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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. 1 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. 2 3.4.1: Extending Polynomial Identities to Include Complex Numbers Key Concepts, continued Identity True for… x+2=2+x This is true for all values of x. This identity illustrates the Commutative Property of Addition. a(b + c) = ab + ac This is true for all values of a, b, and c. This identity is a statement of the Distributive Property. 3 3.4.1: Extending Polynomial Identities to Include Complex Numbers Key Concepts, continued Identity ab + a b +1 =a True for… This is true for all values of a and b, except for b = –1. The expression ab + a = a is not defined for b = –1 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 ab + a a(b + 1) a (b + 1) = = = a. b +1 (b + 1) (b + 1) 3.4.1: Extending Polynomial Identities to Include Complex Numbers 4 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 Key Concepts, continued • Examples of polynomials include: r 2 3 This polynomial has 1 term, so it is called a monomial. x -x 5 This polynomial has 2 terms, so it is called a binomial. 3x2 – 5x + 2 This polynomial has 3 terms, so it is called a trinomial. –4x3y + x2y2 – 4xy3 This polynomial also has 3 terms, so it is also a trinomial. 6 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 + 4x2 is 3. 7 3.4.1: Extending Polynomial Identities to Include Complex Numbers Key Concepts, continued • A quadratic polynomial in one variable is a onevariable 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 x is the variable, a, b, and c are constants, and a ≠ 0. 8 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 x = -b ± b 2 - 4ac . A quadratic equation in 2a 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 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 i =number -1 whose square is –1. We define i so that andi = i 2 =-1 –1. • An imaginary number is any number of the form bi, where b is a real number, i = -1 , and b ≠ 0. 10 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 11 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, i = -1, and b ≠ 0). • Recall that if x2 = a, then x = ± a . For example, if x2 = 25, then x = 5 or x = –5. 12 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, -a = i a. (Note the use of the negative sign under the radical.) • For example, -9 = i 9 = i · 3 = 3i. 13 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 p · q = pq. For example, if p = 4 and q = 9, then p · q = 4 · 9 = 2 · 3 = 6, and pq = 4 · 9 = 36 = 6. • But if p and q are both negative, then p · q ¹ pq. For example, if p = –4 and q = –9, then p · q = -4 · -9 = i 4 · i 9 = 2i · 3i = 6i 2 = 6 · (-1) = -6, but pq = ( -4) ( -9) = 36 = 6. 14 3.4.1: Extending Polynomial Identities to Include Complex Numbers Key Concepts, continued • So, to simplify an expression of the form p · q 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 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: (a + bi)(a – bi) = a2 – abi + abi – b2i 2 (a + bi)(a – bi) = a2 – b2i 2 (a + bi)(a – bi) = a2 – b2(–1) (a + bi)(a – bi) = a2 + b2 Distribute. Simplify. i 2 = –1 Simplify. • Note that a2 + b2 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 Key Concepts, continued • The equation (a + bi)(a – bi) = a2 + b2 is an identity that shows how to factor the sum of two squares. 17 3.4.1: Extending Polynomial Identities to Include Complex Numbers Common Errors/Misconceptions • substituting pq for p · q 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 Guided Practice Example 3 Write a polynomial identity that shows how to factor x2 + 3. 19 3.4.1: Extending Polynomial Identities to Include Complex Numbers Guided Practice: Example 3, continued 1. Solve for x using the quadratic formula. x2 + 3 is not a sum of two squares, nor is there a common monomial. Use the quadratic formula to find the2 solutions to -b ± b - 4ac x2 + 3. x= . 2a The quadratic formula is 20 3.4.1: Extending Polynomial Identities to Include Complex Numbers Guided Practice: Example 3, continued Set the quadratic polynomial equal to 0. 2 x +3=0 2 1x + 0x + 3 = 0 x= x= -0 ± 0 2 - 4 (1) ( 3 ) 2 (1) ± -12 2 Write the polynomial in the 2 form ax + bx + c = 0. Substitute values into the quadratic formula: a = 1, b = 0, and c = 3. Simplify. 21 3.4.1: Extending Polynomial Identities to Include Complex Numbers Guided Practice: Example 3, continued x= x= x= x= ±i 12 2 ±i · 4 · 3 2 ±i · 4 · 3 2 ±2i 3 2 For any positive real number a, -a = i a. Factor 12 to show a perfect square factor. For any real numbers a and b, ab = a · b. Simplify. x = ±i 3 3.4.1: Extending Polynomial Identities to Include Complex Numbers 22 Guided Practice: Example 3, continued The solutions of the equation x2 + 3 = 0 are i 3 and -i 3. Therefore, the equation can be written in the ( )( ) factored form x + i 3 x - i 3 = 0. ( )( ) x 2 + 3 = x + i 3 x - i 3 is an identity that shows how to factor the polynomial x2 + 3. 23 3.4.1: Extending Polynomial Identities to Include Complex Numbers Guided Practice: Example 3, continued 2. Check your answer using square roots. Another method for solving the equation x2 + 3 = 0 is by using a property involving square roots. 24 3.4.1: Extending Polynomial Identities to Include Complex Numbers Guided Practice: Example 3, continued 2 x +3=0 2 x = –3 Set the quadratic polynomial equal to 0. Subtract 3 from both sides. Apply the Square Root x = ± -3 x = ±i 3 2 Property: if x = a, then x = ± a. For any positive real number a, -a = i a. 25 3.4.1: Extending Polynomial Identities to Include Complex Numbers Guided Practice: Example 3, continued 3. Verify the identity by multiplying. (x + i (x + i (x + i (x + i )( 3 )( x - i 3 )( x - i 3 )( x - i ) 3) = x - i ( 3) 3 ) = x - ( -1) ( 3 ) 3) = x + 3 3 x - i 3 = x - xi 3 + xi 3 - i 2 2 2 2 ( 3) 2 2 2 Distribute. Combine similar terms. Simplify. 2 26 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. ( )( ) x 2 + 3 = x + i 3 x - i 3 is an identity that shows how to factor the polynomial x2 + 3. ✔ 27 3.4.1: Extending Polynomial Identities to Include Complex Numbers Guided Practice: Example 3, continued 28 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. 29 3.4.1: Extending Polynomial Identities to Include Complex Numbers Guided Practice: Example 4, continued 1. Solve for x using the quadratic formula. x= The quadratic formula is -b ± b 2 - 4ac 2a . 30 3.4.1: Extending Polynomial Identities to Include Complex Numbers Guided Practice: Example 4, continued 2 3x + 2x + 11 = 0 Set the quadratic polynomial equal to 0. -2 ± 22 - 4 ( 3 ) (11) Substitute values into the x= quadratic formula: a = 3, 2 ( 3) b = 2, and c = 11. x= x= -2 ± 4 - 132 6 Simplify. -2 ± -128 6 3.4.1: Extending Polynomial Identities to Include Complex Numbers 31 Guided Practice: Example 4, continued x= x= x= x= -2 ± i 128 6 -2 ± i · 64 · 2 6 -2 ± i · 64 · 2 6 -2 ± 8i 2 For any positive real number a, -a = i a. Factor 128 to show its greatest perfect square factor. For any real numbers a and b, ab = a · b. Simplify. 6 3.4.1: Extending Polynomial Identities to Include Complex Numbers 32 Guided Practice: Example 4, continued -2 8i 2 x= ± 6 6 Write the real and imaginary parts of the complex number. 1 4 2 x=- ± i 3 3 Simplify. The solutions of the equation 3x2 + 2x + 11 = 0 are 1 4 2 1 4 2 - + i and - i. 3 3 3 3 33 3.4.1: Extending Polynomial Identities to Include Complex Numbers Guided Practice: Example 4, continued 2. 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 = r1 or x = r2. That is, r1 and r2 are the roots (solutions) of the equation. Conversely, if r1 and r2 are the roots of a quadratic equation, then that equation can be written in the factored form (x – r1)(x – r2) = 0. 34 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 1 4 2 1 4 2 - + i and - i. 3 3 3 3 Therefore, the equation can be written in the é æ 1 4 2 öùé æ 1 4 2 öù factored form ê x - ç - + i÷ ú êx - ç - i ÷ ú = 0, 3 øúê 3 øú êë è 3 è 3 ûë û or in the simpler factored form æ 1 4 2 öæ 1 4 2 ö i÷ ç x + + i ÷ = 0. çx+ 3 3 øè 3 3 ø 35 è 3.4.1: Extending Polynomial Identities to Include Complex Numbers Guided Practice: Example 4, continued æ 1 4 2 öæ 1 4 2 ö 3x + 2x + 11= ç x + i÷ ç x + + i ÷ is an identity 3 3 øè 3 3 ø è 2 that shows how to factor the polynomial 3x 2 + 2x + 11. ✔ 36 3.4.1: Extending Polynomial Identities to Include Complex Numbers Guided Practice: Example 4, continued 37 3.4.1: Extending Polynomial Identities to Include Complex Numbers