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Algebra 2: Unit 4 Name_______________________________ Hour__________ Unit 4 Notes: Complex numbers and Higher Degree Polynomials 1 Algebra 2: Unit 4 4.1 Solve Quadratic Equations with Imaginary Solutions Vocabulary: The imaginary unit i is defined as ___________________. o Note then, that ______________. A _____________number written in standard form is _____________. Where a is the ___________ part and bi is the _______________ part. Solving Quadratic Equations. State if the solutions are real or imaginary: Solve the equation. 1. x2 = - 81 2. 2r2 = 24 3. z2 + 6 = 4 4. 3p2 + 7 = -9p2 + 4 5. x2 + 15 = - 35- x2 6. 3x2 + 13 = - 23 Adding and Subtracting: 2 Algebra 2: Unit 4 7. x2 – 3x + 5 = 0 8. 2x2 – 1 = 3x + 4 9. x2 – 6x = -9 10. 3x2 – 3x + 4=0 Summary 3 Algebra 2: Unit 4 4.2 Perform Operations with Complex Numbers Remember i=_________ i i i 2 , i 2 1 Simplify these imaginary numbers: i2 i3 i4 i5 i31 i42 ADDING AND SUBTRACTING COMPLEX NUMBERS Write the expression as a complex number in standard form. Standard Form _____________________ 1. (6 – 3i) + (5 + 4i) 2. (8 + 20i) – (8 – 9i + 7) 3. (3 – 2i) + (6 – 3i) 4. (i3 + 1 + 8i) – (7 + 5i) 4 Algebra 2: Unit 4 MULTIPLYING AND DIVIDING COMPLEX NUMBERS Write the expression as a complex number in standard form. 5. 6i(3 + 2i) 6. (-1 – 5i)(- 1 + 5i) 7. 3i(2 - i)(4 + 2i) 8. (2 + i)(6 - 5i) + (-4 + 2i) Complex Conjugates: (Note, a + bi and a – bi are complex conjugates) Write the expression as a complex number in standard form. To do this, multiply the numerator and the denominator by the ___________________. 9. 6i 3i 11. 6 4i 2i 5 10. 4 9i 2 3i 12. 7i i Algebra 2: Unit 4 13. Challenge: Simplify the following: Summary 4.3 Factoring 3rd and 4th degree Polynomials Factoring we have already done: 6 Algebra 2: Unit 4 Factoring 3rd and 4th degree with a GCF Factor the following 3rd and 4th degree polynomials. 1. 3 − 16 2 − 17 2. 6 4 − 4 3 − 16 2 3. 12 4 − 27 = 0 4. 7 4 −6 3 − 2 Factoring by Grouping: 5. y3 – 7y2 + 4y – 28 6. 23 + 52 + 6 + 15 Factor a quartic trinomial: 7. u4 + 7u2 + 6 8. x4 – 20x2 + 36 7 Algebra 2: Unit 4 Solve by factoring: 9. 3x4 + 15x2 = 18x3 10. 16g4 – 625 = 0 11. 4 − 4 = 0 12. 2 3 − 9 2 = 10 Summary 4.4 Long and Synthetic Division of Polynomials and the Factor Theorem Vocabulary: 5 divided by 4 is 1 ¼ 8 Algebra 2: Unit 4 Long Division with polynomials 1. Divide 3x4 – 5x3 + 4x – 6 by x2 – 3x + 5 2. Divide x3 + 3x2 – 7 by x2 – x – 2 3. Divide ( x 2 x 17) ( x 4) 4. Divide 3 + 3 2 − 7 by 2 − − 2 9 Algebra 2: Unit 4 5. Divide 2 4 + 3 + − 1 by 2 + 2 − 1 6. Divide (8 2 + 34 − 1) ÷ (4 − 1) 10 Algebra 2: Unit 4 7. Divide 2 2 − 7 + 10 by − 5 Using Long Division: Using Synthetic Division: Can be used to divide any polynomial by a divisor of the form x – k 8. Divide 3 − 5 2 − 2 by − 4 9. Divide 4 + 4 3 + 16 − 35 by + 5 10. Divide (2 x 3 x 2 8x 5) ( x 3) using synthetic division ******Factor theorem: A polynomial has a factor (x – k) if and only if the remainder is zero.*********** Determine if the binomial is a factor of the polynomial. 11. 11 Algebra 2: Unit 4 12. Summary 4.5 Completely Factor Polynomials and Find Zeros Factor: Given the polynomial and a factor, factor completely. 1. f ( x) x 3 10 x 2 19 x 30; ( x 6) 2. f ( x) 3x 3 4 x 2 28x 16; ( x 2) 3. f ( x) 2 x 3 15x 2 34 x 21; ( x 1) 12 Algebra 2: Unit 4 Find Zeros: 4. One zero of f ( x) x 3 2 x 2 23x 60 is 3. What are the other zeros? 5. One zero of f ( x) 10 x 3 81x 2 71x 42 is 7. What are the other zeros? 6. One zero of f ( x) x 3 12 x 2 35x 24 is 3. Find the other zeros. Summary 13 Algebra 2: Unit 4 4.6 Graphing Polynomial Functions CUBIC FUNCTIONS (3rd degree) The graph of a cubic polynomial has the basic property that the graph can cross the x-axis in at most 3 points. It may cross in fewer points, or it may not cross at all. Graph of f(x) = ax3 + bx2 + cx + d a is positive a is negative If the degree of a polynomial function is odd, then the left end of the graph and the right end of the graph are in _____________directions. QUARTIC FUNCTIONS (4th degree) The graph of a quartic polynomial can cross the x-axis in at most 4 points. It may cross in fewer points, or it may not cross at all. Graph of f(x) = ax4 + bx3 +cx2 + dx + e a is positive a is negative If the degree of a polynomial function is even, then the left end of the graph and the right end of the graph in the __________ direction. 14 Algebra 2: Unit 4 State the degree of the function and if a is positive or negative 1. 2. 3. 4. Zeros and Crossing x-axis A function of nth degree can cross the x-axis at most n times. That means it can have at most n real zeros. If it doesn’t cross n times, some solutions are just imaginary then. 5. How many times can () = 7 − 3 + 7 − 1 cross the x-axis? 15 Algebra 2: Unit 4 THE MULTIPLICTY OF ZEROS State the zeros of the graph and the multiplicity of them. Also state if the polynomial function is of even or odd degree at least what degree it is. 6. 7. 8. 16 Algebra 2: Unit 4 Sketching Graphs Sketch a rough graph of the polynomial functions based on the degree, value of a, and the zeros and their multiplicity. 9. = ( + )( − ) ( − ) Degree: even or odd a: positive or negative zeros and multiplicity of each: 10. () = ( − )( − )( + )( + ) Degree: even or odd a: positive or negative zeros and multiplicity of each: 17 Algebra 2: Unit 4 11. = −( − ) ( + ) Degree: even or odd a: positive or negative zeros and multiplicity of each: 12. = ( − )( + )( − ) Degree: even or odd a: positive or negative zeros and multiplicity of each: 18 Algebra 2: Unit 4 13. () = −( + ) Degree: even or odd a: positive or negative zeros and multiplicity of each: 14. = −( − )( + ) ( + ) Degree: even or odd a: positive or negative zeros and multiplicity of each: 19 Algebra 2: Unit 4 15. = − − + Degree: even or odd a: positive or negative zeros and multiplicity of each: 16. = + + + given ( + ) is a factor. Degree: even or odd a: positive or negative zeros and multiplicity of each: 20 Algebra 2: Unit 4 Summary 4.7 Write Equations of Polynomial Functions from a Graph Steps to writing an equation from a graph: 1) Locate the ________________ or ________________. 2) Determine the ___________________ of each zero. 3) Write the polynomial in _________________ form. 4) Use a point on the graph to determine the value of ____. If possible, use the __-_______________ for your point. Write equations from the graphs: 1. 21 Algebra 2: Unit 4 2. 3. 4. 22 Algebra 2: Unit 4 5. 6. Write an equation for a polynomial function with the following zeros and point on the graph. Zeros: 2 (multiplicity 1), -1 (multiplicity 1) and 4 (multiplicty 2). Point on Graph (y-intercept): (0, 2) 7. Write an equation for a polynomial function with the following zeros and y-int. Zeros: -3 (multiplicity 2) and 1 (multiplicity 1) Point on Graph: (2, -7) Summary 23