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## Introduction

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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. 3.4.1: Extending Polynomial Identities to Include Complex Numbers**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**Key Concepts, continued**3.4.1: Extending Polynomial Identities to Include Complex Numbers**Key Concepts, continued**3.4.1: Extending Polynomial Identities to Include Complex Numbers**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**Key Concepts, continued**Examples of polynomials include: 3.4.1: Extending Polynomial Identities to Include Complex Numbers**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**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**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**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**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**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**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**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**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**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**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**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**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**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**Guided Practice: Example 3, continued**3.4.1: Extending Polynomial Identities to Include Complex Numbers**Guided Practice: Example 3, continued**3.4.1: Extending Polynomial Identities to Include Complex Numbers**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**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**Guided Practice: Example 3, continued**3.4.1: Extending Polynomial Identities to Include Complex Numbers**Guided Practice: Example 3, continued**Verify the identity by multiplying. 3.4.1: Extending Polynomial Identities to Include Complex Numbers**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**Guided Practice: Example 3, continued**3.4.1: Extending Polynomial Identities to Include Complex Numbers**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**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**Guided Practice: Example 4, continued**3.4.1: Extending Polynomial Identities to Include Complex Numbers**Guided Practice: Example 4, continued**3.4.1: Extending Polynomial Identities to Include Complex Numbers**Guided Practice: Example 4, continued**The solutions of the equation 3x2 + 2x + 11 = 0are 3.4.1: Extending Polynomial Identities to Include Complex Numbers**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**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**Guided Practice: Example 4, continued**✔ 3.4.1: Extending Polynomial Identities to Include Complex Numbers**Guided Practice: Example 4, continued**3.4.1: Extending Polynomial Identities to Include Complex Numbers