TOPIC Competency #2: Properties of Numbers Standard – CN1 Learning Targets & Skills Vocabulary Student goes above and beyond simple mastery to demonstrates a deeper • Absolute value of a 4.0 understanding than a Level 3.0. complex number • Conjugate • Binomial • I • Complex number • Imaginary number • Polynomial • Imaginary part of a complex • Real number number 3.0 I can find the conjugates and moduli of complex numbers • Real part of a complex • Modulus number • Polynomial identity • Explain that a negative number −푥 can be rewritten as Resources (−1)푥. • Complex Conjugates • Explain that a square root can be rewritten as the product of the square roots of two numbers as long as both numbers are not negative. For example, −푎 = −1 ⋅ 푎 = −1 ⋅ 푎, but 푎 = −1 ⋅ −푎 ≠ −1 ⋅ −푎. • Explain that −1 is known as the imaginary number 푖 which is defined as 푖2 = −1. • Explain that the sum of an imaginary number and a real number is known as a complex number and takes the form 푎 + 푏푖 in which 푎 and 푏 are real numbers. • Explain that in a complex number of the form 푎 + 푏푖, 푎 is known as the real part and 푏 is known as the imaginary part. • Explain that complex numbers can be added or subtracted by adding or subtracting the corresponding real parts and the corresponding imaginary parts. For example, 푎 + 푏푖 + 푐 + 푑푖 = 푎 + 푐 + 푏 + 푑 푖 and 푎 + 푏푖 − 푐 + 푑푖 = 푎 − 푐 + 푏 − 푑 푖. • Explain that complex numbers can be multiplied in the same manner as a standard binomial. For example, 푎 + 푏푖 (푐 + 푑푖) = 푎푐 + 푎푑푖 + 푐푏푖 + 푏푑푖2 . • Explain that the conjugate of a complex number 푧 = 푎 + 푏푖 is 푎 − 푏푖 and can be notated as 푧ҧ. • Explain why the sum of a complex number and its conjugate TOPIC Competency #2: Properties of Numbers Standard – CN2 Learning Targets & Skills Vocabulary Student goes above and beyond simple mastery to demonstrates a deeper • Absolute of a complex 4.0 understanding than a Level 3.0. number • Binomial • I • Complex number • Imaginary number • Conjugate • Imaginary part of a complex • Polynomial number 3.0 I can manipulate complex numbers • Real number • Modulus • Real part of a complex • Polynonial identity number • Explain that a negative number −푥 can be rewritten as Resources (−1)푥. • Complex Numbers • Explain that a square root can be rewritten as the product of the square roots of two numbers as long as both numbers are not negative. For example, −푎 = −1 ⋅ 푎 = −1 ⋅ 푎, but 푎 = −1 ⋅ −푎 ≠ −1 ⋅ −푎. • Explain that −1 is known as the imaginary number 푖 which is defined as 푖2 = −1. • Explain that the powers of 푖 cycle through a repeating set of values. For example, 푖0 = 1, 푖1 = 푖, 푖2 = −1, 푖3 = −푖, 푖4 = 1, and so on. • Evaluate powers of 푖. For example, 푖13 = 푖12푖1 = 푖4 3푖1 = 1 3 푖 = 푖. • Explain that the sum of an imaginary number and a real number is known as a complex number and takes the form 푎 + 푏푖 in which 푎 and 푏 are real numbers. • Explain that in a complex number of the form 푎 + 푏푖, 푎 is known as the real part and 푏 is known as the imaginary part. • Explain that complex numbers can be added or subtracted 2.0 by adding or subtracting the corresponding real parts and the corresponding imaginary parts. For example, 푎 + 푏푖 + 푐 + 푑푖 = 푎 + 푐 + 푏 + 푑 푖 and 푎 + 푏푖 − 푐 + 푑푖 = 푎 − 푐 + 푏 − 푑 푖. ( TOPIC Competency #2: Properties of Numbers Standard – CN3 Learning Targets & Skills Vocabulary Student goes above and beyond simple mastery to demonstrates a deeper • Complex number • Factor 4.0 understanding than a Level 3.0. • Complex roots • Fundamental theormem of • Conjugate algebra • Degree of a polynomial • I expression • Imaginary number • Discriminant • Imaginary part of a complex I can solve second-degree polynomial equations that have 3.0 • Real part of a complex number complex roots number • Polynomial equation • Root • Polynomial identity • Polynomial • Quadratic formula • Explain that the fundamental theorem of algebra states that Resources a polynomial of degree 푛 will have 푛 roots, some of which may be complex. • Complex Roots • Explain that the quadratic formula can be used to find the roots of a second-degree polynomial. 2 • State the quadratic formula: 푥 = −푏± 푏 −4푎푐 . 2푎 • Explain why the quadratic formula will always produce exactly two roots for any second-degree polynomial. For example, the roots of any second-degree polynomial 푎푥2 + 2 2 푏푥 + 푐 are 푥 = −푏+ 푏 −4푎푐 and 푥 = −푏− 푏 −4푎푐. 2푎 2푎 • Use the quadratic formula to find the roots of a given second-degree polynomial. 2.0 • Explain that the expression 푏2 − 4푎푐 in the quadratic formula is known as the discriminant. • Explain that if the discriminant of a second-degree polynomial is negative, the polynomial will have complex roots. • Express the complex roots of a quadratic equation in the 2 2 form 푥 = −푏 + 푏 −4푎푐 and 푥 = −푏 − 푏 −4푎푐. 2푎 2푎 2푎 2푎 • Rewrite the square root of a negative number as an imaginary number. For example, rewrite −푎 as 푖 푎. • Explain that the complex roots of any polynomial will always .