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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. .

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    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. 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. 3.4.1: Extending Polynomial Identities to Include Complex Numbers

    3. Key Concepts, continued 3.4.1: Extending Polynomial Identities to Include Complex Numbers

    4. Key Concepts, continued 3.4.1: Extending Polynomial Identities to Include Complex Numbers

    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. 3.4.1: Extending Polynomial Identities to Include Complex Numbers

    6. Key Concepts, continued Examples of polynomials include: 3.4.1: Extending Polynomial Identities to Include Complex Numbers

    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 = –5x1+ 3.) The degree of 4x2+ 8x+ 6 is 2. The degree of x3+ 4x2is 3. 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 ax2 + bx + c, where a ≠ 0. For example, the polynomial 4x2 + 8x + 6 is a quadratic polynomial. A quadratic equation is an equation that can be written in the form ax2 + bx + c= 0, where xis the variable, a, b, and c are constants, and a ≠ 0. 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 ax2 + bx + c = 0 are given by A quadratic equation in this form can have no real solutions, one real solution, or two real solutions. 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 i2 = –1. An imaginary number is any number of the form bi, where b is a real number, , and b ≠ 0. 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. 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 x2 = a, then . For example, if x2= 25, then x = 5 or x = –5. 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, 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 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. 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 a2 + b2is the sum of two squares and it is a real number because a and b are real numbers. 3.4.1: Extending Polynomial Identities to Include Complex Numbers

    17. Key Concepts, continued The equation (a + bi)(a – bi) = a2 + b2 is an identity that shows how to factor the sum of two squares. 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 3.4.1: Extending Polynomial Identities to Include Complex Numbers

    19. Guided Practice Example 3 Write a polynomial identity that shows how to factor x2 + 3. 3.4.1: Extending Polynomial Identities to Include Complex Numbers

    20. Guided Practice: Example 3, continued Solve for x using the quadratic formula. x2 + 3is not a sum of two squares, nor is there a common monomial. Use the quadratic formula to find the solutions to x2+ 3. The quadratic formula is 3.4.1: Extending Polynomial Identities to Include Complex Numbers

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

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

    23. Guided Practice: Example 3, continued The solutions of the equation x2 + 3 = 0 are Therefore, the equation can be written in the factored form is an identity that shows how to factor the polynomial x2 + 3. 3.4.1: Extending Polynomial Identities to Include Complex Numbers

    24. Guided Practice: Example 3, continued Check your answer using square roots. Another method for solving the equation x2 + 3 = 0 is by using a property involving square roots. 3.4.1: Extending Polynomial Identities to Include Complex Numbers

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

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

    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 x2 + 3. ✔ 3.4.1: Extending Polynomial Identities to Include Complex Numbers

    28. Guided Practice: Example 3, continued 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 3x2 + 2x+ 11. 3.4.1: Extending Polynomial Identities to Include Complex Numbers

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

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

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

    33. Guided Practice: Example 4, continued The solutions of the equation 3x2 + 2x + 11 = 0are 3.4.1: Extending Polynomial Identities to Include Complex Numbers

    34. Guided Practice: Example 4, continued Use the solutions from step 1 to write the equation in factored form. If (x – r1)(x – r2) = 0, then by the Zero Product Property, x – r1= 0 or x – r2 = 0, and x= r1or x= r2. That is, r1 and r2 are the roots (solutions) of the equation. Conversely, if r1 and r2are the roots of a quadratic equation, then that equation can be written in the factored form (x – r1)(x – r2) = 0. 3.4.1: Extending Polynomial Identities to Include Complex Numbers

    35. Guided Practice: Example 4, continued The roots of the equation 3x2 + 2x + 11 = 0 are Therefore, the equation can be written in the factored form or in the simpler factored form 3.4.1: Extending Polynomial Identities to Include Complex Numbers

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

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