Questions and Answers
Find all the zeros of the equation $x^5 -3x^4 -15x^3 +45x^2 -16x + 48 = 0$
3, 4, -4, i, -i
Find all the zeros of the equation $x^3 -x^2 +5x - 5 = 0$
i(sqrt(5)), -i(sqrt(5)), 1
Find all the zeros of the equation $7x^2 - 144 = -x^4$
3, -3, 4i, -4i
Study Notes
Fundamental Theorem of Algebra
- States that every non-constant polynomial equation has at least one complex root.
- The number of roots (including multiplicity) equals the degree of the polynomial.
Key Polynomials and Their Zeros
-
Polynomial: ( x^5 - 3x^4 - 15x^3 + 45x^2 - 16x + 48 = 0 )
- Zeros: 3, 4, -4, ( i ), -( i )
-
Polynomial: ( x^3 - x^2 + 5x - 5 = 0 )
- Zeros: ( i \sqrt{5} ), -( i \sqrt{5} ), 1
-
Polynomial: ( 7x^2 - 144 = -x^4 )
- Zeros: 3, -3, 4i, -4i
Important Concepts
- Zeros can be real or complex (including imaginary units).
- The roots can be found through factoring, synthetic division, or using the quadratic formula for lower-degree polynomials.
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Description
Test your understanding of the Fundamental Theorem of Algebra with these flashcards. Each card presents an equation and challenges you to find all its zeros. Ideal for math students looking to strengthen their algebra skills.
