Questions and Answers
What do I have right?
what I got right
How many roots does the equation -2x^3 = 0 have?
3
Which polynomial equation is of least degree and has -1, 2, and 4 as three of its roots?
x^3 - 5x^2 + 2x + 8 = 0
Which polynomial equation of least degree has -2, -2, 3, and 3 as four of its roots?
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What are the roots in the equation x^4 + 3x^2 - 4 = 0?
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What is the least possible degree of a polynomial that has roots -5, 1 + 4i, and -4i?
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Which options CANNOT be the degree of a polynomial that has only imaginary roots and no real root? (Select all that apply)
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What is the least possible degree of a polynomial that has the root -3 + 2i, and a repeated root -2 that occurs twice?
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Which is a possible number of distinct real roots for a cubic function? (Select all that apply)
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How many complex roots does the equation x(x^2 - 4)(x^2 + 16) = 0 have?
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How many real roots does the function y = (x - )(x + )^2 have?
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What are all the roots of the equation (x^2 + 1)(x^3 + 2x)(x^2 - 64) = 0?
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What are the roots of x^4 - 81 = 0, x^4 + 10x^2 + 25 = 0, and x^4 - x^2 - 6 = 0?
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What are the roots in the equation x^3 - 27 = 0?
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Which polynomial equations have -i as one of their roots? (Select all that apply)
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Write a polynomial equation of degree 3 such that two of its roots are 2 and an imaginary number.
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Find all the roots of the equation x^4 - 2x^3 + 14x^2 - 18x + 45 = 0 given that 1 + 2i is one of its roots.
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What are the roots of the equation (x^2 - 1)(x^2 + 2)(x + 3)(x - 4)(x + 1) = 0?
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Study Notes
Fundamental Theorem of Algebra
- Establishes that every non-constant polynomial function over the complex numbers has at least one complex root.
- A polynomial of degree n has exactly n roots, counted with multiplicity.
Roots of Polynomial Equations
- The equation -2x³ = 0 has 3 roots.
- Polynomial x³ - 5x² + 2x + 8 = 0 has roots -1, 2, and 4, indicating it is of least degree 3.
- Polynomial equation (x + 2)²(x - 3)² = 0 has roots -2 (double root) and 3 (double root), indicating a least degree of 4.
Finding Roots
- The equation x⁴ + 3x² - 4 = 0 has roots: -1, 1, 2i, -2i.
- A polynomial must have a degree of 5 to accommodate roots -5, 1 + 4i, and -4i.
- Imaginary roots only: the degree cannot be 3 or 7.
Real Roots in Polynomial Functions
- A polynomial with roots including -3 + 2i and a repeated root -2 has a least degree of 4.
- A cubic function can have possible distinct real roots of 2, 3, or 0.
Complex Roots in Given Equations
- x(x² - 4)(x² + 16) = 0 has 2 complex roots.
- (x² + 4)(x + 5)² = 0 has 4 complex roots.
- x⁶ - 4x⁵ - 24x² + 10x - 3 = 0 has 6 complex roots.
- x⁷ + 128 = 0 has 7 complex roots.
- (x³ + 9)(x² - 4) = 0 has 5 complex roots.
Graphing and Roots
- The graphed polynomial function has 2 real roots and 4 total roots.
- Equation can be expressed as y = (x - a)(x + b)² indicating roots at -a and b.
Solving Polynomial Equations
- For (x² + 1)(x³ + 2x)(x² - 64) = 0, roots include: i, -i, 0, i√2, -i√2, 8, -8.
- For the equation x⁴ - 81 = 0, the roots are 3, -3; for x⁴ + 10x² + 25 = 0, the roots are i√5, -i√5; for x⁴ - x² - 6 = 0, roots include √3, -√3, i√2, -i√2.
Additional Polynomial Equations
- For x³ - 27 = 0, factored as (x + 3)(x² - 3x + 9) = 0.
- Polynomials x³ + 3x² + x + 3 = 0 and x³ - 6x² - 16x + 96 = 0 both have -i as a root.
- Example polynomial of degree 3 with roots 2 and an imaginary number is expressed as y = (x - 2)(x - i)(x + i).
Roots Calculation from Given Root
- For x⁴ - 2x³ + 14x² - 18x + 45 = 0, given 1 + 2i is a root, find all roots by division or substitution methods, leading to further roots calculations.
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Test your knowledge of the Fundamental Theorem of Algebra with these flashcards. Each card contains important questions about polynomial equations and their roots. Perfect for students looking to reinforce their understanding of algebra concepts.
