Grade: Grade 11 Subject: Mathematics Unit: Algebra II Completion Lesson: 3 of 6 SAT: AdvancedMath ACT: Math

Guided Practice

Apply your knowledge of complex numbers and polynomial functions through structured problem-solving with step-by-step guidance.

Learn

In this lesson, you will work through carefully scaffolded problems that reinforce the concepts from the previous lessons on complex numbers and polynomial functions.

Problem-Solving Strategies

  • Read carefully: Identify what the problem is asking before starting calculations
  • Identify the type: Determine if you are working with complex numbers, polynomials, or a combination
  • Show your work: Write out each step to avoid errors and track your thinking
  • Check your answer: Substitute back or use alternative methods to verify

Key Formulas Review

  • Complex number form: a + bi where i = sqrt(-1)
  • Complex conjugate of a + bi is a - bi
  • Polynomial standard form: a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0
  • Factor Theorem: If f(c) = 0, then (x - c) is a factor of f(x)

Examples

Follow along with these worked examples before attempting the practice problems.

Example 1: Complex Number Operations

Problem: Simplify (3 + 2i)(4 - 5i)

Solution:

  1. Use FOIL: (3)(4) + (3)(-5i) + (2i)(4) + (2i)(-5i)
  2. Multiply: 12 - 15i + 8i - 10i^2
  3. Combine like terms and remember i^2 = -1: 12 - 7i - 10(-1)
  4. Simplify: 12 - 7i + 10 = 22 - 7i

Example 2: Polynomial Division

Problem: Divide x^3 + 2x^2 - 5x - 6 by (x + 3)

Solution:

  1. Use synthetic division with c = -3
  2. Coefficients: 1, 2, -5, -6
  3. Result: x^2 - x - 2 with remainder 0
  4. Verify: (x + 3)(x^2 - x - 2) = x^3 + 2x^2 - 5x - 6

Practice

Work through these problems step-by-step. Show all your work.

Problem 1: Simplify (5 + 3i) + (2 - 7i)

Problem 2: Simplify (4 - 2i)(3 + i)

Problem 3: Find the complex conjugate of -2 + 5i and compute the product of the number and its conjugate.

Problem 4: Divide (6 + 2i) by (1 + i). Express your answer in standard form.

Problem 5: Factor completely: x^3 - 6x^2 + 11x - 6

Problem 6: Use synthetic division to divide 2x^3 - 3x^2 + 4x - 1 by (x - 2).

Problem 7: Find all zeros of f(x) = x^3 - 4x^2 + x + 6.

Problem 8: Write a polynomial function with zeros at x = 2, x = -1, and x = 3.

Problem 9: If f(x) = x^4 - 5x^2 + 4, find f(i) where i is the imaginary unit.

Problem 10: Determine the end behavior of g(x) = -2x^5 + 3x^3 - x + 7.

Check Your Understanding

Answer these questions to verify your mastery of the material.

  1. What is the result when you multiply a complex number by its conjugate?
  2. How can you use the Remainder Theorem to quickly check if a value is a zero of a polynomial?
  3. Explain why complex zeros of polynomials with real coefficients always come in conjugate pairs.
  4. What is the relationship between the degree of a polynomial and its maximum number of zeros?

Next Steps

  • Review any problems you found challenging
  • Compare your solutions with the key formulas from the Learn section
  • Move on to Word Problems to apply these skills in context
  • Return to Complex Numbers or Polynomial Functions if you need more review