LAP 2.1 POLYNOMIALS ROOTS, ZEROS, FACTORS, DIVISION
Page 1: Introduction
Topic: 10th Grade Polynomials
Key Concepts: Roots, Zeros, Factors, Division of Polynomials
Page 2: Table of Contents
Sections:
Objectives
Activity
Exercises
Discussion
Page 3: Objectives
Goals:
Determine if a number is a root or zero of a polynomial equation.
Decide if one polynomial is a factor of another through division.
Expected Outcome: At the end of the lesson, mastery of these objectives.
Page 4: Activity - Carry-On Luggage
Situation: A manufacturer ensures that the sum of the length, width, and depth does not exceed 40 inches.
Key Point: Popular luggage models have a length that exceeds depth by 10 inches.
Conditions: Sum = 40 inches; Volume = 2016 cubic inches.
Task: Find dimensions of the luggage.
Page 5: Activity Representation
Variables:
Depth = x
Length = x + 10
Width = 40 - (x + x + 10)
Equation:
Total dimensions equation: 40 = depth + length + width.
Derived equation for width: Width = 40 - (2x + 10).
Page 6: Volume Equation
Volume Calculation:
Volume = depth × length × width.
Equation Formation:
2016 = x × (x + 10) × (30 - 2x)
Simplification leads to:
2016 = x(x + 10)(30 - 2x)
Final polynomial formulation:
2x³ - 10x² - 300x + 2016 = 0.
After dividing by 2:
x³ - 5x² - 150x + 1008 = 0.
Page 7: Polynomial Equation
Definition: This is an example of a polynomial equation.
Leading Coefficient: Identified as 1 (of x³).
Page 8: Discussion - E=mc2
Topic: Philosophy related to polynomial calculations.
Page 9: Example 1 - Roots of Polynomial Equation
Given Polynomial: P(x) = x³ - 5x² - 150x + 1008.
Task: Determine if various numbers are roots.
Tested Number: 10 results in non-zero.
Conclusion: 10 is NOT a root.
Page 10: Continued Example 1 - Testing Numbers
Next Tested Number: 8.
Calculation: 8³ - 5(8)² - 150(8) + 1008 = 0.
Conclusion: 8 is indeed a root.
Page 11: Dimensions of Luggage
Root Context: Since 8 is a root, dimensions of luggage derived as follows.
Page 12: Dimensions Calculation
Calculated Dimensions:
Depth = 8
Length = 18
Width = 14
Verification: The sum of dimensions equals 40; Volume equals 2016.
Page 13: TRY THIS - Testing Roots
New Polynomial: x³ + x² - 6x - 756.
Tested Number: 7.
Outcome: 7 is NOT a root.
Page 14: TRY THIS - Continued Testing
Tested Number: 9 - results show it is a root.
Page 15: Negative Number Testing
Tested Number: -3 results show it is NOT a root.
Page 16: Example 2 - Zeros of Polynomial Function
Polynomial: x³ - 3x² + x - 3 = 0.
Tested Number: -3 does not equal zero.
Conclusion: -3 is NOT a zero.
Page 17: Complex Number Testing
Tested: -i results in being a zero of the function.
Page 18: Further Testing
Polynomials to Test: Confirmed that 2 is indeed a zero.
Page 19: Testing Complex Variants
Tested 2i: Results reveal that 2i is NOT a zero of the function.
Page 20: Exercises
Task: Determine roots and zeros for given polynomials, marked as YES or NO for accuracy.
Page 21: Discussion - Long Division of Polynomials
Page 22: Division Example
Polynomial: x³ - 5x² + 2x - 10.
Divisor: x² + 2.
Outcome: Verification completed with the conclusion that x² + 2 is a factor.
Page 23: Division Process Illustrated
Page 24: Another Division Example
Task: Test for factor x² - 3 of the polynomial x³ - 2x² - 2x + 6.
Conclusion: Remains NOT a factor.
Page 25: Example 4 - Testing Factors
Page 26: Conclusion of Factor Testing
Page 27: Assignments and Testing
Page 28: Exercises Summary - State YES or NO for provided examples.
Page 29: Exercises: Key to Correction
Summary of Answers: Identifying which equations are factors.