 3, continued 22 3.4.1: Extending Polynomial Identities to Include Complex Numbers For any positive real number a, Factor 12 to show a perfect square factor. For any real numbers a and b, Simplify.

23. Guided Practice: Example 3, continued The solutions of the equation x 2 + 3 = 0 are Therefore, the equation can be written in the factored form is an identity that shows how to factor the polynomial x 2 + 3. 23 3.4.1: Extending Polynomial Identities to Include Complex Numbers

24. Guided Practice: Example 3, continued 2.Check your answer using square roots. Another method for solving the equation x 2 + 3 = 0 is by using a property involving square roots. 24 3.4.1: Extending Polynomial Identities to Include Complex Numbers

25. Guided Practice: Example 3, continued 25 3.4.1: Extending Polynomial Identities to Include Complex Numbers x 2 + 3 = 0 Set the quadratic polynomial equal to 0. x 2 = –3Subtract 3 from both sides. Apply the Square Root Property: if x 2 = a, then For any positive real number a,

26. Guided Practice: Example 3, continued 3.Verify the identity by multiplying. 26 3.4.1: Extending Polynomial Identities to Include Complex Numbers Distribute. Combine similar terms. Simplify.

27. Guided Practice: Example 3, continued The square root method produces the same result as the quadratic formula. is an identity that shows how to factor the polynomial x 2 + 3. 27 3.4.1: Extending Polynomial Identities to Include Complex Numbers ✔

28. Guided Practice: Example 3, continued 28 3.4.1: Extending Polynomial Identities to Include Complex Numbers

29. Guided Practice Example 4 Write a polynomial identity that shows how to factor the polynomial 3x 2 + 2x + 11. 29 3.4.1: Extending Polynomial Identities to Include Complex Numbers

30. Guided Practice: Example 4, continued 1.Solve for x using the quadratic formula. The quadratic formula is 30 3.4.1: Extending Polynomial Identities to Include Complex Numbers

31. Guided Practice: Example 4, continued 31 3.4.1: Extending Polynomial Identities to Include Complex Numbers 3x 2 + 2x + 11 = 0 Set the quadratic polynomial equal to 0. Substitute values into the quadratic formula: a = 3, b = 2, and c = 11. Simplify.

32. Guided Practice: Example 4, continued 32 3.4.1: Extending Polynomial Identities to Include Complex Numbers For any positive real number a, Factor 128 to show its greatest perfect square factor. For any real numbers a and b, Simplify.

33. Guided Practice: Example 4, continued The solutions of the equation 3x 2 + 2x + 11 = 0 are 33 3.4.1: Extending Polynomial Identities to Include Complex Numbers Write the real and imaginary parts of the complex number. Simplify.

34. Guided Practice: Example 4, continued 2.Use the solutions from step 1 to write the equation in factored form. If (x – r 1 )(x – r 2 ) = 0, then by the Zero Product Property, x – r 1 = 0 or x – r 2 = 0, and x = r 1 or x = r 2. That is, r 1 and r 2 are the roots (solutions) of the equation. Conversely, if r 1 and r 2 are the roots of a quadratic equation, then that equation can be written in the factored form (x – r 1 )(x – r 2 ) = 0. 34 3.4.1: Extending Polynomial Identities to Include Complex Numbers

35. Guided Practice: Example 4, continued The roots of the equation 3x 2 + 2x + 11 = 0 are Therefore, the equation can be written in the factored form or in the simpler factored form 35 3.4.1: Extending Polynomial Identities to Include Complex Numbers

36. Guided Practice: Example 4, continued 36 3.4.1: Extending Polynomial Identities to Include Complex Numbers ✔

37. Guided Practice: Example 4, continued 37 3.4.1: Extending Polynomial Identities to Include Complex Numbers